This research presents a deep learning based approach to predict stress fields in the solid material elastic deformation using convolutional neural networks (CNN). Two different architectures are proposed to solve the problem. One is Feature Representation embedded Convolutional Neural Network (FR-CNN) with a single input channel, and the other is Squeeze-and-Excitation Residual network modules embedded Fully Convolutional Neural network (SE-Res-FCN) with multiple input channels. Both the tow architectures are stable and converged reliably in training and testing on GPUs. Accuracy analysis shows that SE-Res-FCN has a significantly smaller mean squared error (MSE) and mean absolute error (MAE) than FR-CNN. Mean relative error (MRE) of the SE-Res-FCN model is about 0.25 with respect to the average ground truth. The validation results indicate that the SE-Res-FCN model can accurately predict the stress field. For stress field prediction, the hierarchical architecture becomes deeper within certain limits, and then its prediction becomes more accurate. Fully trained deep learning models have higher computational efficiency over conventional FEM models, so they have great foreground and potential in structural design and topology optimization.
The Artificial Neural Networks (ANN) with the multilayer perceptron (MLP) has been applied in computational mechanics for several years, for structural analysis @cite_23 @cite_4 @cite_33 , materials failure and damage @cite_2 @cite_28 @cite_1 @cite_32 , regression of the material constitutive properties @cite_2 @cite_34 @cite_15 @cite_11 @cite_16 , computational mechanics enhancement @cite_38 @cite_29 , topological design and optimization @cite_36 @cite_24 . However, due to the intrinsic limitation of MLP, the geometry is usually abstracted and simplified into a feature vector that loses the spatial position relationship, making the approach difficult to generalize to complex cases.
@cite_38 @cite_4 @cite_33 @cite_28 @cite_36 @cite_29 @cite_1 @cite_32 @cite_16 @cite_24 @cite_23 @cite_2 @cite_15 @cite_34 @cite_11
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This paper introduces a methodology for neural network global optimization. The aim is the simultaneous optimization of multilayer perceptron (MLP) network weights and architectures, in order to generate topologies with few connections and high classification performance for any data sets. The approach combines the advantages of simulated annealing, tabu search and the backpropagation training algorithm in order to generate an automatic process for producing networks with high classification performance and low complexity. Experimental results obtained with four classification problems and one prediction problem has shown to be better than those obtained by the most commonly used optimization techniques Multilayer perceptrons (MLPs) or neural networks are popular models used for nonlinear regression and classification tasks. As regressors, MLPs model the conditional distribution of the predictor variables Y given the input variables X. However, this predictive distribution is assumed to be unimodal (e.g. Gaussian). For tasks involving structured prediction, the conditional distribution should be multi-modal, resulting in one-to-many mappings. By using stochastic hidden variables rather than deterministic ones, Sigmoid Belief Nets (SBNs) can induce a rich multimodal distribution in the output space. However, previously proposed learning algorithms for SBNs are not efficient and unsuitable for modeling real-valued data. In this paper, we propose a stochastic feedforward network with hidden layers composed of both deterministic and stochastic variables. A new Generalized EM training procedure using importance sampling allows us to efficiently learn complicated conditional distributions. Our model achieves superior performance on synthetic and facial expressions datasets compared to conditional Restricted Boltzmann Machines and Mixture Density Networks. In addition, the latent features of our model improves classification and can learn to generate colorful textures of objects. We investigate the approximation ability of a multilayer perceptron (MLP) network when it is extended to the complex domain. The main challenge for processing complex data with neural networks has been the lack of bounded and analytic complex nonlinear activation functions in the complex domain, as stated by Liouville's theorem. To avoid the conflict between the boundedness and the analyticity of a nonlinear complex function in the complex domain, a number of ad hoc MLPs that include using two real-valued MLPs, one processing the real part and the other processing the imaginary part, have been traditionally employed. However, since nonanalytic functions do not meet the Cauchy-Riemann conditions, they render themselves into degenerative backpropagation algorithms that compromise the efficiency of nonlinear approximation and learning in the complex vector field. A number of elementary transcendental functions (ETFs) derivable from the entire exponential function ez that are analytic are defined as fully complex activation functions and are shown to provide a parsimonious structure for processing data in the complex domain and address most of the shortcomings of the traditional approach. The introduction of ETFs, however, raises a new question in the approximation capability of this fully complex MLP. In this letter, three proofs of the approximation capability of the fully complex MLP are provided based on the characteristics of singularity among ETFs. First, the fully complex MLPs with continuous ETFs over a compact set in the complex vector field are shown to be the universal approximator of any continuous complex mappings. The complex universal approximation theorem extends to bounded measurable ETFs possessing a removable singularity. Finally, it is shown that the output of complex MLPs using ETFs with isolated and essential singularities uniformly converges to any nonlinear mapping in the deleted annulus of singularity nearest to the origin. Multilayer perceptrons (MLPs) with weight values restricted to powers of two or sums of powers of two are introduced. In a digital implementation, these neural networks do not need multipliers but only shift registers when computing in forward mode, thus saving chip area and computation time. A learning procedure, based on backpropagation, is presented for such neural networks. This learning procedure requires full real arithmetic and therefore must be performed offline. Some test cases are presented, concerning MLPs with hidden layers of different sizes, on pattern recognition problems. Such tests demonstrate the validity and the generalization capability of the method and give some insight into the behavior of the learning algorithm. >