We introduce Three-Dimensional Flexible Refinement (3DFlex), a motion-based neural network model for continuous molecular heterogeneity for cryo-EM data. 3DFlex exploits knowledge that conformational variability of a protein is often the result of physical processes that transport density over space and tend to preserve local geometry. From two-dimensional image data, 3DFlex enables the determination of high-resolution 3D density, and provides an explicit model of a flexible protein’s motion over its conformational landscape. Experimentally, for large molecular machines  and small flexible proteins  is a generative neural network model for continuous molecular heterogeneity for cryo-EM data that can be used to determine the structure and motion of flexible biomolecules. It enables visualization of nonrigid motion and improves 3D structure resolution by aggregating information from particle images spanning the conformational landscape of the target molecule. Proteins form the molecular machinery of the cell. They are inherently dynamic, often exhibiting a continuous landscape of conformations, with motion tightly linked to function. Methods that uncover protein motion and the conformational landscape have the potential to illuminate fundamental questions in structural biology, and to enhance the ability to design therapeutic molecules that elicit specific functional changes in a target protein.1. This highlights the need for methods for resolving continuous motion and structure from static 2D images. Local reconstruction via multi-body refinement3 is effective for macromolecules with sufficiently large, rigid subunits, given masks to isolate the subunits. Principal component analysis4 and linear subspace methods, such as 3D variability analysis (3DVA)5 approximate a particle’s space of conformations as a weighted sum of basis density maps. Nonlinear manifold embedding methods7, including deep-learning models such as CryoDRGN8, offer even more expressive power. Such density-based methods are widely applicable, handling compositional and conformational heterogeneity; they provide encouraging evidence that one can estimate structural variation from a single heterogeneous dataset. However, the existing methods have limitations. Local and multi-body refinements are not readily applicable to highly flexible motion or the motion of small subunits (SSUs). Linear subspace models are limited to relatively simple, small motions. Density-based models do not explicitly estimate motion, and as a consequence are not able to aggregate signal across a particle’s conformational landscape to improve the resolution of 3D density in flexible regions.Single-particle cryo-EM collects thousands of static two-dimensional (2D) particle images that, in aggregate, may span the target protein’s 3D conformational space. Cryo-EM therefore holds great promise for uncovering both the atomic-resolution structure and motion of biologically functional moving parts9) and hypermolecules10, address some but not all of these challenges.The development of a computational method that can uncover both fine structural detail and nonrigid protein motion in the presence of continuous flexibility must overcome multiple challenges. It entails joint optimization of many unknowns, including the 3D structure of the density map and a representation of the position of each particle image on the conformational landscape of the protein. It is also unclear how to aggregate signal across all particles, attenuating noise and improving map quality with sufficient regularization to avoid overfitting. Recent methods that do estimate motion, such as e2gmm (ref. We introduce 3D Flexible Refinement (3DFlex), a deep neural network model of continuously flexible protein molecules. 3DFlex is a motion-based heterogeneity model that directly exploits the knowledge that most conformational variability of a protein is a result of physical processes that tend to transport density, preserving local geometry . We formulate 3DFlex as a generative deep-learning architecture that captures conformational variability in terms of a single high-resolution ‘canonical’ 3D density map of the molecule, and a parameterized latent space of deformation fields encoding flexible (nonrigid) motion. The motion model is used to deform the canonical density via convection, yielding all conformations captured by the model. In 3DFlex, the latent coordinates of each particle image, the deformation field generator and the canonical density are jointly learned from image data using a specialized training algorithm, with minimal previous knowledge about the flexibility of the molecule.11, 3DFlex learns a wide range of nonrigid motions, including subunits bending across a span of more than 20 Å. In doing so, the algorithm aggregates structural information from all conformations into a single, optimized density map that resolves high-resolution details in α-helices and β-sheets even in the flexible domains. It estimates motion with sufficient precision to improve the resolution of small flexible parts that are otherwise poorly resolved in conventional and local, focused refinements. We demonstrate this ability with a dataset of TRPV1 ion-channel particles12, where 3DFlex improves the resolution of peripheral α-helices in the flexible soluble domains. Additional experiments on a SARS-CoV-2 spike protein13, an αVβ8 integrin14 and a translocating ribosome15 demonstrate that 3DFlex can reveal structures not resolved by conventional refinement, and can map out conformational variations. With such capabilities, 3DFlex opens up new avenues of inquiry into the study of biological mechanisms and function involving motion.Results on experimental cryo-EM data show that 3DFlex addresses the challenges of uncovering structure and motion of flexible proteins. On a dataset of tri-snRNP spliceosome particles3DFlex is a generative neural network method that determines the structure and motion of flexible biomolecules from cryo-EM images. Central to 3DFlex is the assumption that conformations of a dynamic protein are related to each other through deformation of a single 3D structure. Specifically, a flexible molecule is represented in terms of (1) a canonical 3D map, (2) latent coordinate vectors that specify positions over the protein’s conformational landscape and (3) a flow generator that converts a latent coordinate vector into a deformation field that convects the canonical map into the corresponding conformation. The canonical 3D map, the flow generator, and a latent coordinate vector for each particle image are jointly learned from experimental data.Ii is generated as follows. First, a K-dimensional latent coordinate vector, zi, is input to a flow generator, fθ(zi), with parameters θ. The generator produces a 3D deformation field that is used to convect the canonical 3D density map, V. The convected density, denoted D(fθ(zi), V), is then projected to 2D, contrast transfer function (CTF) modulated and corrupted by additive noise η; that is,Ci is the CTF operator and P(ϕi) is the projection operator for pose ϕi . Fitting 3DFlex to experimental images entails optimizing the flow generator parameters θ, the canonical map V and the latent coordinates zi, to minimize the data log likelihood under the probabilistic model  that outputs a flow vector at each vertex of a tetrahedral mesh. Interpolation within mesh elements, using finite-element methods, yields a flow field that maps density from canonical coordinates to the observed coordinate frame of a given particle image. Thus, 3DFlex conserves density, as do conventional, rigid reconstruction methods. Further, central to the optimization is a regularizer that encourages locally smooth and rigid motion in regions of the canonical map with high density.Computationally determining structure and motion from noisy cryo-EM data is challenging. Hence, there are several important design choices that define an effective model architecture and optimization procedure. Briefly (see 5). During learning, one can optimize the canonical map, the MLP weights and the per-particle latent coordinates simultaneously, or in an alternating form of block coordinate descent. To help encourage a smooth latent representation, we also regularize the latent coordinates by injecting uncertainty into the latent positions at each iteration of network weight updates. Learning is performed using low-resolution particle images to reduce computational cost, and also to enable half-set Fourier shell correlation (FSC) validation of final reconstructions at full resolution  alignments despite conformational heterogeneity in the data. The resulting poses ϕi are fixed, and particle images are downsampled to a limited resolution during training of the 3DFlex model. 3DFlex is run with a real-space mask that excludes solvent in the canonical density V. In the case of membrane proteins, a separate mask is used to enforce zero deformation in the region of detergent micelle or lipid nanodisc.We apply 3DFlex to five experimental cryo-EM datasets: a tri-snRNP spliceosomeUnless otherwise stated, model parameters are set to default values . The defB-factor sharpened. Where noted, resulting maps are locally filtered to aid in visualization. To display conformational changes in figures, we select points in the 3DFlex latent space, for example, zdisplay, and then generate the corresponding convected densities, Wdisplay = D(fθ(zdisplay), V). These densities are rendered overlaid in multiple colors, and with reference position guide markers to help visualize the motion. Supplementary For visualization of the canonical density, the half-maps from 3DFlex are combined, filtered by their FSC curve and 11. The dataset comprises 138,899 particle images (EMPIAR-10073), with box size of 380 pixels of width 1.4 Å. They are first processed through heterogeneous refinement in cryoSPARC (ref. 18) to remove particles missing the ‘head’ region, yielding 102,500 final particles. These are downsampled to a box size of 180 pixels (2.95 Å wide) for training 3DFlex. For the snRNP complex we use a five-dimensional latent space; a larger latent space enables 3DFlex to discover more distinct, intricate motions. A regular tetrahedral mesh with 1,601 vertices and 5,859 cells (approximately 18 Å wide) is automatically generated to cover the input (consensus) map. Given the latent coordinates and flow generator learned by 3DFlex, the full-resolution particle images are used to reconstruct the canonical density  using L-BFGS. Total training time is 18 hours.The U4/U6.U5 tri-snRNP complex represents a large part of the spliceosome, with several moving parts, linkages and flexible domains3DFlex recovers five dimensions of motion Fig. , each ofα-helices can be seen translating several Angstroms while retaining side-chain features. Likewise, a β-sheet in the flexible head region is resolved with separated β-strands, despite the nonrigid motion present. Because the flow generator was trained on downsampled images , these structural features represent additional signal that is resolved from the original data as a consequence of the accuracy of the recovered motion .While recovering 3D motion, 3DFlex also determines high-resolution detail in the canonical map. This occurs through the aggregation of signal across the conformational landscape, with nonrigid alignment between particle images and the canonical map. Individual 3. Such refinement techniques can improve resolution and map quality in domains such as the foot, which remains rigid despite motion relative to the remainder of the molecule. In contrast, 3DFlex does not require manual masking or previous knowledge about the motion of the molecule. It can detect and then correct for nonrigid flexibility across the entire molecule, including the head, which is considerably less rigid than the foot.In regions of substantial motion and flexibility, differences between a static conventional refinement and 3DFlex are dramatic Fig. ; for exa12. We process 200,000 particles of TRPV1 in nanodisc (EMPIAR-10059), downsampled to a box size of 128 pixels of width 1.21 Å for training 3DFlex. A tetrahedral mesh with 1,054 vertices and 3,892 cells (about 14 Å wide) is generated to cover the density. Based on 3DVA (ref. 5), it is evident that TRPV1 exhibits smooth, relatively small deformations. As such, a smaller architecture is sufficient, and helps to mitigate the risk of overfitting; we use a three-layer MLP with 32 units per layer for the flow generator and a 2D latent space. We initialize the latent coordinates to those from 3DVA, which helps avoid poor local minima that can be problematic with small proteins. Once 3DFlex is trained, the original particles (pixel size 1.21 Å) are used to reconstruct the canonical density  to high resolution. Symmetry is not enforced during training but the half-maps are symmetrized post hoc to C4 to enable comparison with the rigid refinement baseline . But with improved alignment at full resolution (pixel size 1.21 Å), gold-standard FSC and local resolution measurements using the two half-set reconstruction in 3DFlex show that it recovers consistent structural information well beyond 4.3 Å. Local resolutions in peripheral helices improve to 3.2 Å revealing helical pitch and side-chain details. The separate half-set reconstructions from 3DFlex allow us to use established validation procedures to measure the improvement derived from nonrigid motion estimation. The FSC curve for the entire density  and N-terminal domain (NTD). We process 2,139 raw cryo-EM movies (EMPIAR-10516) in cryoSPARC to obtain 113,511 particle images . After downsampling , we train 3DFlex with a 3D latent space.The SARS-CoV-2 spike proteinBy default, 3DFlex uses a regular tetrahedral mesh, but the method works with any mesh geometry. As discussed in The resulting 3DFlex model captures multiple coordinated bending motions of the RBDs and NTDs in the S1 region of the spike protein Fig. . The up-αVβ8 integrin is a highly flexible protein involved in cell differentiation during development, and in fibroinflammatory processes and antitumor immunity14. We process 84,266 particles (EMPIAR-10345) of αVβ8 integrin with two Fabs bound. 3DFlex is trained on downsampled images (pixel size 3.15 versus 1.345 Å), with a 2D latent space and a regular tetrahedral mesh with 477 vertices and 1452 cells (22 Å wide).The 5). 3DFlex uncovers joint bending and motion of the flexible arm as well as the core region of the protein and the two Fabs. The local resolution in the core region is also slightly improved by 3DFlex, from 3.6 to 3.5 Å, and local resolutions in other regions  of the Ribosome-EF-G complex in the presence of GDP (EMPIAR-10792). The original study15 separated major conformational states using 3D Classification. We combine particles classified into Hybrid(GDP+Pi), Hybrid(GDP) and Chimeric(GDP) states15 into a single-particle set, discarding the original labels. These images are downsampled (pixel size 2.38 Å), and used to train 3DFlex with a 2D latent space. We use a custom mesh topology, specified by coarse boundaries between the large subunit, SSU, EF-G and transfer RNAs subunits, from which submeshes and a complete fused mesh are automatically generated , a translational GTPaseenerated . The fin15  for accurate rigid alignment3. 3DFlex recovers motion and structure of nonrigidly deformable parts across an entire molecule, with control over the complexity of motion via mesh granularity and smoothness regularizers. Default mesh generation is automatic, but one also has the option to adjust mesh connectivity by separating subunits and domains to better resolve boundaries.Local, focused refinement24 make assumptions about the local energy landscape of a protein around a base state of a molecule to predict flexibility. Methods have been proposed to exploit such models to recover improved density maps from cryo-EM data of flexible molecules25. 3DFlex does not presuppose knowledge of the energy landscape or dynamics of the molecule, but rather learns this from the image data.Techniques such as normal-modes analysis27; notably, 3DVA (ref. 5) computes and visualizes subspace models at high resolution. More advanced techniques use nonlinear manifold embedding31 or deep generative models8 to construct a nonlinear manifold in the space of 3D density. Density-based methods, such as 3DVA and cryoDRGN, capture conformational and compositional heterogeneity, but they do not model protein motion or the preservation of local geometry. As such, they do not enable (nonrigid) alignment and the aggregation of signal across the conformational landscape.Density-based methods have recently emerged to learn continuous heterogeneity. Eigen-methods model the space of 3D conformations as a linear subspace32 represent heterogeneous protein density in a higher-dimensional space, which have the potential to capture nonrigid motion and structure but this has yet to be demonstrated on experimental data. e2gmm (ref. 9) represents a density using a Gaussian mixture model. Changes in 3D density are inferred by a neural network that adjusts the positions and amplitudes of the Gaussian components. Motion is estimated only at the Gaussian centers, in contrast to a dense deformation field with which one can align the entire density map for different conformations. These design choices and other limitations mentioned by Chen and Ludtke9 constrain model resolution, and a method to improve reconstruction quality of the aggregate density beyond the initial rigid consensus reconstruction is not proposed.Methods for continuous heterogeneity with models of motion have begun to emerge. Hypermolecules3DFlex recovers nonrigid motion from single-particle cryo-EM data of large complexes and smaller membrane proteins. The learned motion allows 3DFlex to improve resolution and map quality of flexible regions in the canonical density map by aggregating structural information across a protein’s conformational landscape. While these capabilities can help shed light on biological function, the current method has limitations which highlight interesting directions for future research.One limitation is that 3DFlex was not designed to handle compositional heterogeneity, where density appears and disappears between discrete states. In these cases, 3DFlex may use its model capacity to approximate disappearance as spatial diffusion of density rather than capturing flexibility. Currently, we recommend 3DFlex be run on particle subsets with little compositional heterogeneity. The deformation model of 3DFlex is, however, expected to find deformations between discrete conformational states, aligning them to the canonical map. Note that in the absence of particle data in intermediate conformations, 3DFlex is guided only by its inductive bias and rigidity priors in modeling these transitions.15. Some such changes remain beyond the reach of 3DFlex. Similarly, while 3DFlex improved the resolution of large regions of the ribosome compared to rigid refinement with all particles, similar or better resolution of smaller moving parts can be achieved by repeated application of focused 3D classification and masking15. One potential use of 3DFlex is to provide an interpretable latent representation of the conformational landscape, facilitating particle selection for conventional processing. Focused application of 3DFlex with masks may also be beneficial.In our results on the translocating ribosome, 3DFlex found large, functionally relevant motions. Molecular machines such as the ribosome do undergo these motions during a reaction, but their function also relies on small, intricate changes, such as the motion of a single side chain or loop33, or perhaps fitting atomic models across the latent space to enable further validation.With the ability of methods such as 3DFlex to resolve motion, there is a need for methods to validate that the deformation fields are well supported by the image data. These may be statistical methods, such as FSC36 instead of real-space voxel grids and the tetrahedral mesh encoding of motion. They can be optimized with gradient-based algorithms, but are not limited to spatially uniform resolution. Regularization of 3DFlex, both for the flow generator and the canonical density, can make use of structurally aware priors. For example, backbone models can be used to influence mesh construction and allow fine-grained structures to be resolved. The flow generator can also be expanded to allow for certain motions  to be more naturally encoded, and the architecture can be expanded to handle compositional variability.Beyond the current architectural and optimization choices used in 3DFlex, alternatives may provide gains, one example being the use of neural fields3DFlex is a generative neural network method for determining, from cryo-EM particle images, the structure and motion of flexible protein molecules at atomic resolutions. In what follows, we outline the formulation of the model and the essential design choices of the model architecture and learning procedure. We also discuss hyper-parameter selection for effective use.Central to 3DFlex is the overarching assumption that conformations of a dynamic protein are related to each other through deformation of a single 3D structure. Specifically, a flexible molecule is represented in terms of (1) a canonical 3D density map, (2) latent coordinate vectors that specify positions over the protein’s conformational landscape and (3) a flow generator that converts a latent coordinate vector into a deformation field that convects the canonical map into the corresponding protein conformation. The canonical 3D map, the parameters of the flow generator and a latent coordinate vector for each particle image are the model parameters that are initially unknown. They are jointly learned from experimental data.Ii is generated as follows. First, the K-dimensional latent coordinates zi of the particle are input to the flow generator fθ(zi). The generator provides a 3D deformation field, denoted ui(x), where x is a 3D position and θ denotes the parameters of the generator. The deformation vector field and the canonical 3D density map V are input to a convection operator, denoted D, which outputs a convected density, denoted Wi. The 2D particle image Ii is then a CTF-corrupted projection of Wi, plus additive noise η; that is,Ci denotes the CTF operator and P(ϕi) is the projection operator for pose ϕi, specifying the rigid transformation between the microscope coordinate frame and the coordinate frame of the canonical map.Under the 3DFlex model Fig. , a singlθ, the canonical density map V and the per-particle latent coordinates zM1:, to maximize the likelihood of the experimental data under the probabilistic model . This iThe 3DFlex framework entails several important design choices that define the architecture of the 3DFlex model. Computationally determining structure and motion from noisy cryo-EM data is a challenging problem. As such, discussion of the design choices below provides insight into the working model, reflecting our exploration of different designs and hyper-parameter settings during the development of 3DFlex.z is the low-dimensional latent coordinate vector for a given image, and the output is a 3D flow field u(x). The number of hidden units per layer  and the number of layers  are adjustable hyperparameters. The final layer is linear (without biases or nonlinear activation). The default 3DFlex architecture is a six-layer MLP with 64 units per hidden layer, which works well on a wide range of experimental datasets we have used. Larger models are more expressive but may also be more prone to overfitting. Accordingly, smaller networks are often useful for proteins with smooth motions, such as TRPV1 in the section We use a fully connected deep neural network  with rectified linear activation functions for the flow generator. The input The latent space in 3DFlex represents the conformational landscape. Different latent positions correspond to different deformations of the canonical map. 3DFlex defaults to a 2D latent space but allows the user to explore other options. For more complex motions, a large latent space allows for discovery of multiple dimensions or types of motion. Typically, we use latent dimensions between 2 and 6; in our experience it is useful to start with two dimensions, and then incrementally increase the latent dimension to explore more complex models.I and the current estimate of the 3D map, one wants to infer the posterior distribution over latent coordinates z, where high probability coordinates are those for which the flow generator and canonical map explain the image well. Equivalently, these are the latent coordinates that yield low values of the negative log likelihood  for each input particle image. In probabilistic terms, given an image 37, is amortized variational inference, in which a single feed-forward neural network (the encoder) is used to approximate the posterior for any image. Given an input image, the encoder outputs the mean and covariance over latent coordinates, essentially ‘inverting’ the generative model to predict the most likely latent coordinates for that image. This approach has been used by deep-learning based heterogeneity methods9. In the context of 3DFlex, the encoder would be trained jointly with the flow generator and the canonical map, to maximize the likelihood of the particle images.Determining the exact posterior distribution is, however, intractable for problems such as 3DFlex. So, instead, we turn to approximate inference. One approach, commonly used in variational auto-encoders (VAE)38. In the context of 3DFlex, an effective encoder network must be able to invert CTF corruption, 3D-to-2D projection, and 3D deformation to infer a latent state given a noisy 2D image. Motion must be learned in tandem with the generative model, so when the flow generator shifts a particular subunit up or down, the encoder must simultaneously learn the same motion and how it appears from 2D viewing directions to infer the position of the subunit in an image. This learning task is difficult, especially given the high noise levels in experimental images. Indeed, we did not find amortized inference effective for resolving high-resolution structure and motion.VAEs are usually stable to train and inference is fast, requiring just a single pass through the encoder network. They also incorporate a prior over latent coordinates that helps to regularize the latent space, encouraging smoothness. Nevertheless, amortized inference can be problematic. Primarily, it can be difficult for the encoder network to accurately approximate the posterior defined by the generative modelIn 3DFlex, we instead adopt an auto-decoder model, where we perform inference by optimizing a point estimate of the latent coordinates independently for each image, taking advantage of the structure of the generative model directly. Although more computationally expensive than amortized inference with an encoder network, this direct inference is more precise. This allows 3DFlex to capture structure and motion with sufficient detail to resolve flexible protein regions to higher resolution than is possible with previous methods.K of the latent space is small enough, it is also possible to use coordinate descent. We found the latter approach to be simpler and equally effective in our experiments.The generative model for 3DFlex is end-to-end differentiable, and so one can compute gradients of the data likelihood with respect to the latent coordinates for each image, and then use gradient-based optimization to perform inference. When the dimensionality z given I, one could explicitly approximate the posterior p(z ∣ I). In doing so, one captures uncertainty in z that can be used to help regularize the model and encourage smoothness of the latent space; that is, ensuring that nearby latent coordinates yield similar deformation fields, and hence similar conformations. In 3DFlex, we find that directly adding noise to the point estimate during training produces a similar positive effect. This method can be likened to variational inference with a Gaussian variational family with a fixed covariance, and has been used to regularize deterministic auto-encoders39. Finally, in addition to noise injection, we use a Gaussian prior on latent coordinates with unit variance to help control the spread of the latent embeddings for different particles within a given dataset, and to center the distribution of latent embeddings at the origin in the latent space.Instead of computing a point estimate for 41). On the other hand, the convection of density between conformations is more naturally expressed in real space, where structures in the canonical density map V need to be shifted, rotated and potentially deformed to produce densities consistent with the observed particles.Algorithms for single-particle reconstruction commonly represent 3D maps and 2D images in the Fourier domain. Working in the Fourier domain reduces the computational cost of CTF modulation and image projection (via the Fourier-slice theorem). It also allows maximum-likelihood 3D reconstruction with known poses in closed-form .In 3DFlex, we represent the canonical density Wi) to canonical coordinates. Convection then requires interpolating the canonical density V at positions specified by the flow field. For density conservation, the interpolated density must be modulated by the determinant of the Jacobian of the mapping, which is challenging to compute and differentiate. Instead, the flow in 3DFlex, ui, is a forward mapping from canonical coordinates in V to the deformed coordinates in Wi. This approach naturally conserves density, as every voxel in V has a destination in Wi where its contribution is accumulated through an interpolant function. The convected density at location x can be written asui = fθ(zi), k is an interpolation kernel with finite support and the summation is over 3D spatial positions y of the canonical map. Here, divergence and convergence of the flow field must be treated carefully to avoid undesirable artifacts such as holes, Moiré patterns and discontinuities. We found high-order  interpolation and strong regularization (below) useful to ensure accurate interpolation and artifact-free gradients.Convection of density is an essential element of 3DFlex, modeling the physical nature of protein motion, thereby allowing high-resolution structural detail from experimental data to backpropagate through the model. There are several ways to construct a convection operator. One is to express the flow field as a mapping from convected coordinates  in the canonical map, along with flow fields that translated these aberrations by large distances to explain noise in the experimental images. This problem was especially pronounced with smaller proteins, higher levels of image noise and membrane proteins containing disordered micelle or nanodisc regions . Overfitting also occurs when the regularization is not strong enough to force the model to separate structure from motion. For example, rather than improve the canonical density with structure common to all conformations, the model sometimes learned to deform a low-resolution canonical density to create high-resolution structure .To address these issues, 3DFlex exploits previous knowledge of smoothness and local rigidity in the deformation field. In particular, it is unlikely that natural deformations would involve large discontinuities in regions of high density; for example, an α-helix should not be sheared into disjoint pieces. It is also unlikely that deformations will be highly nonrigid at fine scales in regions of high density; at the extreme, bond lengths should not stretch or compress substantially. With these intuitions, we tried simple regularizers acting on flow fields defined at each voxel, such as limiting the frequency content of the flow field or penalizing its curvature. However, these regularizers were difficult to tune and did not prevent overfitting reliably.3DFlex instead models flow generation using finite-element methods. A tetrahedral mesh covering regions of high density is generated in the canonical frame, based on a preliminary consensus refinement or input by the user  below. Tjth tetrahedral element for image i, denoted uij(x) can be written as a linear mapping:A and vector b are uniquely determined from 3D flow vectors at the element vertices. We quantify local nonrigidity in terms of the distance between A and the nearest orthogonal matrix (in a mean squared-error sense43). In particular, we measure the squared deviation of the singular values of A from unity. Letting ℓth singular value of Aij, we express the local rigidity regularization loss aswj are weights defining the strength of the prior within each mesh element, based on the density present within the jth mesh element. The densest elements have weight 1.0 and empty elements have a lower weight, by default 0.5. This weighting ensures that deformation fields are encouraged to compress and expand empty space around the protein. Weight customization is possible but not necessary; the default weights were sufficient for all of our experiments.We also encourage local rigidity of the flow in each mesh element. The deformation field within the 3DFlex can take as input any mesh construction and topology. The size and connectedness of the mesh become hyperparameters of the model and provide inductive bias in estimating deformations. By default, 3DFlex automatically generates a regular mesh of tetrahedral elements of a fixed size and spacing made to cover the density of the input consensus refinement, where all elements share vertices with their direct neighbors.One can also create an customized mesh by modifying an automatically generated mesh, for example with variable sized mesh elements, overlapping elements or where it is not necessary that neighboring elements share vertices. The last case can be thought of as introducing ‘cuts’ into the mesh at the faces where neighboring elements do not share vertices. At these faces, the deformation model has the freedom to move adjacent mesh elements in opposite or shearing directions.44). These segments and the consensus map are input into the 3DFlex mesh utility. The utility generates a base regular tetrahedral mesh over the extent of the consensus map  during deformation. Likewise for custom meshes, the separation of subdomains does not need to be ‘exact’ as the canonical voxel density values and structure within each region of the mesh are still learned from the data by 3DFlex.45 or stochastic gradient descent with Nesterov acceleration46 with minibatches of at least 500 images, due to the high levels of image noise. Inference of the latent coordinates for each image in a minibatch is performed before computing gradients with respect to the canonical density and flow parameters. The loss function  is a weV is constrained to be low-pass, with an upper frequency band-limit that increases over iterations. The frequency and learning rate schedule, and λrigid, can be tuned for each dataset in our current implementation, but default values work well in most cases. Optimization is done with a box size, N = NL, that is typically smaller than the raw size of the particle images NH. In this way, 3DFlex optimization only has access to relatively low frequencies from the observed images . This is useful as it makes optimization faster, it allows one to use the resulting alignment of higher frequencies as a way to validate the quality of the deformation and makes it easier to rule out overfitting.During optimization, we use frequency marching, learning the model in a coarse-to-fine manner. The canonical density V to be a low-pass filtered version of a consensus nonuniform refinement map in cryoSPARC17, given the same particle images. The parameters of the flow generator are randomly initialized. The latent coordinates are initialized to zero by default. Optionally, one can also use different initial values. With smaller, low signal-to-noise particles, such as TRPV1 for example, we find that initializing with latent coordinates from 3DVA (ref. 5) in cryoSPARC18 improves results.To initialize training, we set the canonical density V and flow generator parameters θ leads to overfitting after thousands of gradient iterations, despite strong regularization. However, we find that a form of block coordinate descent provides a stable way to optimize 3DFlex. By default, when initializing from zero for latent coordinates, we first perform five epochs of optimizing θ and latent coordinates with V fixed. When initializing from input latent coordinates, we perform the first five epochs with the latent coordinates fixed. Then, we proceed to epochs alternating between optimization of V and optimization of θ and latent coordinates, repeating until convergence.We find that simultaneously training the canonical density With the ability of 3DFlex to capture motion and latent coordinates of each particle image, it becomes possible in principle to attempt recovery of high-resolution detail in flexible parts of protein molecules that would otherwise be blurred using conventional refinement methods.N = NL. Once optimization has converged, we freeze the flow generator parameters θ and the latent coordinates N = NH. We partition the particles using the same split that was used in the consensus refinement (from which we obtained the poses {ϕi}). For each half-set we initialize the canonical density V to zero, and re-optimize it at full box size NH, with the other parts of the model fixed. In the same way as established reconstruction validation methods33, the two resulting half-maps can be compared via FSC. Correlation beyond the training-time Nyquist resolution limit indicates that consistent signal was recovered from both separate particle sets, as opposed to spurious correlation or overfitting of the model. This correlation serves to validate the improvement in reconstruction of flexible regions with the learned deformation model.3DFlex is optimized at a small box size, V at high resolution under the full 3DFlex model for each half-set. We initially experimented with localized reconstructions (in Fourier space) but encountered issues with nonrigidity and curvature. We also found that minibatch stochastic gradient descent methods for directly optimizing V did not yield high quality results, potentially because noise in minibatch gradient estimates is problematic for this task. Instead, we were able to solve the problem using full-batch L-BFGS (ref. 16) in real space. This approach is substantially more expensive than Fourier-space reconstruction, and requires many iterative passes over the whole dataset. However, it is notable in that it allows 3DFlex to solve high-resolution detail in all flexible parts of the protein simultaneously, without making explicit assumptions of local rigidity or smoothness.To this end, we need to optimize Symmetric particles will not necessarily maintain symmetry during conformational changes, since the motion of individual subunits or the overall structure can break symmetry. In the current method, we do not enforce symmetry during training of 3DFlex. However, in processing of TRPV1 we do assume C4 symmetry in the initial rigid consensus refinement. To facilitate comparison with rigid reconstruction baselines where symmetry was enforced, we can apply symmetrization to the reconstructed half-maps after high-resolution reconstruction under 3DFlex. This makes the assumption that the canonical density of the molecule is indeed symmetric, and it is only the conformational changes that break symmetry. The symmetrized half-maps can then be compared using gold-standard FSC with half-maps from rigid reconstruction where symmetry was enforced, as we showed for the TRPV1 dataset in Fig. Further information on research design is available in the Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41592-023-01853-8.Supplementary InformationSupplementary Note, Figs. 1 and 2 and Table 1.Reporting SummarySupplementary Video 1Video of 3DFlex results for snRNP.Supplementary Video 2Video of 3DFlex results for TRPV1 (motion).Supplementary Video 3Video of 3DFlex results for TRPV1 (reconstruction).Supplementary Video 4Video of 3DFlex results for SARS-CoV-2 spike.Supplementary Video 5Video of 3DFlex results for Integrin.Supplementary Video 6Video of 3DFlex results for Ribosome.


Fiber Metal Laminates (FMLs) have garnered considerable attention and are increasingly being utilized in the development of protective armors for explosion and ballistic scenarios. While most research has focused on assessing the response of FMLs to single impacts, real battlefield situations often require shielding structures to endure multiple impacts. Thus, this study revolves around the creation of hybrid FMLs designed for shock shielding purposes. The primary focus is on how these laminates withstand repetitive impacts from high-intensity shock waves, aiming to pinpoint the optimal sequence that offers the highest resistance against multiple shock impacts. To establish effective shielding, a multi-layered FML configuration is employed. This configuration incorporates AA6061-T6 facing plates, ballistic-grade synthetic materials like aramid/epoxy ply, and ultra-high molecular weight polyethylene (UHMWPE)/epoxy ply. Additionally, a paperboard/epoxy lamina is introduced to induce functional grading based on layerwise shock impedance mismatches. Shock impact experiments are conducted using a shock tube equipped with helium as the driver gas. Critical shock parameters, including Mach Number, positive impulse, and peak overpressure, are meticulously evaluated. For validation purposes, a numerical model is employed to project the damage profile as a function of radial distance across different laminate sequences. The study unveils that ply deformations are strongly influenced by the arrangement of core layers, particularly the positions of the paperboard and UHMWPE layers within the core structure. To contextualize the findings, the shock impact results obtained from this study are compared with those from prior experiments that employed nitrogen-driven shocks. Sandwich structures are at the forefront of latest research with advancements involving use of novel core materials, modified adhesives, and clamping mechanisms for holding the layers together21. This shift in the paradigm of creating hybrid sandwich structures through the fusion of various material types, ushered in the era of Fiber-Metal Laminates (FMLs)22.In line with the ever-evolving armoury of weapons worldwide, the safety and defense requirements too have become stringent and demanding. The materials used for the shielding structures have undergone a drastic transformation from the early 2000s to the current. In the past four decades, engineers designing the shielding material for protection against impact from high velocity projectiles and shockwaves have resorted to a variety of materials— metallic24. FMLs are a sub-class of sandwich structures in terms of construction. Popular uses for FMLs include structural, impact resistance and shielding applications. In recent years, FMLs like ARALL, CARALL and GLARE have been the focus of in-depth research on mechanical properties, impact resistance to ballistic and blast loads, energy absorption capacities, and thermal resistance25. Including ballistic grade fabrics as sandwiched plies along with metallic layers can improve the resistance against the blunt impact by shock waves in addition to protection against ballistic impacts12. The interlocking weave pattern between fiber yarns (warp and weft) facilitates the distribution of shock energy over a broader area, effectively dissipating the energy across the fabrics. When subjected to shock impact, the high-strength fiber yarns within the woven pattern undergo stretching and deformation, efficiently converting the kinetic energy into mechanical work.FMLs combine lightweight, impact-resistant fiber-reinforced polymer plies with high strength and high stiffness (and correspondingly higher density) metallic layers to provide the best qualities of both materials28. This has been achieved through an explosive-based direct blast loading, assisted by a ballistic pendulum or the use of appropriate shockwave generating equipment (such as shocktubes) in35. When the blast wave or shock wave is incident on a target structure, a part of it is transmitted into the structure while another part is reflected. As the transmitted blast wave travels through the structure, it may undergo subsequent reflections at ply interfaces, which depends on the shock impedance mismatch between the adjacent layers. Transmitted high pressure waves lead to positive impulse while reflected pressure wave are responsible for a negative impulse in the corresponding layer. In one of the works, Latourte et al.9 examined the failures and deformation mechanisms of sandwiched and monolithic panels with polyvinyl chloride (PVC) foam core subjected to blast loading. Zhou et al.20 investigated the air-blast reaction of a FML made of stainless steel faceplate and backplate, sandwich structure containing Divinycell H-series PVC foam core, with suitable adhesive for bonding the layers. They observed local dishing of the faceplate, dome bending at the backplate, core crushing, and delamination at the faceplate/core and backplate/core interfaces. Menkes and Opat1 proved that substantial inelastic deformation (Mode 1), rupture at the boundary due to tension (Mode 2), and the supports being sheared (Mode 3) are the three main failure mechanisms in beams subjected to blast loads. Nurick et al.16 studied the blast-loading response of sandwich panels made of thin aluminium plates  approach employing the MATERIAL DESIGNER module. The stackups are built on the ANSYS COMPOSITE PREP/POST (ACP) to obtain the laminate properties. The loads applied in the numerical model are derived from the outcomes of the experiments. The numerical model predicts the deformation profiles of the FMLs subjected to shock wave impact, combining the approaches of RVE, Stackup construction, and helium-driven shock parameters from the shock tube experiments, which serves as the novelty in this work. The dimensionless deformation for the laminate computed by the numerical model is compared with that of the experimental results. Cone beam computed tomography (CBCT) is used to study the shock-impacted specimens of the sequences, and identify the failure modes and mechanisms of the FMLs. The relationship between the ordering of the layers within the core based on shock impedance matching, and the deformation profiles, failure modes, and shock shielding capabilities is studied. FMLs displaying least deformation offer the best protection against shockwave impact. Thus, this study enhances the understanding of mechanical response of FMLs to repeated shockwave impacts, and the numerical model can be used in other material systems. The inclusion of partially auxetic layers within the core and the location of the ply also influences the response of the FMLs. The research enhances knowledge on FML behavior under dynamic loads and provides valuable insights for designing impact-resistant materials. The findings hold significance for aerospace, defense, and automotive industries. However, the numerical model may fail to predict the FML responses at close proximity to the blast epicentre, since the shockwave propagation is three-dimensional.The experimental and numerical approaches used in the multiple shockwave impact effect studies on the FMLs can be inferred from Fig. 37. Subsequent to the metallic plates, ballistic grade aramid fabric was strategically positioned to provide impact resistance against blast debris and shrapnel that might breach the metallic plates. Likewise, ballistic grade UHMWPE would serve a similar role on the rear side of the core configuration. The metallic plates were made of AA6061-T6 sheets, to serve as the faceplate and backplate , core layers comprising UHMWPE, aramid, and paperboard plies, and epoxy binder. The epoxy resin used in the work was CT/AH?951 (ratio of the resin to hardener was 100:10 by weight). The resin and hardener were supplied by M/s Composites Tomorrow, India. The arrangements of the core layers were altered based on functional grading taking into account the shock impedance mismatch as a metric. Overall, five sequences were taken up for the shock impact investigation.The laminates were constructed using a multi-layered sandwich configuration, in which metallic plates sandwiched the core layers made of ballistic grade fabrics of aramid, UHMWPE, and a separate material class paperboard-epoxy ply for the purpose of shock tuning. Metallic plates play the role of high strength, high shock impedance layers, which attenuate the transmitted shock pressures propagating into successive layers of lower shock impedanceAA6061-T6 plates (0.7 mm thick) were procured from Hi-Tech Sales Corporation in Mangalore, India. The AA6061-T6 plates were subjected to mechanical abrasion in order to give them a rough finish that would facilitate proper wetting by the resin. The aramid plies (0.3 mm and 0.6 mm thick), UHMWPE plies (0.25 mm thick and 0.5 mm thick), and epoxy binder were sourced from Composites Tomorrow in Gujarat. In addition, paperboard sheet (0.25 mm thick) was supplied by Vijay Papers, India. Figure 26, the pressure at the contact face increases to a peak overpressure value before declining as seen in Fig. P’, the flow area ‘A’ over a time period ‘dt’. The positive impulse P(t) being the pressure at any time instant ‘t’, PAccording to the Friedlander curvementclass2pt{minim in Eqs.  is achie48, the specifications and the experimental settings have been described. Helium gas was chosen in the current study as the driver. Being an inert gas with an adiabatic index of 1.67 and the second highest mobility (next to hydrogen gas), it is capable of generating higher Mach numbers on the same shock tube with aluminium diaphragms . The specimen was located 50 mm behind the End pressure (EP) transducer as seen in Fig. He), beyond a certain pressure, the scribed diaphragm broke, sending shockwaves toward the target plate as demonstrated in Fig. A high-pressure shock tube was used for the shock impact experiments. In our previous worka’ depends on the adiabatic index of air ‘T’ (= 27–28 °C), and the gas constant ‘R’. Shock Mach Number ‘M’ is then expressed as a ratio of the shock speed to the acoustic speed. During each shock tube experiment, the data acquisition device yielded the pressure time data which was used to construct the pressure-time plots. The pressure-time plots helped obtain the key metrics: Shock speed P–t plot) and the decay time ‘The data acquisition device comprising a three channel oscilloscope (Make: HANTEK) and signal conditioner (Make: PCB Piezotronics) collected the pressure-time information for the shockwave. Each specimen of a particular sequence was first subjected to a single shock impact, after which measurements of the deformation profile were taken, and the specimen was then exposed to a second shock impact in the shocktube. Repeated measurements of the deformation profile were taken. The shock Mach number can be obtained using Eqs. , 5) and and5) anTo measure the faceplate and backplate deformations, the specimens were placed on suitable fixtures mounted on a surface plate and a dial guage was used to measure the deformation values by moving the tip across the diameter of the specimen as shown in Fig. 43. Regarding the modeling of aramid-epoxy, UHMWPE, and paperboard layers, they have been modeled as elastic until failure. As no penetration or perforation damage was observed, the damage model, strength model, and failure models of the plies were not taken into account12.To develop the numerical model, the individual behaviour of the constitutive plies in the five sequences had to be taken up. The elastic behaviour of AA6061-T6 was considered as isotropic and homogeneous while the other plies displayed orthotropic elasticity. To account for the plastic behaviour of aluminium alloy, a von Mises yield criterion was taken up48. An isotropic RVE was used for the AA6061-T6 layer, while a bidirectional (BD) RVE was chosen for the Aramid-epoxy ply due to its bidirectional fabric structure. For the UHMWPE-epoxy ply, a unidirectional (UD) RVE was employed, matching its unidirectional fabric orientation. Regarding the paperboard-epoxy ply, a chopped fiber RVE was selected in MATERIAL DESIGNER, where the fiber orientation was random. This randomness can be attributed to the paperboard’s composition, consisting of cellulosic fiber networks with fiber lengths ranging from 0.5 to 2 mm, forming a non-woven network held together by hydrogen bonding at various points51. The RVE unit cells for the materials are shown in Fig. 40 with following assumptions:The individual plies of the laminates apart from AA6061-T6 behaved orthotropically (modelled elastic until failure), with the primary material axes positioned along global x and y axes, respectively.D’).Each ply has a constant thickness “The interfaces of the plies are flawlessly bonded, and any porosity within the plies is neglected.Since circular laminate plates are used as test specimens for shocks, Fig. a, by app30, over a decay time of 7–7.6 ms. Hence, the average pressure 3. The values of ‘pDriver gas pressure (Helium) acts during the positive impulse phase for the single shock impactIn a N-ply laminate . The maximum transverse deflection after the second shock impact ‘Using the RVE approach on the MATERIAL DESIGNER module of ANSYS R20.0,the orthotropic properties of the plies were computed with a similar approach to one of our recent worksmentclasspt{miniman by Eq.    48. An iUsing the i-CAT CBCT scanner, the specimens that had been subjected to the twin shock hits were examined independently. In the cone-beam technique, the X-ray head rotates continuously in a 360°scan, focussing on the specimen placed on a suitable fixture. Figure 48. Afterwards, there is a gradual decrease to the initial pressure value, over a decay time ‘tAs the diaphragm separating the driver and driven sections of the shocktube ruptures, a compression shockwave travels through the driven side and at the same time, a rarefaction wave through the driver side. The pressure on the driven side of the shocktube increases to P33.The least backplate deformation . For the results of the numerical model, the deformation was predicted for the entire laminate without differentiating the faceplate and backplate deformations. Hence, the relative deformation given in Eq. , while substantial difference between the faceplate deformation values between experimental and numerical results was observed. Also, the discrepancies in the spatial deformation values for the sequences can be attributed to the formation and propagation of the shock wave inside the circular section at the juncture of diaphragm fracture. The shock velocity profiles resulted in lower deformation for the lower sectional radii, and at the centre of the shock tube, maximum shock velocity was responsible for high central deformation across sequences.The face and the back layers distort in the direction of the shock impact, hence, a dimensionless cumulative deformation , while the AA6061-T6 backplate protruded significantly due to the shock impact as seen in Fig. Mach numbers (The decreased deformations displayed by SRSP-II, SRSP-IV, and SRSP-III in comparison to SRSP-I, indicate that using paperboard-epoxy ply as a central layer to induce shock impedance mismatch can positively influence the FML response.The maximum backplate deformation among the sequences, observed in SRSP-V was caused by the absence of the AA6061-T6 faceplate. As a result, the presence of a metallic face layer of high shock impedance is essential for FMLs for effective shock energy attenuation.The numerical model yielded moderately accurate results for faceplate deformation in SRSP-I, SRSP-II, SRSP-III, and SRSP-IV sequences. However, a notable disparity was observed in SRSP-V between the numerical and experimental deformation. Moreover, the numerical model consistently underestimated backplate deformations across all sequences.There was an excellent agreement between the cumulative dimensionless deformation from shocktube experiments and the relative deformation predicted by the numerical model across all sequences, which proves that the numerical model can be successfully utilized to capture the shockwave impact responses for laminates.As for the shock impact effect on the materials of the FMLs, successive impacts cause the deformation of the FMLs to increase drastically, indicating that thin-FMLs can sustain up to a maximum of twin shock impacts.Helium-driven shocks severely deformed the specimens, with the second shock impact leading to the delamination of AA6061 plates, as inferred from the CBCT scanned images of the specimens. The backplates showed core delamination while the faceplates showed delamination at the peripheral annulus region.Among all the sequences, SRSP-II and SRSP-IV emerged as ideal laminate sequences for use as effective shielding configurations against multiple shock impacts, as they showed the least backplate deformation essential to safeguard any structure lying beyond the shield. The functional grading of the sequences based on shock impedance matching could be advantageous for shock impact resistance, and could be extended to other material systems in multi-layered sandwich structures.The primary objective of investigating the response of different stackup sequences to repeated helium-driven shock impacts was successfully achieved. The experimental results, encompassing actual deformation, maximum deformation, and dimensionless cumulative deformation, were examined. These findings were compared against the outcomes generated by numerical simulations. By juxtaposing the deformation profiles of various stackups subjected to helium-driven twin shock impacts with those from nitrogen-driven twin shock impacts, noteworthy distinctions were observed. Beyond the evident distinctions in higher Mach numbers and quicker decay times associated with helium-driven shocks, it became evident that such shocks intensified depression, bulging, and delamination failures within the laminates. The shock-impacted specimens were subjected to CBCT scanning and the modes of failure in each of the laminates were scrutinized. The following conclusions were drawn :


When t/d decreased from 8.6 to 2.4, σb decreased from 1458 MPa to 1132 MPa, σs decreased from 917 MPa to 730 MPa, and the elongation decreased from 0.31 to 0.28 at ultra-low temperatures. When t/d decreased from 8.6 to 2.4, the surface roughness of the specimen increased from 0.769 to 0.890 at room temperature and increased from 0.648 to 0.809 at ultra-low temperatures. During the microplastic deformation process of Ni-Co-based superalloy thin sheets, the coupled effects of surface roughening caused by free surface grains and hindered dislocation movement induced by grain boundary resulted in strain localization, which caused fracture failure of Ni-Co-based superalloy thin sheets.With the development of product miniaturization in aerospace, the nuclear industry, and other fields, Ni-Co-based superalloys with excellent overall properties have become key materials for micro components in these fields. In the microforming field, size effects significantly impact the mechanical properties and plastic deformation behavior of materials. In this paper, micro-tensile experiments at room temperature and an ultra-low temperature were carried out to study the effects of initial microstructure and deformation temperature on the deformation behavior of Ni-Co-based superalloy thin sheets. The results show that as the ratio of specimen thickness to grain size ( Product miniaturization is one of the main trends in advanced manufacturing technology. Micro-electro-mechanical systems (MEMS) emerged under this situation. Microforming technology has the characteristics of large volumes, high efficiency, high precision, low costs, and no pollution and has become a key technology for the manufacturing of thin sheet microstructure parts ,2,3. Supt/d, and a “smaller is weaker” size effect appeared. In the microforming of metal thin sheets, the strain coordination ability between grains decreased with decreasing t/d, resulting in a decrease in the homogeneous plastic deformation ability. Nie et al. [t/d. As t/d decreased, the fracture mechanism changed from ductile fracture to brittle fracture. Zhao et al. [t/d for pure copper foils with thicknesses ranging from 100 to 600 μm.To name a few, previous studies ,22,23,24e et al.  conducteo et al.  conducteo et al.  conducteo et al.  conducteo et al. ,30 foundExisting research shows that ultra-low temperatures play a very good strengthening and toughening role in face-centered cubic (FCC) metal ,32,33. TTo date, research on size effects mainly focuses on room temperature and high temperatures. Ultra-low temperature forming is still a relatively new and potential forming method to improve the mechanical properties of metal materials. The deformation characteristics of materials at ultra-low temperatures are very different from those at room temperature, such as mechanical properties, microstructure evolution, fracture mechanism, etc. Therefore, this study carried out micro-tensile tests on Ni-Co-based superalloys at room temperature and an ultra-low temperature. The microstructural evolution behavior during micro-tensile deformation was characterized and analyzed using electron backscatter diffraction (EBSD) and scanning electron microscopy (SEM). The impacts of grain and ultra-low temperatures on the micro-tensile properties and deformation mechanism of Ni-Co-based superalloy thin sheets are clarified. This study provides a theoretical basis for the forming process of Ni-Co-based superalloy thin sheet micro components and has practical application value in promoting its application in the microforming field.2H5OH+20 mL HCl+3 gCuSO4, and the metallographic structure was observed using an OLYMPUS microscope .Cold-rolled and annealed Ni-Co-based superalloy thin sheets were selected as the experimental material. The main mass percentages of Ni-Co-based superalloys were 25% Co, 10% Cr, 4% Mo, 1% W, 2% Al, 4.5% Ti, 1% Nb, and the rest was Ni. To obtain specimens with different initial grain sizes, solution heat treatment schemes were designed, namely 1150 °C/4 h, 1150 °C/6 h, 1150 °C/8 h, 1200 °C/2 h, 1200 °C/4 h followed by water quenching (WQ). WQ was to maintain the microstructure at high temperatures. To prevent Ni-Co-based superalloy from oxidation during the heat treatment process, the vacuum heat treatment method was used. The solution specimen was ground with 400–1000 grit SiC metallographic water sandpaper. The specimen was mechanically polished using a diamond metallographic spray polishing agent with a particle size of 0.5~2.5 μm until there were no obvious scratches on the surface of the specimen. Chemical etching was carried out in a mixed solution of 20 mL C−1. The ultra-low temperature tensile device included a cryogenic chamber, a temperature control device, and a liquid nitrogen tank, as shown in 2H5OH+10% perchloric acid. A scanning step size of 0.5 μm and a binning of 4 × 4 were used for EBSD  characterization. The fracture morphology of the specimens was characterized and analyzed using SEM .The dog-bone-shaped micro-tensile specimens with a gauge width of 2 mm and a gauge length of 6 mm were prepared along the rolling direction (RD) using electrical discharge machining (EDM). A 30 kN electronic universal testing machine  was used to conduct the micro-tensile test, and the strain rate was set to 0.01 sα represents the proportion of grains in the surface layer of the specimen; t represents the thickness of the specimen (μm); W represents the width of the specimen (μm); d represents the average grain size of the specimen (μm).At the mesoscopic scale, materials no longer behave as isotropic strain-hardening materials. The surface layer model is currently a widely recognized theory in the microforming field ,45. The α was 25.1%. However, when the grain size was 84.8 μm, α reached 86.1%, which led to an obvious softening effect of flow stress.t/d. When t/d decreased from 8.6 to 2.4, the tensile strength σb decreased from 1221 MPa to 1090 MPa, and the yield strength σs decreased from 793 MPa to 622 MPa at room temperature. When t/d decreased from 8.6 to 2.4, σb decreased from 1458 MPa to 1132 MPa, and σs decreased from 917 MPa to 730 MPa at ultra-low temperatures. The engineering stress of the micro-tensile specimen with a grain size of 84.8 μm increased with increasing engineering strain. However, the increasing degree of engineering stress was slightly lower compared to those of micro-tensile specimens with other grain sizes. This was because the surface layer grain deformation was less constrained due to the existence of the free surface, and dislocations were not easily accumulated in the surface layer grains, resulting in a reduction in the strain hardening ability of the surface layer grains and lower deformation resistance. At the same t/d, the stress of the specimen increased significantly with decreasing deformation temperature. It could be inferred that the ultra-low temperature was beneficial in improving the strength of the materials. The elongation of the alloy was better due to the matrix γ phase of the face-centered cubic (FCC) structure. When t/d decreased from 8.6 to 2.4, the elongation decreased from 0.26 to 0.21 at room temperature, and the elongation decreased from 0.31 to 0.28 at ultra-low temperatures. As the proportion of the surface layer grains increased, the number of grains contained in the thickness direction decreased, weakening the deformation coordination ability. Thus, the elongation of the Ni-Co-based superalloy decreased with decreasing t/d. At the same t/d, the elongation of the specimen increased significantly with decreasing deformation temperature. In summary, the strength and elongation of materials under the ultra-low temperature conditions were significantly better than those at room temperature. Therefore, ultra-low temperatures had a strengthening and plasticizing effect on the Ni-Co-based superalloys.The engineering stress–strain curve has universal significance because the curve reflects various deformation processes such as plasticity, yield, and fracture that occur in the material under the action of external forces. Plastic strains can be characterized by local micro-orientations in the material . KAM repAs shown in When the grain size was small, the number of grains in the thickness direction of the specimen was large, and the deformation of the grains could be distributed to adjacent grains to coordinate the deformation, making the plastic deformation relatively homogeneous. As the grain size increased, the number of grains in the thickness direction of the specimen decreased, and the proportion of surface layer grains increased, which produced significant size effects. Therefore, as the proportion of the surface layer grains increased, the deformation coordination ability of materials weakened during the micro-tensile deformation process. At the same time, soft-oriented grains were easy to deform, while hard-oriented grains were difficult to deform. The material tended to follow the direction of minimum acting. The reduction in grains contained in the thickness direction of the thin sheet tended to form a “weak and weak” bond, causing strain concentration in the specimen and premature fracture.As the deformation temperature increased, the KAM value of the Ni-Co-based superalloy specimen decreased, and the deformation ability decreased. When the strain was 0.19, obvious concentrated areas with larger KAM values appeared in the surface grains and internal grains of the specimen at room temperature. However, the areas with larger KAM values were more evenly distributed throughout the grain at ultra-low temperatures. The KAM value distribution of the fractured specimen in The plastic deformation and ductile fracture at the mesoscopic scale are very different from those at the macroscopic scale. Surface roughening occurs during the plastic deformation of polycrystalline materials, which affects the subsequent plastic deformation behavior and formability of the material. The evolution of specimen surface roughness is mainly affected by the initial microstructure and deformation conditions. To explore the effects of grain size and deformation temperature on material flow behavior during micro-tensile deformation, the surface roughness was measured using a confocal laser scanning microscope (CLSM)  on the gauge length part of the specimen with a scanning step of 0.3 μm and an observation area of 640 μm × 640 μm.Differences in crystal orientation between adjacent grains led to surface roughening at the mesoscopic scale . PolycryAdditionally, the surface roughness of specimens at ultra-low temperatures is lower than that at room temperature. This may be due to the fact that an ultra-low temperature reduces the dislocation activation energy and inhibits dislocation recovery, which increases the dislocation density within the grain. Compared with the deformation at room temperature, the deformation at ultra-low temperatures is borne more within the grain, thereby reducing the overall rotation of the grain. Thus, the deformation becomes more homogeneous, resulting in less surface roughening.η of the specimen decreased from 0.902 to 0.816, with a decrease rate of 9.5%. The increase in the proportion of the surface layer grains promoted the development of local necking and reduced the deformation uniformity.As shown in η decreased from 0.920 to 0.889 with a decrease rate of 3.4%. The difference in deformation between the surface grains and the internal grains was reduced, indicating that the specimen deformed more homogeneously at ultra-low temperatures. High local misorientation was mainly concentrated inside the grains at ultra-low temperatures, and the storage capacity of dislocations was enhanced. The weakening of strain concentration at grain boundaries resulted in a delayed onset of cracks. Thus, ultra-low temperatures enhanced the ductility of Ni-Co-based superalloys.As shown in η) [After high-temperature solution treatment, there were no second-phase particles in the matrix γ phase. The deformation of surface layer grains was less constrained due to the presence of free surfaces. The surface layer grains were easy to rotate to promote the movement of dislocations, thereby reducing the flow stress and hindering the occurrence of void nucleation in the surface layer grains. The rotation of the grains caused uneven surface deformation. The rotation of grains was considered as a possible explanation for surface roughening during plastic deformation ,50. The η) ,52 cause(1)Different grain sizes were controlled with high-temperature solution treatment, and the average grain size of the Ni-Co-based superalloy increased from 23.3 μm to 84.8 μm and the feature size decreased from 8.6 to 2.4 with increasing solution temperature.(2)t/d decreased from 8.6 to 2.4, the tensile strength σb decreased from 1221 MPa to 1090 MPa, the yield strength σs decreased from 793 MPa to 622 MPa, and the elongation decreased from 0.26 to 0.21 at room temperature. When t/d decreased from 8.6 to 2.4, σb decreased from 1458 MPa to 1132 MPa, σs decreased from 917 MPa to 730 MPa, and the elongation decreased from 0.31 to 0.28 at ultra-low temperatures.The flow stress decreased significantly with increasing grain size and increased significantly with decreasing deformation temperature. When (3)t/d and decreased with decreasing deformation temperature. Ultra-low temperatures enhanced the plastic deformation ability of the alloy. When t/d decreased from 8.6 to 2.4, the surface roughness of the specimen increased from 0.769 to 0.890 at room temperature and increased from 0.648 to 0.809 at ultra-low temperatures.The surface roughness of the specimen increased with decreasing (4)The coupled effects of surface roughening caused by free surface grains and hindered dislocation movement induced by grain boundaries during plastic deformation resulted in strain localization, causing fracture failure of the Ni-Co-based superalloys.This study carried out micro-tensile tests on Ni-Co-based superalloy thin sheets. The effects of grain size and deformation temperature on micro-tensile mechanical properties and microstructure evolution were analyzed, and the micro-tensile fracture mechanism was revealed. The main conclusions are as follows:


State of health is a critical state which evaluates the degradation level of batteries. However, it cannot be measured directly but requires estimation. While accurate state of health estimation has progressed markedly, the time- and resource-consuming degradation experiments to generate target battery labels hinder the development of state of health estimation methods. In this article, we design a deep-learning framework to enable the estimation of battery state of health in the absence of target battery labels. This framework integrates a swarm of deep neural networks equipped with domain adaptation to produce accurate estimation. We employ 65 commercial batteries from 5 different manufacturers to generate 71,588 samples for cross-validation. The validation results indicate that the proposed framework can ensure absolute errors of less than 3% for 89.4% of samples (less than 5% for 98.9% of samples), with a maximum absolute error of less than 8.87% in the absence of target labels. This work emphasizes the power of deep learning in precluding degradation experiments and highlights the promise of rapid development of battery management algorithms for new-generation batteries using only previous experimental data. Estimation of Li-ion battery state of health is crucial but requires time- and resource-consuming degradation tests for development. Here, authors propose a deep-learning method that enables accurate estimations without additional tests, ensuring absolute errors of less than 3% for 89.4% of samples. The global market of LIBs displays staggering figures in 2020, up to 142.8 GWh on the side of electric vehicles, and it is expected to exceed 91.8 billion dollars5 in the next few years. While LIBs are being popularized at a phenomenal rate, their prolonged applications are facing tough challenges. As is the case with machines, LIB components such as electrodes and separators experience varying levels of degradation. These negative spillovers lead to capacity and power fade7 and thereby imperil the assets8. To ensure safe and efficient battery management, obtaining an accurate battery state of health (SOH) is of vital importance.Lithium-ion batteries (LIBs) offer high energy density, fast response, and environmental friendliness9 or by the increase in the internal resistance10. Although these variables are easily measurable, battery degradation is also accompanied by capacity loss, whose accurate determination impacts other battery management tasks such as driving range estimation and life prediction. Thus, SOH, defined as the ratio between the present capacity and the initial capacity, is drawing broad attention12. However, the capacity measurement requires completely charging or discharging the batteries with specific protocols13, which is not practical for batteries in use. This motivates the SOH estimation from daily operating data.Battery SOH has been defined in various forms. It can be defined by the service time15, electrochemical features16, acoustic features17, mechanical features19, and thermal features20. Such arduous data collection and feature engineering steps impede the development of SOH estimation methods. Recently, deep neural networks (DNNs)22 have demonstrated the automatic extraction of black-box features from raw operating data, showing impressive SOH estimation performance. However, experimental collection of the target-labeled data is time-consuming and resource-intensive23, and creating massive datasets for different types of LIBs is rarely sustainable.Existing SOH estimation studies are generally devoted to extracting features correlated with SOH degradation and mapping them to the SOH. These methods require lifelong battery degradation data with measured SOH labels of the target LIBs . On this basis, many features have been crafted based on our understanding of battery degradation, such as the electrical features24. For example, a DNN for charging curve prediction trained using one type of battery can adapt to other types of batteries by fine-tuning using a small number of new samples25. A growing body of literature27 applies retraining or fine-tuning techniques for SOH estimation of various types of batteries. Generally, these works require at least 25–30% of labeled lifelong data in the target domain. Therefore, target domain degradation experiments are still needed. These approaches relying on conventional experiments cannot keep pace with the battery upgrading and have established a barrier to the development of battery technologies.A paradigm of generalization for DNNs is transfer learning, which is one promising solution to lighten the burden of data collection. Transfer learning transfers knowledge learned by a DNN in a training dataset (the source domain) to a different dataset (the target domain) using a small number of target-labeled data30. This can help the rapid development of battery management systems (BMSs) for new-generation batteries using only existing experimental data, saving considerable time and resources. It is also expected to motivate the utilization of large-scale field data without labels. Indeed, issues for target label-agnostic cases have long been noticed. Cross-domain learning in the absence of target labels has been proven to be equivalent to a dual training task, i.e., learning to predict the source labels while closing the gap between the source and target domains32. Research in the field of visual recognition33 has shown that even though there are no target-labeled data for training, a DNN jointly trained for classification and domain invariance can conduct classification precisely across visually distinct domains. This confirms that DNNs can accomplish the tasks in target label-agnostic cases. However, LIB SOH estimation, which has a massive demand for target labels, has yet to benefit from it.Approaches without the need for additional target-labeled data to estimate SOH are attractiveIn this work, we propose a deep learning-based framework to estimate battery SOH without relying on target labels for training. The proposed framework integrates the estimates of a swarm of DNNs into a reliable SOH estimate rather than relying on a single DNN. Individual DNNs are trained to learn cross-domain knowledge according to source labels and domain invariance of degradation features. DNNs with good performance in the swarm are selected for reliable estimation. We further reveal the influence of the sample distribution in the source domain on SOH estimation and propose to improve estimation performance by trimming the sample distribution in the source domain. We adopt two self-developed and three public battery degradation datasets for cross-validation. The validation covers 80 cases, encompassing 71,588 samples collected from 65 cells. We demonstrate that the proposed framework can achieve an absolute error of less than 3% for 89.4% of samples (less than 5% for 98.9% of samples), with a maximum absolute error of less than 8.87%. To provide references for the selection of hyper-parameters, we also investigate the influence of the crucial hyper-parameters on the estimation performance. These results highlight the potential of deep learning in supplanting the time-consuming battery degradation experiments, and further rapid development of BMS for new-generation batteries using existing experimental data.We develop a SOH estimation framework composed of a swarm of DNNs Fig. . This frN DNNs  CNN layers to extract their feature vectors. After that, feature vectors of the source domain are flattened and fed into a terminal fully connected (TFC) layer to generate their SOH estimates. These estimates are used together with the source domain labels to calculate the source domain loss. On the other hand, feature vectors of target-domain samples are flattened to a middle fully connected (MFC) layer for reconstructing their feature vectors. These reconstructed feature vectors play two roles. The first is to quantify the domain gap together with the source domain feature vectors. The second is to provide estimates of target domain samples (treated as the pre-estimates of each trained DNN) in the estimation procedure, where the reconstructed feature vectors are further fed into the same TFC as the source domain for regression. By simultaneously minimizing the SOH estimation loss of source domain samples and the gap between the TFC inputs of the two domains, each sub-training transfers the source domain knowledge to the target domain.Each DNN in the proposed framework has identical hyper-parameters Fig.  but is iThe estimation procedure of the proposed framework, unlike the training procedure, is to select a swarm of the trained DNNs to participate in the estimation Fig. . First, 2, LCO)39, a blend of LCO and lithium nickel cobalt oxide 7, nickel manganese cobalt (Li(NiMnCo)O2, NMC)40, nickel cobalt aluminum , and lithium iron phosphate . Note that the exact composition of the cathode active materials cannot be further provided here as this information is not available in the existing literature. The specifications of these five types of LIBs and their experimental costs are compared in Table 43. Details regarding the datasets can be found in Table SOH estimation in target label-agnostic cases spans different applications, manufacturers, and chemistries. To reflect such situations, we employ 10,757 samples collected from 65 commercial LIB cells produced by five different manufacturers for validation. The eventual datasets cover five kinds of widely-used cathode active materials, including lithium cobalt oxide , respectively. As a result, each combination contains four cases with four different lower SOH bounds , random forest45 (RF), support vector regression46 (SVR), and CNN47. Their hyper-parameter settings can be found in Table To verify the advancement of the proposed framework, we gather all validation cases and compare our accuracy with that of four popular methods, including Gaussian process regressionWe first show the performance of the four existing methods when the target domain labels are available. Having enough target labels for learning the target domain, the existing methods show high accuracy with MAEs of less than 1%. However, the target labels in practice come at the cost of numerous workforce and energy. Developing a battery degradation dataset requires 644–8473 hours of degradation experiments  to investigate its influence. The distributions of pre-estimation root mean square errors (RMSEs) of DNN swarms in these cases are reported in Fig. W* of the TFC layer and these feature vectors along the channels (theorized in the DNN explanation section). We first dissect an individual DNN without domain adaptation at 10 equal-interval cycles to visualize the evolution of the explanation map as a function of degradation in Fig. After evaluating the swarm-driven strategy, we investigate the other crucial strategy of the proposed framework, i.e., domain adaptation. The feature vectors, before being fed to the terminal fully connected layer, are the subject of domain adaptation. Nevertheless, we do not focus on them but define an explanation map to visualize their contributions to the SOH estimate. The explanation map is defined as the weighted sum of the unflattened weight vector We further investigate the impact of some crucial hyper-parameters on the estimation performance of the proposed framework, including the size of the DNN swarm, activation functions, number of channels, and number of layers of the CNN. The size of the DNN swarm is set to 1, 50, 100, …, and 300, respectively, and the results are shown in Fig. The present study can be improved in the future. First, as a data-driven approach, the proposed framework does not assume specific properties and dimensions of the input data. Hence, the proposed framework can be applied to a wider variety of battery materials, other SOH metrics, and input signals. Second, our preliminary trimming strategy can be developed with more advanced techniques to optimize estimation performance. Finally, the proposed framework does not assume specific application scenarios. It thus can be explored to apply to the big data containing a large amount of battery real-world operation history. The proposed framework is promising to help maximize the potential of big data, which generally lacks labels.Existing techniques for battery SOH estimation are highly dependent on the labeled degradation data of the target battery, resulting in an enormous expenditure of time and resources for data collection. In this work, we devise a target label-agnostic solution to battery SOH estimation based on deep learning. This framework selectively integrates the estimations of a swarm of DNNs into a reliable SOH estimate rather than relying on a single DNN. Each DNN is trained for source labels and domain invariance of degradation features simultaneously. A trim strategy is proposed to regulate the skewness of the source domain sample distribution to improve the accuracy.As a case study, we take the partial charging curve as the input of the proposed framework. For validation, we combine two experimentally generated datasets and three public datasets for cross-validation, resulting in 80 cases covering 71,588 samples. We first demonstrate that the proposed framework can achieve absolute errors of less than 3% for 89.4% of samples (less than 5% for 98.9% of samples), with a maximum absolute error of less than 8.87% in the absence of target battery labels. Compared with the existing methods, the proposed framework reduces the MAE and maximum absolute error by more than half. These results illustrate the successful application for various domains. Furthermore, we dissect the DNN and visually explicate that the proposed architecture of DNNs can effectively minimize the domain gap. The analysis of the swarm of DNNs unveils a correlation between the mean, standard deviation, and errors of DNNs’ estimates and clarifies how our framework can select the DNNs for SOH estimation in target label-agnostic cases. Finally, we investigate the impact of the crucial hyper-parameters on the estimation performance, and provide references for hyper-parameter selection to apply the proposed framework better.In summary, our work highlights the potential of deep learning in supplanting time- and resource-consuming battery degradation tests. We envisage that the proposed framework will motivate the utilization of large-scale historically collected but unlabeled LIB data . It can also enable the rapid development of BMS for new-generation batteries using only existing experimental data.V(t) and current I(t) are stored by BMS at a time step t, and the partial charging curve qψ can then be captured by setting a voltage sampling window:ψ indicates whether qψ belongs to the source domain S or the target domain T. Q denotes the initial capacity, which is used to normalize the partial charging curve of different types of LIBs. Vmin is the lower voltage limit. The voltage sampling window is gridded by a given voltage step ΔV and ranges from Vmin to Vmin + KΔV.Partial charging curves of LIBs are employed as the input of DNNs for SOH estimation. In a constant-current charging process, the voltage nbin bins with a uniform width (set to 2% in this work) according to their labels. Using the number of samples ni,origin in the i-th bin as an upper bound, the number of samples in this bin ni,trim that need a trim is then optimized by:nremain represents the number of the remaining samples after trim. nremain source domain samples randomly selected from the i-th bin. gk denotes the component of the objective function, k∈{1,2}, and αk is the weight corresponding to the gk. In this work, α1 and α2 are set to 0.2 and 2, respectively. μ3(∙) is the skewness operator defined as the third standardized moment:μ and σ are the mean and standard deviation of y, respectively. In this optimization, minimizing g1 makes the skewness of the trimmed source domain approach its target value μ3* (set to zero in this work) to avoid asymmetry, while minimizing g2 ensures that as few samples as possible are discarded. Two constraints on the optimization are defined to generate a new source domain with a comparable range and unimodal distribution:ε is the maximum difference in the number of samples among the bins, which is set to 4.5% in this work.To improve estimation performance, we generate a more balanced source domain by trimming the distribution of samples in the original source domain. Specifically, the original source domain samples are first grouped into x, x ∈ {1, 2, …, N}, the gridded input is first processed by serially stacked 1D CNN layers for feature vector extraction, which can be formulated as:l+1Xψ and lXψ denote the output and input of the (l + 1)-th 1D CNN layer, respectively. ⊗ represents the valid cross-correlation operator. l+1w,l+1b,l+1k, and l+1str are the weight, bias, kernel size, and stride of the (l + 1)-th 1D CNN layer, respectively. lC and l+1C are the numbers of input and output channels of the (l + 1)-th 1D CNN layer, respectively. l+1L and lL are the lengths of the l+1Xψ and lXψ, respectively. (ϒ−1) denotes the total number of the 1D CNN layers.The proposed framework is composed of a swarm of DNNs. For the DNN #ϒXT and MFCXT are the target domain input and output of the MFC layer, respectively. WMFC and bMFC are the weight and bias vectors of the MFC layer, respectively.A MFC layer is exclusively designed for the target domain after the shared CNN layers to reconstruct the extracted feature vectors. The output of the terminal 1D CNN layer is flattened by the channel and then input to the MFC layer, which can be described as:ϑψ denotes the SOH pre-estimate of each DNN. ϒXS is the source domain output of the ϒ-th 1D CNN layer. WTFC and bTFC are the weight vector and the bias of the TFC layer, respectively.A shared TFC layer is designed at the terminal of the DNN for both source and target domains to regress SOH. The source domain output of the terminal 1D CNN layer is flattened and then fed to the TFC layer, while the target domain output of the MFC layer is directly provided to the TFC layer. This layer can be expressed as:The rectified linear unit (ReLU) activation function, which takes the maximum value between 0 and its input as output, is designed to follow each 1D CNN layer and fully connected layer. The sigmoid activation function is applied before outputting the estimate to ensure that the SOH pre-estimate is between 0 and 1.Fψ to visualize the domain adaptation of the proposed framework, which can be formulated as:W* is the unflattened weight vector of the TFC layer. ζXψ is the unflattened feature vector or the feature vector before flattening. The sigmoid activation function is also applied before the output of this equation. Thus, Fψ is equivalent to a TFC layer output minus bTFC and cancels the sampling point-wise weighted sum. Combining Fψ at various SOH can produce an explanation map for visualizing the contribution of the feature vectors to the lifelong SOH estimation.We define a vector N DNNs are independently trained, and in the present study, we set N = 300. Each DNN is parameterized by the labeled data from the source domain and the unlabeled data from the target domain. The widely-used Adam algorithm48 is employed to optimize the parameters iteratively. The learning rate is set to 0.001. To realize cross-domain transfer learning, we define a loss function E containing three components, which can be formulated as:fi denotes the component of the loss function J, i ∈ {1, 2, 3}, and κi is the weight corresponding to the fi. This work sets κ1, κ2, and κ3 to 1, 0.1, and 1, respectively. *ϑS denotes the available label of the source domain sample. ns and nt are the number of samples from the source and target domains, respectively. ||·||H represents the norm of the reproducing Hilbert space in terms of the embedding kernel ϕ(·), and the Gaussian kernel is employed as the ϕ(·). ϑT0 denotes the pre-estimate of the target domain at the first cycle, and Z is the number of these pre-estimates. f1 evaluates the mean squared error between each element of the source domain in the pre-estimate and the available labels, which is designed for training each DNN to learn SOH estimation from labeled source domain samples. f2 evaluates the domain invariance, and the maximum mean discrepancy (MMD) is employed as a criterion to measure the distance between the high-dimensional degradation features of the source domain samples and those of the target domain samples after reconstruction. f3 is also the measure of the mean squared error but only for the target domain samples at the first cycle. This is because the partial charging curves of the first cycle  are easily obtained , and their labels can be treated as 1 to improve the learning of the target domain samples.First, In each sub-training, samples from the source domain are divided into a training set and a validation set. Two-thirds of the source domain samples are used as the training set, and the rest are the validation set. Each sub-training is terminated when the RMSE of the validation and training sets is less than both 5%, or when the number of epochs reaches 2000. The minimum number of epochs is set to 500. All the samples are divided into mini-batches for training with a mini-batch size of 20. All DNNs are trained based on an NVIDIA Tesla V100 GPU in this work.x of the final choices can be formulated as:Q1 and Q3 denote the lower quartile operator and the upper quartile operator, respectively. The expected pre-estimates of the retaining DNNs can be treated as the final SOH estimation for the target domain. In this study, we employ root mean square error RMSE, absolute error AE, and its average value MAE to evaluate the SOH estimation:yi and ŷi are the measured value and the final estimate of the SOH for sample i, respectively. M represents the total number of samples of interest.The proposed framework selectively integrates the pre-estimates of the DNNs to generate a reliable SOH estimate. We employ mean and standard deviation to evaluate the pre-estimates of each trained DNN. An efficient metric, quartile, is used to select DNNs according to these measures. The retaining DNNs are the final choices of the proposed framework for each estimate, and the indexes Batteries from datasets #3 and #5 are tested in thermal chambers at 20 and 45 °C, respectively. An ARBIN BT2000 battery test system is employed to cycle the batteries. In each cycle of the battery from Dataset #3, the charge strategy is to charge at a constant-current rate of 0.3 C until the voltage reaches 4.2 V and then hold at 4.2 V until the charging current drops below 0.03 A; the discharge strategy is to discharge at a constant-current rate of 2 C. In each cycle of the battery from Dataset #5, the charge strategy is to charge at a constant-current rate of 1 C until the voltage reaches 3.65 V and then hold at 3.65 V until the charging current drops below 1.35 A; the discharge strategy is to discharge at a constant-current rate of 1 C. The charging curves extracted from the constant-current charging phase of the cycles are integrated into the datasets.Supplementary InformationPeer review file