article: Humans can easily describe, imagine, and, crucially, predict a wide variety of behaviors of liquids—splashing, squirting, gushing, sloshing, soaking, dripping, draining, trickling, pooling, and pouring—despite tremendous variability in their material and dynamical properties. Here we propose and test a computational model of how people perceive and predict these liquid dynamics, based on coarse approximate simulations of fluids as collections of interacting particles. Our model is analogous to a “game engine in the head”, drawing on techniques for interactive simulations (as in video games) that optimize for efficiency and natural appearance rather than physical accuracy. In two behavioral experiments, we found that the model accurately captured people’s predictions about how liquids flow among complex solid obstacles, and was significantly better than several alternatives based on simple heuristics and deep neural networks. Our model was also able to explain how people’s predictions varied as a function of the liquids’ properties (e. g. , viscosity and stickiness). Together, the model and empirical results extend the recent proposal that human physical scene understanding for the dynamics of rigid, solid objects can be supported by approximate probabilistic simulation, to the more complex and unexplored domain of fluid dynamics.
  We study the judgments people make about complex physical dynamics—liquid flow—in rich, realistically rendered scenes. This is very different from the highly simplified and abstract stimuli studied in some of the classic work on intuitive psychology in the cognitive psychology literature. Also, our focus on people’s ability to make sophisticated quantitative physical predictions in complex everyday scenes [3,7] may appear incongruous next to classic studies of intuitive physics that emphasized ways in which people’s judgments about simple scenarios of objects in motion can be inconsistent with even very basic principles of Newtonian mechanics [12]. In part this is just a difference in emphasis: We are interested in explaining how people make successful predictions in complex everyday interactive settings, capacities that no robot, computer vision or AI system currently comes close to, rather than the mistakes people make in much simpler settings which are important for formal physics education but not necessarily for everyday physical interaction with the world. It is important to note, however, that even in previous work which emphasized the failures of people’s physical intuitions, it was often the case that a majority or plurality of participants’ judgments in many of the classic studies were consistent with Newtonian principles or approximate Newtonian simulations [12–15]; it is just the striking patterns of error which occurred in some cases that were presented as the main findings. With some exceptions [13], the classic literature often focused on textbook-style problems, typically presented with static, illustrated diagrams, and requiring participants to draw out future trajectories on paper. In contrast, recent literature has focused on more perceptual and perceptual-motor judgments, typically with rich dynamic displays and implicit inferences about what will happen next over short time scales (but see also, e. g. [16] for older examples). Recently, [5] even replicated [17]’s results showing that people’s drawings of predicted ballistic motion are often very inaccurate, but found that when those same participants were tasked with an interactive task of intercepting a moving object or releasing it to fall with a desired trajectory, their predictions were consistent with Newtonian dynamics. Most relevant to the present work, [14] found that subjects were better at judging the trajectory of water exiting a hose than a ball exiting a curved tube, hinting toward a crucial difference in physical predictions with fluids when people can access representations they have developed from their everyday interactions. These and other findings [18] are beginning to coalesce on an explanation of why people’s physical intuitions appear to vary greatly in their accuracy: higher-level, deliberate reasoning about physics differs markedly from physical knowledge available to lower-level perceptual and motor systems. We speculate that higher-level systems, which are more flexible, not specific to physical reasoning, and based more on learning, may not benefit from the perceptual conditions and reward incentives that give rise to accurate physical knowledge in lower-level systems. Hence, our underlying hypothesis is that there are multiple systems supporting physical scene understanding, one of which is physical simulation. The purpose of this work is not to explore the boundaries between the different cognitive mechanisms for understanding physical systems, but rather to provide the first examination of mental simulation as an account of people’s ability to predict fluid flow in realistic situations, and to develop computational models of these simulation abilities that can capture to a reasonable first approximation how people might make these predictions. Previous artificial intelligence research on reasoning about fluids has often pursued qualitative [19–23] or logical [20,24] approaches, which are computationally efficient, and were developed prior to more recent hardware and algorithmic advances that support fluid simulation in real-time on mobile devices. Qualitative and logical models have been especially useful in situations that ask for explicit reasoning about a range of physical systems that people do not have much direct perceptual-motor experience with (such as thermodynamic or hydrodynamic systems or electrical networks). More recently, [20] developed a system of reasoning using first-order logic to draw conclusions about transferring liquid from one container to another in theoretical, two-dimensional scenarios. A key advantage of this approach is that it can be applied even when no numerical information about container geometries, relative distances, initial conditions or volume of the water, pouring speed, etc. is available. At the same time, however, it is unclear how these qualitative or logical models could account for the fine-grained quantitative judgments that participants make in the perceptual settings we study here. A final class of classic AI approaches to intuitive physics is analogical simulation, e. g. , [25] devised an analogical particle-based fluid simulator, whose particles “communicate” with each other based on a set of eight rules (see Fig 4C). This approach combines some aspects of qualitative reasoning and quantitative, physics-engine-based simulations, and can be seen as a simpler, more heuristic version of physically based simulations. Most recently, deep neural networks have been proposed as a very different approach to making physical predictions on short time scales [26]. Both analogical simulation and neural network approaches can be applied to our tasks, and we evaluate simple versions of both on our experimental data, in addition to the more quantitative approximate simulation models we have focused on. Proponents of qualitative approaches have also argued that solving problems that involve predicting fluid motion “to a high degree of accuracy involves computational fluid dynamics” and “it is quite unlikely that we are capable of performing such a prodigious feat mentally” [27]. But it is still unclear to what extent qualitative approaches can account for people’s sophisticated capacity for understanding and interacting with a wide variety of fluids and fluid-like materials in everyday settings. Furthermore, there are strong reasons to believe fluid simulation in the brain is not prohibitively expensive. Particle simulations can be run in parallel on GPUs, and as we argue here, the brain may be able to get by with just a coarse approximation for its purposes. Such simulations require computation that is on par (or less) than that used by today’s deep neural networks for computer vision [28], which are thought to be reasonable models of parts of biological visual processing [29]. In addition, [30]’s recent “interaction networks” and [31]’s “neural physics engine” models introduce neural networks which operate on objects and relations, and can learn very accurate quantitative physical simulators from data. In light of this, as well as the mounting evidence that human predictions and inferences about rigid physical objects is supported by simulation, we believe it is plausible that humans perform quantitative simulations, and not justifiable to dismiss this possibility out of hand. Some psychological work in vision has looked at how humans perceive liquid viscosity, and has also been skeptical of the sort of computations we propose. For example, [32,33] argue that perception of liquids relies on superficial cues, and show that people use certain statistical motion cues to predict viscosity. Furthermore, they argue that, “Given that the liquid image motion is a result of…complex physical processes, it seems practically impossible for the visual system to infer the underlying physical movements of particles from observed movements using inverse optics computations. ” We do not disagree that surface appearance cues can be important in how people perceive liquid properties. However, such cues alone cannot directly explain how people can predict the motion of liquids, which is the focus of our experimental work. The remainder of this paper is organized as follows. We first present our simulation model and the particle-based simulation method it uses, followed by several alternative models, including simpler simulation alternatives, as well as deep neural networks. We then present Experiment 1, which demonstrates that participants are coarsely able to predict the complex physics of both scenes involving low-viscosity liquids (e. g. , water) and scenes involving high-viscosity liquids (e. g. , honey), and explores the roles of damping (see Intuitive fluids engine) and uncertainty in our model’s correspondence to participants’ judgments. Experiment 2 then focuses more directly on people’s sensitivity to viscosity. Finally, we discuss implications of both experiments and directions for future work.
  All subjects gave informed consent and were treated according to protocol approved by MIT’s IRB (COUHES 1011004131). For some model parameter settings in Experiment 1, both water and honey models made similar predictions, which limited the model’s ability to assess people’s sensitivity to viscosity and stickiness. In Experiment 2, we designed new stimuli to elicit highly different outcomes between low- and high-particle-friction and stickiness IFE simulations. See Fig 5 for examples of Experiment 2’s more strongly contrastive stimuli.
  We first asked: How accurate are participants’ predictions about flowing liquids? In all analyses we calculated the means across participants’ judgments for each scene, and estimated Pearson correlations between those mean judgments and each model’s predictions as a measure of how well the model fit the human data. In Experiment 1a, the correlations between the mean participant responses and the water and honey ground truth models were r = 0. 76[0. 74,0. 78] and r = 0. 83[0. 81,0. 84], respectively (the interval in brackets is a 95% CI, estimated by a bootstrap analysis with 10,000 resamples [42]). This is somewhat surprising because one would expect water simulations to better account for judgments about water than honey simulations (see Experiment 2). By contrast, Experiment 1b showed that participants’ judgments about water and honey were best-explained by their corresponding ground truth models: r = 0. 48[0. 43,0. 53] for water ground truth and r = 0. 73[0. 69,0. 76] for honey ground truth. Importantly, our results show a significant correlation (two-sample Kolmogorov-Smirnov p <. 0001) between individual participants’ judgments and the ground truth in all our experiments (see Fig 6). The null hypothesis was generated by drawing random samples from a beta distribution fit to participant data, which captured the variance of their judgments but not any stimulus-specific structure. This result shows that participants cannot simply do well by chance. Next we asked: How well are participants’ judgments explained by the IFE models versus alternative models? Overall, the results show that the IFE did well in explaining the data, but we also found that the honey IFEs (and honey ground truth) did moderately well in accounting for responses on water trials. Thus, from Experiment 1 alone, it is not clear to what extent people were sensitive to the different fluid properties. In addition, our alternative models also performed moderately well in some cases, compared to the IFE. Experiment 2 addresses both of these issues. Experiment 1 scenes were not designed to guarantee that honey and water resulted in very different outcomes, and thus responses for water versus honey in many scenes should be similar. In Experiment 2, we design scenes for which water and honey simulations have maximally different outcomes, and therefore, if people’s predictions are sensitive to different liquid properties, we should be able to detect this. Furthermore, the Experiment 2 scenes are a more stringent test of the alternative models, whose physics do not as closely match the ground truths for water or honey as the IFE’s does. Correlations between the data in Experiment 1a and uncertain IFE were r = 0. 92[0. 91,0. 93], r = 0. 76[0. 74,0. 78], and r = 0. 81[0. 79,0. 82] for water, Model 1 honey, and Model 2 honey, respectively. Experiment 1b had correlations of r = 0. 74[0. 71,0. 78], r = 0. 79[0. 76,0. 83], and r = 0. 86[0. 83,0. 88] for water, Model 1 honey, and Model 2 honey, respectively. These correlations are summarized in Fig 7. Fig 8 summarizes how IFE fits vary across different values of ζ and α. The IFE outperformed all alternative models in Experiment 1a, but was slightly outperformed by MarbleSim in Experiment 1b. MarbleSim was also highly competitive in water trials. SimpleSim was somewhat competitive in water trials and highly competitive in honey trials, and the gravity heuristic was competitive on honey trials, but less competitive in water trials. Table 2 summarizes the numerical results of the alternative models, which are also depicted visually in Fig 7C (with the exception of the gravity heuristic, which is shown in Fig 9). Table 2 also summarizes the agreements between alternative models and the ground truth predictions for each liquid. MarbleSim’s best-fitting parameter values for water and honey trials respectively were (N = 25, ζ = 2) and (N = 75, ζ = 6). Both N and ζ were fit independently for water versus honey trials. We will refer to these best-fit parameter values as “water MarbleSim” and “honey MarbleSim”, respectively. SimpleSim’s best-fitting parameter values for water and honey trials respectively were (g = 0. 1, m = 0. 1) and (g = 0. 0, m = 0. 04). Both g and m were fit independently for water versus honey trials. We will refer to these best-fit parameter values as “water SimpleSim” and “honey SimpleSim”, respectively. To implement SimpleSim’s uncertainty over m, which was analogous to the IFE’s and MarbleSim’s uncertainty over ζ, we average model predictions over the range [m − σm, m + σm], where σm/ (mmax − mmin) = σ/ (ζmax − ζmin), and max and min represent the maximum and minimum values explored for those parameters. Thus, since mmin = 0, mmax = 0. 2, ζmin = 0, ζmax = 20, and σ = 4, then σm = 0. 04. Here, we only report results for this value of σm. The ConvNet was initially trained on the Experiment 1a ground truth (“water ConvNet”), and a separate model (“honey ConvNet”) was then made by fine-tuning the original network on a smaller number of honey ground truth simulations, in order to better capture human judgments on Experiment 1b. As seen in Table 2, the water ConvNet successfully learned to approximate ground truth predictions, but its correlation with people in water trials was comparable to ground truth itself. The honey ConvNet was less successful in learning the ground truth, but its correlations with people in Experiment 1b were also comparable to those with the ground truth. Like with the honey ground truth, the honey ConvNet’s predictions were highly correlated with people in Experiment 1a. Experiment 2 further addresses this issue, by establishing whether people actually make different predictions for water and honey. We also found hints that people’s predictions were sensitive to the physical attributes of the liquids, which could be predicted by the IFE. In both Experiment 1a and 1b, participants were at least partially sensitive to the physical attributes of viscosity and stickiness of each liquid: the ground truth model whose particle friction and stickiness corresponded to Experiment 1a was a better fit to Experiment 1a’s participants’ responses than those of Experiment 1b (see Fig 7). Experiment 1b’s participants’ responses were better fit by the ground truth model with Experiment 1b’s physical attributes. However, the fit between Experiment 1a responses and the honey ground truth was quite high, an issue which we address in Experiment 2. In both experiments, including damping in the simulation models improved fits with people’s judgments. Furthermore, the uncertain IFE model had modestly higher correlations in most cases than the deterministic IFE (results not presented here). However, future work should further investigate the role of uncertainty in this kind of task. We also tested our previously presented uncertainty model [37], which assumed veridical physics plus uncertainty about the initial location of the fluid. We found this model to be more consistent with the data than ground truth, but it did not account as well for the data as either the deterministic or uncertain damped IFE models we present here. In Experiment 1a, split-half correlations reveal participants were highly consistent with each other (r = 0. 96[0. 94,0. 97]). Experiment 1b participants were less consistent with each other (r = 0. 81[0. 73,0. 86]) than Experiment 1a, which might be attributable to less familiarity with the liquid, since Experiment 1a participants saw a liquid that behaved similarly to real water, but the liquid in Experiment 1b, while comparable to a highly viscous honey, was less similar to most common liquids. In addition, real honey has highly variable viscosity, so any experience people have with honey is less consistent than with water. Our results show that people’s judgments on trials of a particular fluid type were positively correlated with the appropriate model, and near zero or anti-correlated with the other fluid type (see Fig 7B). Correlations with the IFE model were calculated using the same model parameters from Experiment 1. The correlations for participants on water trials were r = 0. 57[0. 48,0. 64] with the water IFE, r = −0. 10[−0. 20,0. 01] with the Model 1 honey IFE and r = 0. 08[−0. 03,0. 19] with Model 2 honey IFE. On honey trials, correlations were r = 0. 00[−0. 07,0. 08] with the water IFE, r = 0. 81[0. 77,0. 85] with the Model 1 honey IFE, and r = 0. 82[0. 78,0. 86] with Model 2 honey IFE. Correlations with ground truth for water trials were r = 0. 41[0. 35,0. 47] and r = −0. 03[−0. 08,0. 03] for water and honey respectively. Correlations with ground truth for honey trials were r = −0. 69[−0. 71, −0. 68] and r = 0. 69[0. 68,0. 7] for water and Model 1 honey respectively. As shown in Fig 9, the gravity heuristic and SimpleSim accounted just as well for honey trial data as the honey IFE models, but were not as well correlated with water trials. Numerical results for all alternative models are also summarized in Table 3, and visualized in Fig 7D. Parameters for MarbleSim and SimpleSim were fixed to their corresponding values in Experiment 1. Results show that, in contrast to Experiment 1, MarbleSim did poorly on honey trials and SimpleSim did poorly on water trials, compared to the IFE. As in Experiment 1, the ConvNet model was trained initially on water trials, and then fine-tuned on a smaller number of honey trials, resulting in two separate networks. (Note that separate networks needed to be trained for Experiment 2, because there was a cup rather than a divider at the bottom.) The ConvNet results are shown in Table 3. Like Experiment 1, the water ConvNet was quite successful in predicting ground-truth physics, but its correlation with the water trial responses were still lower than the water IFE. By contrast, the honey ConvNet was not as successful in predicting the ground truth on our stimuli. We believe that this is because the scenes we picked as stimuli were not drawn from the same distribution as those in the ConvNet training set. Specifically, the stimuli were drawn from a skewed distribution, which filtered for scenes with highly divergent outcomes for water versus honey, whereas the training set as a whole was not. We believe this problem did not affect the water ConvNet as much, because of differences in behavior between the water and honey. Honey may exhibit different “regimes” of behavior: if only a low proportion of the particles get stuck on an obstacle surface, the simulation may have a very different outcome than if slightly more particles get stuck. It is possible that with a very large training set, the network would be more accurate on our honey stimuli. However, we stopped seeing incremental improvement when the honey ConvNet test set (in both experiments) included more than about 25% of the full set of 10000 training examples. The results of Experiment 2 show that people are sensitive to the physical parameters (both stickiness and viscosity) of each fluid. Furthermore, our data show that people can accommodate their predictions systematically to different fluids with different physical parameters, but they may do so in one of two ways: by adjusting parameters in the simulation corresponding roughly to the physics, like stickiness and viscosity, or by approximating these physical variations using a single simple parameter, damping. The latter is cognitively appealing for its representational simplicity, but needs further investigation as a candidate cognitive model. But while participants had good fits with both honey IFE models on honey trials, the water IFE model’s fit to participants’ responses was significantly lower than in the previous experiment, and responses were less accurate with respect to ground truth. In addition, subjects’ split-half correlations were much lower (r = 0. 72[0. 53,0. 82], as compared to r = 0. 96[0. 94,0. 97] in Experiment 1 water trials, r = 0. 81[0. 73,0. 86] in Experiment 1 honey trials, and r = 0. 87[0. 81,0. 92] in Experiment 2 honey trials). One potential explanation of this performance gap is that the task was more intrinsically difficult, as the cup was less wide than each bin in Experiment 1. That is, Experiment 2 scenes may require more precision in simulation (i. e. more samples) to match the accuracy levels in Experiment 1. As a measure of how much precision was required, we compared IFE prediction variance across tasks. That is, we looked at the deterministic IFE predictions over a range of ζ values to see how much they varied. If there is high variance, it means the model is less “certain” of the outcome, and therefore the task is more difficult. The ranges of ζ examined were the same as those used by the uncertain IFE in the results presented above (i. e. 1 ≤ ζ ≤ 9 for water and Model 1 honey, 7 ≤ ζ ≤ 15 for Model 2 honey). Table 4 shows the mean variance across scenes for each experiment and IFE fluid type. The mean variance for water in Experiment 2 was significantly higher than the mean variances for the other two IFE models in both experiments (unpaired t-test, all p-values near zero), supporting the notion that Experiment 2 water stimuli were intrinsically more difficult than the other three tasks. (Note that the human performance was higher on Experiment 1 (both water and honey trials) and Experiment 2 honey trials, but lower on Experiment 2 water trials, qualitatively matching the pattern of model variances.) However, further work will be necessary to understand why the model had higher agreement with ground truth in Experiment 2 water trials than human responses.
  Here we introduced a computational cognitive model of how humans judge fluid dynamics in complex everyday settings. Our proposed model, the “Intuitive Fluids Engine” (IFE), posits that humans run particle-based mental simulations to predict how a fluid will flow in the world. Our experimental results show that humans are able to make coarse, but often accurate, predictions about fluid dynamics, and that these judgments are better explained by the IFE than several alternative models, including simpler particle-based simulations and models that do not explicitly simulate liquid dynamics. In two experiments, we found that people were able to predict fluid flow in a range of novel environments with multiple obstacles, varied fluid properties, and different task dynamics, appropriately taking into account complex geometric structure and fluid properties such as stickiness and viscosity. Solving these prediction tasks accurately would be very computationally intensive for an agent who has perfect knowledge of the ground truth physics and attempts to model the fluid flow with high precision. Our model explains the difference between this ideal and human performance as the result of several approximations: human mental simulations use only a very limited number of particles to represent the fluid, and only an imperfect approximation to the physics inherent to all particle-based simulators; in addition, particle momentum in mental simulations may be “damped” to varying degrees, especially for more viscous liquids. More generally, the success of this model is consistent with the broader view that many aspects of human intuitive physics can be understood in terms of approximate probabilistic simulation [3], extending this thesis from an account of how people understand and interact with simple rigid-body systems to much more complex fluid systems. A crucial feature of our IFE model is how coarsely it approximates the true physics of fluids. Though our particle-based model is in the same family as SPH methods used for approximate simulations in physics, engineering, computer graphics and video games, its details and implementation differ importantly from those used in other disciplines. The rules that govern the particle-particle and particle-rigid interactions, the parameters of the simulated materials, and the granularity of the mass density approximation are highly simplified relative to those used in typical engineering applications, in ways that render the simulation non-veridical even though it still looks reasonably natural. In particular, very small numbers of particles, by the standards of typical SPH methods (e. g. , 50 in our model, versus the hundreds of thousands or millions of particles typical in graphics and engineering, or the thousands typical in video games), gave the best fit to people’s judgments. And when simulating viscous fluids, a variant of our model that lumps various “stickiness” parameters into a single damping parameter fits human judgments well compared to the variant that keeps them separate. Though it requires further exploration, representational simplifications like these may point toward ways that human minds simulate complex real-world physics with cognitively plausible and efficient computations. The only exception to the general pattern of agreement between physical simulation and people’s judgments occurred in the “Water” condition of Experiment 2, where a minority of participants had no correlation with ground truth. Our analyses showed that trials in this condition were objectively more difficult for a simulation-based approach than those in other conditions, but it is also possible that there are interesting qualitative differences between these trials to be explored further. Future work should more systematically investigate people’s patterns of success and failures in predicting qualitatively different kinds of fluid motion, and we expect that this will yield more insights into the nature of the computations and representations that people use in predicting fluid flow. A common critique of quantitative simulation-based cognitive models of intuitive physics [27,43] is they are impractical or unrealistic, due to the computation costs and the detailed physical knowledge implicitly embedded in these simulation algorithms. The simulation-based model we propose here should alleviate some of these concerns: Even extremely simplified simulations, with very coarse physical approximations, can nonetheless capture the fluid dynamics well enough for everyday judgment purposes. At the same time, these models can capture to a reasonable extent the surprisingly accurate quantitative predictions of fluid flow that people are able to make. The alternative models we presented aimed to provide a benchmark, against which to compare the performance of our IFE results. We explored simpler, particle-based simulation models, and found that they captured certain aspects of people’s judgments, but no single model could account as fully for people’s predictions as the IFE. Nonetheless, simpler simulation models can do reasonably well here, which suggests that the brain may implement something not quite as sophisticated as our IFE. Taken together, the success of our IFE and alternative simulation models provides strong evidence for relatively sophisticated mental simulation in people’s heads, but point toward the need to further explore simpler simulation alternatives that are still sophisticated enough to account for the wide range of accurate predictions that people can make. Our SimpleSim model performed especially well on honey trials in both experiments, achieving correlations similar to those of the IFE. A possible explanation for this success is that honey is an easier fluid to simulate heuristically. Supporting this notion, SimpleSim in Experiment 2 had a maximum correlation with the honey ground truth of r = 0. 80 (occurring at g = 0. 1, m = 0. 06), but with the water ground truth, the maximum correlation was just r = 0. 20 (occurring at g = 0. 0, m = 0. 2). (Recall that Experiment 2 scenes were specifically designed to elicit different outcomes for water versus honey, and are thus the best test of whether any parameter setting of SimpleSim could approximate water, as compared to honey.) This result makes intuitive sense, since both real honey and particles in SimpleSim tend to fall straight down near obstacle edges, by contrast to water, which splashes energetically, in different and less predictable directions. We believe that the moderate success of the heuristic simulation model can be attributed in part to the above: it is intrinsically well-suited for simulating sticky, viscous liquids falling under gravity. SimpleSim’s relative success in Experiment 1 water trials can largely be attributed to its close agreement with honey ground truth physics, since honey ground truth and honey IFE also had high correlations with those trials. Thus, the real test of whether SimpleSim is a good model for water trials is Experiment 2, where it does poorly relative to water IFE, water ground truth, and water MarbleSim. We also explored a non-simulation, neural network model that learned a fast mapping between images of the starting configurations and the numerical outcomes. First, our neural network model shows that it is possible for a connectionist architecture to learn an accurate mapping from a still image representing initial conditions to a single number representing the simulated outcome for our task. Since our networks were trained on ground truth physics, they learned close approximations to the ground truth. It is perhaps unsurprising, then, that they had similar correlations with human data compared to the ground truth simulations. If people learn a similar kind of fast mapping between starting conditions and final outcomes, our results suggest there may be other biases or constraints involved, which we have not captured here. (We also tested whether any intermediate points in training, prior to convergence, resulted in higher correlations with subjects, and found this not to be the case). Previous AI models for explicit reasoning about fluids (e. g. [20–23,25]) have been studied in fundamentally different contexts than those we examined here. For example, [21] and [22] assume laminar flow that is highly constrained by container geometry (e. g. a piston cylinder). These models were designed to answer binary questions (Is the fluid flowing up or down? Is pressure increasing or decreasing?), rather than to predict the location of liquid during or after a splash. It is not clear how such qualitative or logical modeling approaches could be adapted to the present experiment, which involves fluids in rapid dynamical flow, or whether they could capture our participants’ quantitative judgments, but this is an important topic for future work. Perhaps the most similar previous model is [25], which could be seen as a simpler particle-based approach and in part motivated the gravity heuristic and SimpleSim models here. Their approach may be adaptable to the present experiments, but it is not immediately clear how to modify it such that it could distinguish between different liquids, such as water and honey. Future work should explore how the precision, temporal duration, and other structural and parametric features of mental simulation for fluids might be implemented. For example, how does the precision vary as a function of how far into the future one must mentally simulate? [44] showed that for stability judgments, where the question is how far a fluid-containing vessel can be tilted before the fluid spills out and only one-time step predictions are needed (in contrast to our judgments here which required looking ahead many time-steps), IFE models can be extremely quantitatively accurate as accounts of people’s judgments. How closely do the attributes represented by the mind correspond to the actual physical characteristics of a liquid? Physically accurate simulation models can account for people’s judgments, but we also showed that a simpler simulation model (SimpleSim) also did reasonably well, even if the more physically accurate simulation models did better overall. This raises the question as to whether other simple simulation models may do better than SPH in accounting for people’s judgments. In order to address this question, it is crucial to design good experiments that can distinguish between different simulation models. For example, a simple extension to the current work could be to find scenes that distinguish as sharply as possible between the predictions of MarbleSim or SimpleSim and those of the IFE. In addition, using more dynamic and interactive scenes, as well as varying how far ahead one must mentally simulate may be useful in distinguishing between potential models. Can our general approach extend to capture other classes of physical intuitions that go beyond rigid-body dynamics? Particle-based models can provide reasonable simulations for liquids and gases, as well as collections of solid elements (e. g. , piles of sand) and composite materials (e. g. , mashed potatoes or Play-Doh) whose dynamics share similarities with liquids. In fact, work by [45] provides preliminary evidence that particle-based simulation models can capture people’s judgments about sand in a very similar task to the one we present here. What are the limits of simulation-based models–what kinds of non-rigid dynamics can people make coherent predictions about, and might some be better explained by alternative approaches such as qualitative reasoning? Do people begin to use some form of qualitative reasoning as simulation becomes too computationally demanding or in situations where they have insufficient prior experience? There is evidence that internal forward models in the motor system are involved in predicting dynamics that cannot be reenacted by the body [46], which might help explain how mental simulations are implemented. Establishing connections between such work and our approximate simulation framework could lead to a deeper understanding of people’s capabilities and limitations when simulating physics. Another important question is: How could intuitive physics engines be represented in the brain? [47] report fMRI evidence for physics-specific processing in an array of cortical regions related to the “multiple demand” system. But no models yet have connected the computational models to their neural substrates. Inspired by the recent successes of deep learning approaches in AI, a plausible candidate might be a recurrent neural network with dynamic, parallel, and distributed structure [48]. To date, neural network models have only been able to capture simple and restricted classes of rigid-body dynamics [26,49], but as work on multimodal image annotation, linguistic modeling, and more general purpose distributed algorithm learning systems (e. g. , LSTMs [50,51], the “Neural Turing Machine” [52]) advances, more richly structured scenes and physical dynamics might become accessible to these frameworks. A more general question for development, and computational cognitive psychologists is: Where does people’s knowledge of liquids come from? Five month-olds can distinguish between solids and liquids in novel contexts after observing their distinct patterns of movement [53,54], which suggests either a very data-efficient experience-based learning process or innate biases. Perhaps our simulation-based model’s core components can be measured in young children, or its more complex features can be observed as they emerge and mature. Approximate simulation is a powerful framework that may explain how people understand a wide variety of complex physical processes they encounter in everyday life, such as rigids, non-rigids, and fluids. This work offers the first computational model of how people make intuitive predictions about fluids, and provides evidence that a particle-based simulation model can account for human predictions about liquids with different substance attributes.
summary: Although most people struggle to learn physics in school, every human brain is a remarkable “intuitive physicist” when it comes to the quick, unconscious judgments we make in interacting with the world. Almost effortlessly, and with surprisingly high quantitative accuracy, we can judge when a plate placed near the edge of a table might be at risk of falling, or how far a partially filled glass of water can be tipped before the water is in danger of spilling. What kinds of computations in the brain support these abilities? We suggest an answer based on probabilistic inference operating over particle-based simulations, the same class of approximation methods used in video games to simulate convincing real-time interactions between objects in a virtual environment. This hypothesis can potentially account for people’s quantitative, graded judgments in diverse and novel situations including a wide array of materials and physical properties, without positing a large number of separate systems or heuristics. Here, we build on previous evidence that a system of approximate probabilistic simulation supports judgments about rigid objects (e. g. judging the stability of towers of blocks, as in the game Jenga), and ask whether people can also make systematic and accurate predictions about flowing and splashing liquids, such as water or honey. We show that it is possible to capture people’s quantitative predictions using a computational model that approximates the true underlying fluid dynamics to varying degrees of coarseness, and find that people’s responses are most consistent with a very coarse approximation; while typical engineering applications might use tens or hundreds of thousands of particles to simulate a fluid, the brain might get by with roughly a hundred particles. Furthermore, we find that people consistently underestimate the potential energy of a splashing liquid in our virtual scenes, and that our model captures this behavior.
keywords: medicine and health sciences
  classical mechanics
  fluid mechanics
  engineering and technology
  liquids
  diet
  animal products
  simulation and modeling
  engines
  viscosity
  nutrition
  materials science
  materials physics
  research and analysis methods
  chemical properties
  physical chemistry
  food
  honey
  chemistry
  fluid dynamics
  continuum mechanics
  agriculture
  fluid flow
  physics
  mechanical engineering
  fluids
  states of matter
  biology and life sciences
  physical sciences
title: Modeling human intuitions about liquid flow with particle-based simulation