A Bayesian-Symbolic Approach to Learning and Reasoning for Intuitive Physics

1 INTRODUCTION . Imagine a ball rolling down a ramp . If asked to predict the trajectory of the ball , most of us will find it fairly easy to make a reasonable prediction . Not only that , simply by observing a single trajectory people can make reasonable guesses about the material and weight of the ball and the ramp . It is astonishing that while the exact answers to any of these prediction and reasoning tasks requires an indepth knowledge of Newtonian mechanics and solving of some intricate equations , yet an average human can perform such tasks without any formal training in physics . Even from an early age , humans demonstrate an innate ability to quickly learn and discover the laws of physical interactions with very limited supervision . This allows them to efficiently reason and plan action about commonsense tasks even in absence of complete information ( Spelke , 2000 ; Battaglia et al. , 2013 ) . Recent studies suggest that this ability of efficient physical reasoning with limited supervision is driven by a noisy model of the exact Newtonian dynamics , referred as the intuitive physics engine ( IPE ; Bates et al. , 2015 ; Gerstenberg et al. , 2015 ; Sanborn et al. , 2013 ; Lake et al. , 2017 ; Battaglia et al. , 2013 ) . As sample-efficient physical reasoning is recognized as a core domain of human common-sense knowledge ( Spelke & Kinzler , 2007 ) ; therefore an important problem in artificial intelligence is to develop agents that not only learn faster but also generalize beyond the training data . This has lead to a surge in works aimed at developing agents with an IPE or a model of the environment dynamics ( Amos et al. , 2018 ; Chang et al. , 2016 ; Grzeszczuk & Animator , 1998 ; Fragkiadaki et al. , 2015 ; Watters et al. , 2017 ; Battaglia et al. , 2016 ; Sanchez-Gonzalez et al. , 2019 ; Ehrhardt et al. , 2017 ; Kipf et al. , 2018 ; Seo et al. , 2019 ; Baradel et al. , 2020 ) . Among these , neural-network based learned models of physics ( Breen et al. , 2019 ; Battaglia et al. , 2016 ; Sanchez-Gonzalez et al. , 2019 ) tend to have good predictive accuracy but poor sample efficiency for learning . On the other hand , symbolic models ( Ullman et al. , 2018 ; Smith et al. , 2019 ; Sanborn et al. , 2013 ; Bramley et al. , 2018 ) are sample efficient but fail to adapt or accommodate any deviation from their fixed physics engine . Inspired by humans ’ highly data-efficient ability of learning and reasoning about their environment , we present Bayesian-symbolic physics ( BSP ) , the first fully Bayesian approach to symbolic intuitive physics that , by combining symbolic learning of physical force laws and statistical learning of unobserved properties of objects , enjoys the sample efficiency of symbolic methods with the accuracy and generalization of data-driven learned approaches . In BSP , we pose the evolution of the environment dynamics over time as a generative program of its objects interacting under Newtonian mechanics using forces , as shown in figure 2 . Being a fully Bayesian model , we treat objects and their properties such as mass , charge , etc . as random variables . As force laws are simply functions of these properties under the Newtonian assumption , in BSP we replace data-hungry neural networks ( NN ) with symbolic regression ( SR ) to learn explicit force laws ( in symbolic form ) and then evolve them deterministically using equations of motion . But a naive SR implementation is not enough : a vanilla grammar that does not constrain the search space of the force-laws can potentially have far worse sample efficiency and accuracy than a neural network . Therefore , we also introduce a grammar of Newtonian physics that leverages dimensional analysis to induce a physical unit system over the search space and then imposes physics-based constraints on the production rules , which help prune away any physically meaningless laws , thus drastically speeding up SR. Our main contributions are threefold : • We introduce a fully differentiable , top-down , Bayesian model for physical dynamics and an expectation-maximization ( EM ) based algorithm , which combines Markov chain Monte Carlo ( MCMC ) and SR , for maximum likelihood fitting of the model . • We introduce a grammar of Newtonian physics that appropriately constrains SR to allow data-efficient physics learning . • Through empirical evaluations , we demonstrate that the BSP approach reaches human-like sample efficiency , often just requiring 1 to 5 observations to learn the exact force laws – usually more than 10x fewer than that of the closest neural alternatives . 2 RELATED WORK . At a high level , the logic of physics engines can be decomposed into a dynamics module and a model of how the entities interact with each other depending on their mutual properties . These modules can be further divided into more components depending on how the module is realized . Using this break-down , we can categorize different models of physics based on what components of the model are learned . In figure 1 , we compare some of the recent models of physics that are of closely related to our work . Starting on the right end , we have fully learned , deep neural-network approach used by Breen et al . ( 2019 ) that do not use any prior knowledge about physics and therefore learn to predict dynamics completely in purely data-driven way . In the middle are hybrid models that introduce some prior knowledge about physical interaction or dynamics in their deep network based prection model . These include interaction networks ( INs ; Battaglia et al. , 2016 ) , ODE graph networks ( OGNs ) and Hamiltonian ODE graph networks ( HOGNs ; Sanchez-Gonzalez et al. , 2019 ) . Since these approaches employ deep networks to learn , they tend to have very good predictive accuracy but extremely bad sample efficiency and therefore require orders of magnitude more data to train than humans ( Ullman et al. , 2018 ; Battaglia et al. , 2016 ; Sanchez-Gonzalez et al. , 2019 ) . On the other end of the spectrum ( left ) are the fully symbolic , rule-based physics models and engines ( Smith et al. , 2019 ; Allen et al. , 2019 ; Wu et al. , 2015 ; Ullman et al. , 2018 ) . While these methods are suitable for reasoning tasks , they lack the flexibility of the data-driven , learned models as they can not generalize or adapt to any changes in the environment that their fixed physics engine simulates . For example , in such fixed models , inference can fail on physically implausible scenes and may require additional tricks to resolve such issues ( Smith et al. , 2019 ) . Symbolic regression has been used for general physics learning in many prior works ranging from Schmidt & Lipson ( 2009 ) that used SR to discover force laws from experimental data to the more recent work of Cranmer et al . ( 2020 ) on distilling symbolic forces from INs using genetic algorithms . Even more recently , Udrescu & Tegmark ( 2020 ) proposed an interesting framework AI Feynman , which recursively simplifies the SR problem using dimensional analysis and symmetries discovered by neural networks to discover the underlying physics equation that generated the data . The focus of these prior work has been to discover the underlying physical equations that directly leads to the observed data but unlike our approach , they do not target to allow for reasoning in physical environments , that is common task of interest in intuitive physics studies . 3 BAYESIAN-SYMBOLIC PHYSICS . Our framework , Bayesian-symbolic physics , combines symbolic learning of physical force laws with Bayesian statistical learning of object properties such as mass and charge . The environment is modelled by a probabilistic generative model governed by Newtonian dynamics . Physical laws are learnable symbolic expressions that determine the force exerted on each object , based on the position and properties of other objects . These properties might not be observed , so they are treated as latent variables and learned in a Bayesian fashion . The physical laws themselves have a prior distribution to organize the search space and discourage the model from learning physically meaningless laws ; we call this distribution a grammar of Newtonian physics . To learn with incomplete data , we are inspired by results from Ullman et al . ( 2018 ) , humans are able to simultaneously predict the trajectory and update their inference about the properties of the object under new observation . This motivates an EM-based learning and inference method to fit BSP models . In the E-step , we obtain the distribution of entity properties by sampling from their posterior distribution using the current guess of force laws , and in the M-step , we use the samples from the E-step to perform SR to update the force functions . This enables BSP models to learn and reason in environment with incomplete information . 3.1 GENERATIVE MODEL OF THE ENVIRONMENT . We represent each entity i ∈ { 1 . . . N } by a vector of properties zi , such as mass , charge , coefficient of friction , and shape , some of which may be unobserved . At each time step t , a state vector sit = ( p i t , v i t ) is associated with each entity i , where p i t ∈ Rd and vit ∈ Rd are position and velocity vectors respectively and d is the dimensionality of the environment , typically 2 or 3 . Let { τ i } Ni=1 , where τ i = pi1 : T : = ( p i 1 , . . . , p i T ) , be the set of observed trajectories from an environment with N entities . Together with a prior on z , the generative process of BSP defines a joint probability distribution p ( D , z ; F ) over the observed trajectory data D and latent properties z given the force function F .1 An example of the generative process of a three-body problem is shown in figure 2 . The state transition of an entity in a Newtonian system depends not only on its properties and current state but also on its interaction with other entities in the environment . Therefore , we define a pairwise interaction function F ( zi , si , zj , sj ) , where i , j ∈ { 1 . . . N } ; we interpret F as the force applied to i due to its interaction with j . Then , the trajectory τi of each entity is generated by a transition function T that consumes the current state and all of its interactions as sit+1 = T ( sit , F ( z i , sit , z 1 , s1t ) , . . . , F ( z i , sit , z N , sNt ) ) . ( 1 ) As forces are additive , forces on entity i can be easily summed to get the total force applied as f it = ∑N j=1 F ( z i , sit , z j , sjt ) . Similar to Sanchez-Gonzalez et al . ( 2019 ) , we use numerical integration to simulate the Newtonian dynamics , updating s with the acceleration obtained from f it . 1As physical dynamics are typically sensitive to initial states , we assume the noise-free initial states are given either as part of the data D or as a point-mass prior over the initial state , thus are omitted in the notation . Specifically , we choose the Euler integrator since its update rules correspond to the basic relations between position , velocity and acceleration . With these specifications , equation 1 becomes ait = f i t/m i , vit+1 = v i t + at∆t , p i t+1 = p i t + v i t+1∆t , ( 2 ) where mi is the mass of the recipient of the force f it and ∆t is the step size of the Euler integrator . Finally , we add Gaussian noise to each trajectory { τ i } Ni=1 , that is , D : = { τ̃ i } Ni=1 where τ̃ i : = ( p̃i1 , . . . , p̃ i T ) , p̃ i t ∼ N ( pit , σ2 ) and σ is the noise level . For clarity , Appendix A provides the complete generative process represented by a probabilistic program .

Summary: The paper addresses the problem of sample-efficient inference for symbolic physical rules. In the literature, there exists neural-network based models for learning a physical engine which have good predictive accuracy but poor sample efficiency, as well as symbolic models which are highly sensitive to deviations from their fixed physics engine. To be able to overcome issues as such, authors propose a generative model along with a symbolic regression framework, in which forces are produced from a probabilistic context free grammar that is designed to mimic simple Newtonian physics. This particular grammar is parameterized by a few latent variables related to unobserved properties of the physical environment, such as mass and charge. Finally, they develop an Expectation-Maximization algorithm, in order for estimating these latent variables as well as inferring the underlying physical laws of the system.