A Bayesian-Symbolic Approach to Reasoning and Learning in Intuitive Physics
Authors: Kai Xu
Akash Srivastava
Dan Gutfreund
Felix Anthony Sosa
Tomer Ullman
Joshua B. Tenenbaum
Charles Sutton
Venue: NeurIPS 2021 Poster (2021)
Keywords: intuitive physics
physics learning
Bayesian inference
symbolic regression
data efficiency

Abstract: Humans can reason about intuitive physics in fully or partially observed environments even after being exposed to a very limited set of observations. This sample-efficient intuitive physical reasoning is considered a core domain of human common sense knowledge. One hypothesis to explain this remarkable capacity, posits that humans quickly learn approximations to the laws of physics that govern the dynamics of the environment. In this paper, we propose a Bayesian-symbolic framework (BSP) for physical reasoning and learning that is close to human-level sample-efficiency and accuracy. In BSP, the environment is represented by a top-down generative model of entities, which are assumed to interact with each other under unknown force laws over their latent and observed properties. BSP models each of these entities as random variables, and uses Bayesian inference to estimate their unknown properties. For learning the unknown forces, BSP leverages symbolic regression on a novel grammar of Newtonian physics in a bilevel optimization setup. These inference and regression steps are performed in an iterative manner using expectation-maximization, allowing BSP to simultaneously learn force laws while maintaining uncertainty over entity properties. We show that BSP is more sample-efficient compared to neural alternatives on controlled synthetic datasets, demonstrate BSP's applicability to real-world common sense scenes and study BSP's performance on tasks previously used to study human physical reasoning.

A Bayesian-Symbolic Approach to Reasoning and Learning in Intuitive Physics
Kai Xu University of Edinburgh contact@xuk. ai
Akash Srivastava MIT-IBM Watson Al Lab akash. sraastssivvstt com
Dan Gutfreund
Felix A. Sosa
Tomer Ullman
MIT-IBM Watson Al Lab dgutfre@us. imm. com
Harvard University
Harvard University
fsosašfas. harvard. edu
tomerul 1aaerrmammmm@mmmammmmmmmm com
Joshua B. Tenenbaum
Charles Sutton
Massachusetts Institute of Technology abt@mit.edh
University of Edinburgh & Google AI c. w...tntede.d.o ac uk
Abstract
Humans can reason about intuitive physics in fully or partially observed envi- ronments even after being exposed to very limited set of observations. This sample-efficient intuitive physical reasoning is considered a core domain of human common sense knowledge. One hypothesis explain this remarkable capacity, posits that humans quickly learn approximations to the laws of physics that govern the dynamics of the environment. In this paper, we propose Bayesian-symbolla framework (BSP) for physical reasoning and learning that is close to human-level sample-efficinncy and accuracy. In BSP, the environment is represented by top- down generative model of entities, which are assumed to interact with each other under unknown force laws over their latent and observed properties. BSP models each of these entities as random variables, and uses Bayesian inference to estimate their unknown properties. For learning the unknown forces, BSP leverages sym- bolic regression on novel grammar of Newtonian physics in a bilevel optimization setup. These inference and regression steps are performed in an iterative manner using expectation maximmzatiinn..nn.....mmmmmammo allowing BSP to simultaneously learn force laws while maintaining uncertainty over entity properties. We show that BSP is more sample-efficient compared to neural alternatives on controlled synthetic datasets, demonstrate BSP's applicability to real-world common sense scenes and study BSP's performance on tasks previously used to study human physical reasoning!
Introduction
Imagine ball rolling down ramp. If asked predict the trajectory of the ball, most of will find it fairly easy to make a reasonable prediction. Not only that, simply by observing a single trajectory people 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 in-depth knowledge of Newtonian mechanics and solving of some intricate equations, yet average human can perform such tasks without any formal training in physics. Studies suggest that from early age humans come understand physical interactions with very limited supervision, and can
Source code as well training and testing data can accessed httpss:/bsppp.... xuk. ai/
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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efficiently reason and plan actions in common sense tasks, even in absence of complete information (Spelke) 2000 Battagha et al., 2013). For example, with limited data, 4 or years old children are capable of learning the physical laws behind magnetism (Bonawitz et al., 2019). Physical reasoning considered a core domain of human common-sense knowledge (Spelke & Kinzler 2007). Recent studies suggest that the ability to efficiently learn physical properties and interactions with limited supervision is driven by a noisy model of Newtonian dynamics, referred to 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)) This has led to a surge in research aimed at developing agents with an IPE, or Grzeszezuk & Animator
model of the environment dynamics (Amos al. 2018 Chang al. 2016
1998
Fragkiadaki et al. 2015 Battagla et al. 2016; Watters et al. 2017]
Sanchez-
et al.
2019 Ehrhardt et al. [ Kipf al. 201 8 Seo et al.) 2019 Baradel et al.] [2020). These efforts
have created methods that either trade-off data-efficiency, by using deep neural networks (NNs).
for high predictive accuracy (Breen et al. 2019 Battaglia et al. [0066] (Sanchez-Gonzalez et al. 2019)
or trade-off flexibility to learn from data for data-efiiiency by using symbolic methods (Ulmman
et al. 2018 (mith et al. 2019 Sanborn et al. 2013 Bramley al. 2018).
Inspired by the highly data-efficient ability of humans to learn and reason about their physical environment with incomplete information, we present Bayesian-symbolic physics (BSP), Bayesian-symbolic model with an pnpectation-maximizaan the sample efficiency of symbolic methods with the accuracy and generalization of data-driven approaches, using statistical inference of unobserved object properties and symbolic learning of physical force laws. In BSP, we model the evolution of the environment's 's dynamics over time as a generative program of entities interacting under Newtonian mechanics. As a probabilistic method, BSP treats the properties of entities, such as mass and charge, as random variables. Since Newtonian force laws are functions of these properties, in BSP we replace data-hungry NNs with symbolic regression (SR) learn explicit force expressions, and then evolve them deterministically using equations of motion. A naive SR implementation here is not enough though due to two issues. One is that if operates on a vanilla grammar that does not constrain the search space over force-laws, it can potentially have worse data data-efficiency than a NN. Therefore, we introduce a grammar of Newtonian physics that leverages dimensional analysis to induce physical unit system over the search space and impose physics- bayscce-based constraints on the production rules. This prunes physically meaningless laws, therefore, drastically speeding up SR. Another issue that the symbolic force expressions usually contain global constants, e.g. the gravitational constant, to learn, and common ways to deal with this challenge turn out to be inefficient especially in an EM setup. We tackle this challenge by using SR in a bilevel optimization framework in which a lower-level gradient based optimization step is used to optimize the constant. In short, our three main contributions are:
(EM) algorithm that combines
We introduce a combines approximate inference methods and SR, for maximum likelihood learning.
model for physical dynamics and an EM based algorithm, which
We introduce a grammar of Newtonian physics that appropriately constrains SR for data-efficient learning, based on priors from dimensional analysis and physics-based constraints.
Empirically, we show that BSP reaches human-like data-efficiency, often requiring just 1 to 5 synthetic scenes learn the underlying force laws much more data efficient than the closest neural alternatives. We then illustrate how BSP can discover physical laws from real-world common sense scenes from Wu al. (2016). Finally, we study BSP on tasks previously used to study human phys- ical reasoning in Ullman et al. (2018) and discuss the similarity and differences with human results.
2
Related work
Many symbolic and data driven models of learning and reasoning about physics can be broken down into smaller components that are either learned or fixed. In figure[] we compare some of the closely-related recent work on physics learning. Starting on the right end, we have fully learned, deep NN approaches such as that used by Breen et al. (2019). This approach does not use any prior knowledge about physics, and learns to predict dynamics in purely data-driven way. In the middle are hybrid models that introduce some prior knowledge about physical interactions or dynamics, in their NN-based prediction models. These include interaction networks (INs: Battaglia et al.. 2016). ODE graph networks (OGNs), and Hamiltonian ODE graph networks (HOGNs; Sanche7- et al. 2019). Since these middle approaches use deep NNs, they tend to have very good predictive accuracy. yet poor sample complexity, requiring orders of magnitude more data train than humans
2

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More adaptive but less data-cfficient
Fixed Symbolic Force Laws
Learnable Symbolic Force Laws
Neurally Learned Interaction
Neurally Learned
Legend
Interaction
Fixed
Deep Neural Networks
Physics model
Physics
Summation
Summation
Aggregation
Aggregation
Interaction model
Engine
How each entity
Newtonian Dynamics
Newtonian
Hamiltonian Dynamics
Neurally Learned
receives interactions
Dynamics
Dynumics model
Dynamics
Figure 1: From left to right are rule-based to purely data-driven models of physics. Examples for
each column are (1) |Smith et al. (2019), (2) Ullman al. (2018), (3) BSP (Ours). (4) (H)OGN
Sanchez-Gonzalee et al. (2019), (5) IN Battagla et al. (2016 and (6) Breen et al. (2019).
(Ullman et al. 2018 Battaglia al. 2016 Sanchez-Gonzalez al. 2019). On the other end of the spectrum (left) are fully symbolic, based physics models and engines (Smith et al. 2019: Allen et al.1,20191 Wu al., 2013 Ullman al., (018). While these methods are suitable for reasoning tasks, they lack the flexibility of data-driven, learned models as they cannot generalize or adapt to changes in the environment that their fixed physics engine cannot simulate. For example, inference can fail on physically implausible scenes, and may require additional workarounds such as 'low probability events' outside of the dynamics (Smith al. 2019).
Symbolic regression has been used for general physics learning in prior research, ranging from Schmidt & Lipson 2209)'ss 's work on discovering force laws from experimental data, to the more recent work of Cranmer et al. (020) on distilling symbolic forces from INs using genetic algorithms. More recently, [drresuu & Tegmark (2020) proposed AI Feynman, which recursively simplifies the SR problem using dimensional analysis and symmetries inferred by neural networks, to discover the underlying physies equations of the data. The focus of these types of work has been discover underlying laws and equations based on direct input-output data. The focus of BSP, on the other hand, on physics learning based on indirect signals from the environment; this is task of interest in both intuitive physics studies with humans, and for human-like AI. Further, most symbolic approaches learn physics while assuming all properties in system are known, which renders them inapplicable to environments with incomplete information. Some neural approaches focus on addressing such limitations in end-to-end fashion (Zheng al. 2018 Veerapaneni al. 2020 Janner al. 2019)
Bayesian-symbolic physics
BSP represents the physical environment using generative model that evolves under Newtonian dynamics (section 3.T. In this model, physical laws are treated as learnable symbolic expressions and learned using symbolic regression and specialized grammar of Newtonian physics that confines the search space and prevent the model from learning physically meaningless laws (section (3.2). BSP does not require all properties of the entities to be fully observed. It models these properties as latent random variables and infers them using Bayesian learning. To fit BSP on data, with incomplete information, we propose EM algorithm that iterates between Bayesian inference of the latent properties, and SR which gives maximum likelihood estimation of the force expressions (section 3.3
3.1 Generative model of the environment
In BSP's generative model, we represent each entity E {1...N) by a vector of intrinsic physical properties z' (such as mass, charge, and shape), and time dependent state vector s (pi v\) which describes the evolution of its position p E Rd and velocity vž E Rd under Newtonian dynamics. Here, refer to the dimensionality of the environment, and is typically 2 or Let (7*) be set of observed trajectories from an environment with N entities, where 7' Pi.r (pi. p Then, together with the prior on 2, for an observed trajectory data, D, the generative model BSP defines joint probability distribution p(D, 2; F)) over D and latent properties z, given the force
3

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mi + Mjj mi Pj. Dimensional
my+ my. Paj
m m2 Pij my jg,, my Mgj
Translation invariance
my x Hg. my 7 Mg. analysis
my x Hg. m4 + Mj-
||p: lp: - Pylla
[pP:l22 llp. - Pyll2
||p/ife llp; Pylla
Figure 2: Illustration of how the dimensional analysis and translation invariance priors help constrain the search space. Each box contains subset of valid and illegal (stroked) sub-expressions.
function F The state transition of an entity in a Newtonian system depends on its properties and current state as well as its interaction with other entities. So, in BSP the force on entity i at time 23, I is defined as fj Σ"= F(2', = sy), where F(=*, st, z², s() is the interaction force between entities i and j. Then, the trajectory т; of entity i is generated by transition function T that consumes the current state and the resultant force to compute 1/41 = T (s;, fi). Similar to Sanchez-Gonzales al. (2019), we use numerical integration to simulate the Newtonian dynamics inside T. Specifically, we choose the Euler integrator and expand T as
a = ff//m', V²+1 = v + at P = =P; + V At,
(1)
where m² is the mass of the recipient of the force ff and At is the step size of the Euler integrator.
Finally, we add Gaussian noise to each trajectory {T'} that is, D [7') where F* := (Pi..... P4), D ~ N (ppi, o²) and a is the noise level. See appendix A. for the details of the complete generative process and illustrative examples.
3.2 A grammar of Newtonian physics
In order to attain good data efficiency, we choose to learn the pairwise force F(z',s', 20 s between entities i and j using symbolic search. This approach can be inefficient if the search space of possible functions is too large, or inaccurate if the search space is too small. So, we constrain the function I to be a member of a context-free language with grammar G, which we call the grammar of Newtonian physics. We design the grammar to be expressive enough to represent a large variety of potential force laws, while incorporating some simple, general constraints to improve the efficiency of symbolic search. Here we describe G informally; for the formal description, see figure (appendix (A.2).
We consider the following terminal nodes in G: the masses m,, m,, of the entities, their friction coefficients p4: Myj, shapes 8i, 8j. positions pi, Pj. velocities v., V2 the contact point i.e. the position (if any) which they touch, and finally a set of K learnable constants {ck ] In cases of no contact, is set as the middle position of the two objects, i.e. (square). +, x, + 11-112 (1.2-norm), normalise() 1) and protect (-, +). which projects a vector onto the unit ball Fl The grammar also allows forces to be conditioned on Boolean expression, in order to support conditional forces that only apply when a condition is true, c.g., when two objects collide. We provide & two primitive functions that encode the output of the perception system: doesCollide for collision detection and isOn to check if an entity is on a surface. These functions output integers 0 or 1. A rule the grammar then allows a force expression to be multiplied by conditional, so that BSP can learn expressions that represent when a conditional force should be applied.
(p: Pb) /2. We include the operators: (.)²
Naively supporting all possible expressions for any combination of terminals would make SR highly inefficient, and even lead to physically impossible force laws. Therefore, we introduce two simple and general types of prior knowledge inspired by physies: dimensional analysis and translation invariance. Figure 2 shows examples of expressions that are excluded by each. First, inspired by dimensional analysis in natural sciences, where the relations between different units of measurement are tracked (Brescia 2012), we built the concept of units of measurement into the nonterminals of G. The units we consider are kilogram (Kg) for mass, meter (Meter) for distance, and meter per second ( (MeterSec) for speed. With this unit system in place, we only allow addition and subtraction of symbols with the same units, avoiding physically impossible aab-expressions such as Kg Meter
2 As physical dynamics are typically sensitive initial states, we assume the noise-free initial states are given either as part the data or can be accurately estimated in pre-procee pre-processing step, e.g. by using consecutive positions, thus are omitted in the notation.
In our work consider maximally three forces to be learn in the same time, thus setting K = 3; more learnable constants and/or entity properties can be easily added to the grammar if needed.
'Constraining the grammar using units shares a similar spirit to type inference in type- synthesis Feser et al. 2015 Osera & Zdancewic 2015). which erucial improve search by avoiding illegal programs.

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Importantly, this can lead to force laws with unit Newton (N) Second, the grammar ensures that all force laws are consslationinvariinn that is, independent on the choice of the origin of the reference frame. To do this, the grammar forbids the direct use of absolute positions Pi. Pj and and only allows their differences to be used expressions!"
Finally, some care is needed to ensure the grammar is unambiguous. For example, if we used rule like Coeff + Coeff x Coeff then the grammar could generate many expressions that redundantly represent the same function. This would make search much more expensive. Instead, rewrite this rule in an equivalent right-branching way, e.g. Coeff + BuseCoeff x Coeff'- This significantly reduces the search space without changing the expressivity of the grammar. Overall, although the grammar puts basic constraints on plausible physical laws, it is still expressive: there are more than 7 million possible trees up to depth while even the expression of universal gravitation has depth of 7; the number of expressions up to depth 7 in G is intractable to count.
3.3 Learning algorithm
Following the EM approach, our learning method alternates between an E-step, where object property distributions are estimated given the current forces via approximate inference, and an M-step step, where forces are learned given object property distributions via SR (section D. For the E-step, we consider two standard inference options: importance sampling (for any prior) and Hamiltonian Monte Carlo (for continuous priors only). Appendix [A.3.] discusses some details on applying them in BSP. Note appendix A.3] also provides all pseudo-code code for algorithms introduced in this section.
Implementation In our work, we implement the generative models as probabilistic programs using the Turing probabilistic programming language (Ge et al. 2018) in Julia. As such the E-step is simply done by Turing's 's built-in samplers. For the M-step, we use the cross-entropy implementation from the ExprOptimization.] package which allows users to define grammars with intuitive syntax.
3.3.1 Symbolic regression with bilevel optimization for learnable constants
Symbolic regression (SR) function approximator that searches over a space of mathematical expressions defined by a context free grammar (CGF) (Koza) 1994). In our work, we use the cross-entropy method for SR. The method starts with a Probabilistic context-free grammars (PCFG) that assumes a uniform distribution over the production rules (PCFGs extend CFGs by assigning each production rule a probability). At each successive iteration, samples n trees (up to depth d) from the current PCFG, evaluates their fitness by loss function L, and uses the top-k trees to fit a PCFG via maximum likelihood for the next iteration. This process returns the learned force law at the end of the training. More formally, to learn force laws, we need to find an expression E L(G), where L(g) is the language generated by G. and values for the learnable constants c == { J that define the force function fe.c. The loss used by the cross-entropy method involves computing the log- log-likelihood of the generative model. As the observed trajectory is generated sequentially given an initial state, the computation of the likeliiod term cannot be parallelized. and can be computationally expensive in practice. Therefore, following Battaglia et al. (2016) and [Sanchez-Gonzeze vectorized version of the log- likelihood that basically performs simulation in each time stamp in - (og.((Pi41 T (st, f.fff...),,))) where T is expanded following equation and Si == (pi, vi). Clearly, LL differs from its sequential corresponding, as the input for the integrator contains noise at each step. However, similar to previous work, we found it is not an issue when learning forces by regression.
et al. (201)), we use
parallel LL(e, G 2, D) =
In order to prevent overfitting by finding over-complex expressions, we add a regularization term weighted log-pro proorabbbility under uniform PCFG prior of G to the negative log- likelihood; to arrive at our final loss per trajectory L(e, 2; D) - (Le, D) + X log Po(e). Here Pa is the that controls the regularization. The loss for
uniform PCFG of G, and X is the hyper-paramerte multiple trajectories is just summation of L over individual trajectories. The continuous constants e require care as they can take any value. To handle this, we use hilevel optimization (Dempe) 2002),
When forming the unit N. constants can in fact has arbitrary units. But if there is any sub-expue tub-exxression like Kg Meter, which is disallowed by our grammar, the final expression would not have any proper unit. This is consistent with how such variables are pre processed in neural network approaches (Breen et al. Usually the mean of pair of positions are subtracted from the pair make them translation translatooniivariinnt invariant.
2019
5

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where the upper-level is the original symbolic regression problem, and the lower-level is an extra optimization for constants. This means we optimize the constants before computing the loss of each candidate tree within the cross-entropy iterations. The defined loss for each expression in SR is then ((e; 2, D) =[e, arg min, L(e, 2, D). In BSP, we use the L.-BGGS optimizer to solve the lower-level optimization.
Our way of handling learnable constants is related other SR methods and bilevel optimization. Traditionally, constants are either randomly generated from predefined, fixed integer set or a continuous interval, or for evolutionary algorithms, they can be mutated and combined during evolution to produce constants that fit better; such constants are often referred as ephemeral constants (Davidson al., 2001). Compared to these methods, the benefit of our formulation is that the evaluation of each tree candidate depends on the symbolic form only as the constants are optimized- away, making the search more efficient. Note that although the literature has not explicitly considered our way of constant learning as bilevel optimization problem, similar strategies are also used in (Cerny et al. 2008 Kommenda et al. 2013) Quade al., 2016). In contrast to recent use of bilevel optimization in meta-learning, e.g. (Finn al., (017), our method is simpler: As our upper-level optimization is gradient-free, we do not need to pass gradient from the lower-level to the upper-level.
4 Experiment: Learning force laws in fully observed environment
In this section, we evaluate the BSP Lata-limited setting when the properties are fully observed.
Synthetic datasets (SYNTH). We created three synthetic datasets for controlled evalua- tion: NBODY (n-body simula- tions with 4 bodies), BOUNCE (bouncing balls) and MAT (mats with friction); see figure 3 for an illustration. NBODY populated by placing a heavy body with large mass and no ve- locity at [0, 0). and three other bodies random positions with random velocities such that, they orbit the heavy body in the middle in the absence of the other two bodies. The gravitational constant is set such that the system is stable for the duration of the simulation. The ground truth force to learn is the gravitational force between bodies. BOUNCE is generated by simulating elastic collisions between balls in a box. The ground truth force to learn is the collision resolution force. MAT simulates fiction-based interaction between discs and a mat. We populate this dataset by rolling dises over mats and applying friction force when they come into contact. We randomized the initial states of the discs as well as the sizes, friction coefficients, and positions of the mats. The ground truth force to learn is the force of friction.
Rectee
Dec3
(a) NBODY
(b) BOUNCE
(c) MAT
Figure 3: Example scenes from SYNTH. Entities in gray are static.
All scenes are simulated using physics engine with a time-discretization of 0.02, for 50 frames. We generate 100 scenes per dataset, and hold-out 20 of them for testing. Appendix C.I provides the ground truth force expressions used to generate each dataset under our grammar.
4.1 Data-efficiency: Symbolic vs neural
Baselines For the experiments in this section we use four different neural baselines: (i) A specialized instance of the OGN model (Sanchez-Gonzalez al. 2019) that only outputs the partial derivative of the velocity variable, unlike the original model that also outputs the partial derivative of the position variable. This is because under Newtonian dynamics, the partial derivative of the position variable is simply the velocity. (ii) An Interaction Network (IN) (Battaglia et al., 2016) (ii) A multi-layer perceptron- baaed force model (MLP (Force)) that directly outputs the force, and (iv) A multi-layer perceptron-basee -baed position model (MIP (Position)) that outputs the next position. See appendix C.2 for details of the neural architecture, training and parameterization setup for all the baselines. Lastly, reference, we also include the performance of zero-force baseline (F)), which corresponds to

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(Forca)
0.30
BSP
(Forcel
6,0030
FFrcce
OIGN
5g
-
OGN
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5g
-
OGN
IN
0.25
PRASSE
0.20
0.0020
0.15
0.10
6.0010
0.05
0,0000
10
10
No. of scenes in training
No. of scones in training
No. of scones training
(a) NBODY
(b) BOUNCE
MAT
Figure 4: Comparison of neural baselines and BSP, using predictive emror on held out scenes given varying number of training scenes. Some baselines are not displayed due to very poor performance; see figure 16 appendix C.4 for the version with all methods displayed.
the constant velocity baseline in (Battaglia et al.) 2016). Note that all neural baselines as well as BSP are provided with symbolic representations for fair comparisons.
In order to compare the symbolic M-step of BSP against the neural baselines in terms of data- efficiency, we report the per-frame prediction accuracy on held-out datasets, as a function of the amount of training data. We use noise-free trajectories in this evaluation. Since the neural baselines cannot be trained if the properties For each dataset, we hold out 20 scenes for evaluation. We randomly shuffle the remaining 80 scenes, and use the first k scenes to fit the models. Because an average person can perform physical learning task similar to the ones we use with fewer than 5 scenes (Ullman et al. 2018), we only vary k from 1 to 10 in our experiments. We use the normalized root mean squared error (nRMSE), per frame per entity, between the predicted location of the entity and its actually location, as the performance metric. We repeat each of the experiments five times with different training set. These results are shown in figure 4 where the line plots are median values and the error bars are the 25% and 75% quantiles. Note that the ground truth force F has an RMSE of 0 per frame. As can be seen, for most values of k across of the three datasets, the symbolic M-step of BSP is more data-efficient than the neural baselines. The exceptions are in the MAT for This is likely due to specific bad local minima that may exist in the limited training data.
not fully observed, we provide all properties as observed data.
and in the MAT dataset for l: 2,3, 4.
For NBODY and MAT, BSP can find the ground truth force function with scene and 10 scenes respectively. BOUNCE is the only case where our method fails to find the true law within 10 scenes, we include the typical inferred force law in appendix C.3.1 as well the predicated trajectories of some selected scenes for inspection. In appendix C.3.2 we also demonstrate that this inferred force law closely approximates the true force law and so can generalize to other scenes.
For BOUNCE, the neural baselines cannot reach the performance of Fo even after 10 scenes for training. This is known issue with neural network approaches when learning collisions, as the inherently sparse nature of the collision interaction does not provide enough training signal (Battaglia) al. 2016). The rank of performance between neural baselines also supports our discussion around higure] I Object-centeric modeling (OGN and IN) tends to have better performance than the rest by decomposing the transition into interaction and dynamics. Predefined dynamies with numerical integration (OGN and MLP force) have better performance than their correspondences with learned dynamics (IN and MLP position) by a notion of "force" " (as the Euler integrator is used in this case).
In short, the symbolic regression with proper priors that constrain the search space in BSP leads to significantly better performance in terms of data efficiency across the three datasets studied. Together with the performance rank of neural models, our experiments shows how different levels and forms of prior help with data-efficient learning.
4.2 Ablation study of priors in the BSP grammar
To demonstrate the impact of the grammar of Newtonian physics on the overall data efficiency of BSP, we consider two ablations of our grammar: (i) G01, which is G without the dimensional analysis prior
and Goo, which is G without both priors in (i) and (ii); we also refer to G as G1 in this section. For reference, for maximum depth of 5, Goo contains 8,593,200 expressions, Got contains 7,935,408
7

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Ga Og
Ga -ole Gg
G
G
nRAASE
NRASEE
ARIMSE
wap Ox
1.0%
0.002
0.000
No, scenes in training
No. of scanes training
No. scenes training
(a) NBODY
(b) BOUNCE (y-axis in log scale)
(c) MAT
Figure 5: Ablation study of priors using predictive error on held out scenes given varying number of
training scenes. Comparison between G1 and Gor shows the effect of the dimensional analysis prior and comparison between Gor and Goo shows the effect of the translation invariance prior.
Deal
SEAP
Comeato
02
12
(a) i Fo
(b) 1: 1: Ft
(c) P3 F
(d) Expression tree for F3.
Figure 6: Results of the EM algorithm NBODY. figure 6ajto figure shows the posterior of mass
for Entity I in Scene I with the corresponding force function for different EM iteration i. In figure 6b the force function F 239.99
mim,
P
The constant in figure 6d is 2.04e3.
PP: PP-Ppllaa
expression and Gn1 contains 75,816. For maximum depth of 6, G11 contains 771,120, and the number for other variants is intractable.
We repeat the experiment from section 4. for all three grammar variants, and report the results in figure As can seen, both priors in BSP's grammar contribute to data efficiency of M-step while dimensional analysis has more impact. This is aligned with the analysis of the number of expressions per grammar above, showing that data efficiency improves as the number of possible expressions decreases. There is also case in which the priors do not show advantages: MAT, in which the friction law is simple enough (the shallowest expression among all) thus easy to find even without priors.
Experiment: Learning force laws in partially observed environments
We now evaluate BSP's performance in environments with some unknown intrinsic entity properties] We do not consider the neural baselines in this section, as they did not show competitive performance compared BSP, even in the fully observed environment.
5.1 EM performance SYNTH
We first demonstrate BSP's ability to jointly learn and reason about the environment by recovering the true force law when some properties unobserved. As illustrative example, we use three scenes from the NBODY dataset (with four entities per scene), such that the true masses are given, the M-step can successfully learn the force law. We assume that the mass of the heavy entity is known but the masses of other the three, lighter entities are unknown with a uniform prior (0.02, 02,00 9). We the EM algorithm fit the same generative model that simulates the data using BSP. Figure shows the posterior distribution over mass and the force function at initialization (figure ba). middle (figure(b)) (6). and convergence (figure 6c). In this run, after 3 iterations, our algorithm successfully recovers the true force function. We repeat this experiment ten times with randomly sampled scenes. For eight of them, BSP successfully recovers the true force law. Appendix D.I.I provides another demonstrative example on MAT.
For quantitative analysis, run the EM algorithm each of the three scenarios from SYNTH
we report nRMSE on a fixed testing set of 20 with 5 random scenes repeated 5 times. In table I
'I is worth to mention that there is an identifiability issue for joint reasoning reasoning-learming tasks with limited data, which we elaborate in appendix A.4

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NBODY
BOUNCE
MAT 25% Q
Median
25% Q
75% Q
Median
25% Q 75% Q
Median
75% Q
BSP
8.05e-1
7.83e-1
1.07e0
3.16e-2
3.11e-2
3.94e-2
5.39e-4
3.58e-4
7.63e-4
Fo
1.77e0
5.98e-2
1.20e-3
Table 1: Test predictive performance (nRMSE) on partially observed SYNTH scenarios (using 5 ran- dom scenes for training and from different runs). Note BSP consistently beats the constant baseline.
scenes. Notice that the performance has large variance due to the fact that not all randomly selected 5-scene subsets provide enough training signal. In all three scenarios, BSP consistently outperforms the zero force baseline and can successfully recover the true force law in some random subsets.
5.2 Real world data: Physies 101
While SYNTH benchmark interesting, a strength of our approach is the ability generalize to real world data. To demonstrate this, we use the PHYS101 dataset (Wu et al. 2016). a dataset of real world physical scenes. We consider two scenarios, FALL and SPRING (shown figure 7 As BSP works on symbolic inputs, we pre pro-psseess raw videos using standard tracking algorithms from OpenCV to extract observations in numerical form; see the appendix for pre- pre-processing details.
Mode
25%
75% Q
BSP
4.30e-2
3.45e-2
5.09e-2
Fo
5.05e-2
4.61e-2
5.49e-2
normaliee
botth
5
morm
BSP
9.22e-3
8.25e-3
9.87c-3
Fo
2.19e-2
2.17e-2
2.21e-2
Figure 7:
Example
Table 2: Test predictive performance (nRMSE) for FALL (top) and SPRING
frames for FALL (left) Figure Learned force expression and SPRING (right) for FALL (left) and SPRING (right) (bottom)
from FALL, where object is dropped on table. A typical force expression that BSP discovers is c x m, x my x normalize(p: - Pj), as shown in figure 8 suggesting that BSP is able to learn the correct form of gravitational force law. Here, the direction of p; Pj points towards the table and my is the mass of the table, which together with serves the constant g in Fg myg. In another solution frequently found by BSP, it learns the direction as normalize (v:) as the velocity is always downwards. BSP can also learn global forces directly if constant vectors [L, and [0, 1] are provided, which is done in the next section.
FALL We first train BSP single
SPRING After learning the gravity from FALL, in SPRING, assuming that the original length of the spring is known, train BSP a single scene to evaluate if it can learn the Hooke's 's law F=kr. Here & is the tensor coefficient, and x is the displacement of the spring. An example force law that BSP learns in this case is x (norm(p: PP: P, 12) x (p; Pj) norm(p, Py), as shown in figure 8 clearly suggesting that BSP can learn Hooke's 's law.
Finally, to quantitatively evaluate BSP's performance on PHYS101, we select fixed set of 4 scenes from each scenario and use for training and for testing, repeating for all the permutations of the 4 scenes. The aggregated test performance in terms of normalized RMSE is given in table 2 BSP only slightly outperforms compared to the zero force baseline on the FALL. This is because the FALL contains only limited number of frames (< 10) and the trajectories do not diverge too far away from what zero force would predict. For SPRING, BSP outperforms the baseline significantly, closely estimating with ground truth time period of the harmonic motion. We also provide qualitative evaluation via a time-series plot of the change of the block's 's vertical position with time in the appendix.
5.3 Does BSP perform similarly to humans?
In this section, we compare BSP's performance against humans' on the experiment done in (Ullman al. 2018). For this purpose, we use the U..LMAN dataset from this study, which consists of 60 videos in which set of discs interact with each other and mats within bounded area, exemplified

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in figure 9 While similar to SYNTH, ULLMAN has lot more diversity in the scenes (stimulus) and reasoning tasks. The foree laws in ULLMAN are similar those in SYNTH but they have different constants and the scenes are generated from a completely different simulator, In the original experiment, each participant is presented with videos. Each of the videos is from a different "world", such that the object properties (for each color) and force laws are different in every video. For each video, the participant is asked 13 multiple choice questions related to the mass of discs ("Mass"), roughness of mats ("Friction"" and types of global ("Global") and pairwise forces ("Pairwise"). For example, "How massive are red objects?" where the options to choose from are "Light", "Medium" and "Heavy". Please refer to appendix D33 for the complete set of questions and options.
Human
BSP
Chance
Mass
43%
40%
33%
Friction
44%
39%
33%
Global
68%
55%
20%
Pairwise 62%
50%
33%
Figure 9: Example from ULLMAN
Table 3: Accuracy per question category
To be consistent with the setup in (Ullman al. 2018), we assume that the friction and collision forces are known apriori. Thus the goal is to apply BSP on a single scene and infer the properties by learning the expressions for the residual global and pairwise force. The properties to infer are the mass for the discs, the friction coefficients for the mats and a latent property q that controls the pairwise interaction. To accommodate for the global force, we added two constant vectors [1, and [0. 1] the grammar; properties related to the known forces are also removed from the grammar. In comparison to models studied Ullman et al. (2018), BSP aims to learn the force expressions explicitly, rather than inferring binary variables to turn on/off predefined force components. Appendix providde details on the learning tasks and setup of BSP. We perform 3 runs of BSP on each of the scenes and use the learn- ing results to answer the same set of 13 questions presented to participants in (Ullman et al., 2018).
Table 3 summarizes the accuracy for humans and BSP on the four question categories. As it can be seen, BSP's performance is worse than that of humans' but convincingly better than chance. Considering the difficulty of inferring 9 properties and learning the targeted force law using only 1 scene, this may not be surprising. There are intriguing similarities between the answers given by BSP and human participants. Both display the same relative order of accuracy across question types "Global", > "Pairwise" > "Friction" > "Mass" while BSP's performance is still inferior to humans'. We hypothesize that humans may have much prior experience with similar physical scenes to answer these questions or they may answer these questions in different way than explicitly learning the forces. Fully addressing the similarity and difference between BSP and humans requires more analysis that is out of the scope of this paper.
display
6 Limitation and Future Work
BSP relies on two main assumptions. First, we assume it has access to the grammar of Newtonian physics. However, how this knowledge is acquired, e.g. through evolution, is not addressed. Second, BSP works on symbolic inputs and assumes perceptual modules are given. This requires extra pre- processing steps when applying BSP to perceptual data, e.g. videos from PHYS 101 and ULLMAN. BSP's performance also depends on the quality of such pre-processing step. It will be interesting to address these two limitations by formulating computational framework in which the grammar and the perceptual modules are learned, either in separate phases or jointly with symbolic force learning.
7 Conclusion
We present BSP, a Bayesian approach to learning symbolic physics which, to our knowledge, the first to combine symbolic learning of physical force laws and statistical learning of unobserved attributes. Our work enables data-efficient symbolic physics learning from partially-observed trajectory data and paves the way for using learnable IPEs in intuitive physics by providing a computational framework to study how humans' iterative aasoning-learning is mentally performed.
10

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Acknowledgment
We thank David D. Cox, John Cohn, Masataro Asai, Cole Hurwitz, Seungwook Han and all the reviewers for their helpful feedback and discussions. This work is supported by the DARPA Machine Common Sense (MCS) program.
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13


Reviews:
[{"review_id": "gKWoxsOtWt", "reviewer": "Reviewer_9Hgp", "summary": "This paper is proposing a new Bayesian symbolic framework that can assist in being able to perform types of physical reasoning over items and environments. The motivation in the paper is that humans are able to understand the physical properties of a system given very few data samples therefore we should be able to develop machine learning systems that can determine the underlying physical properties of the environment with similar data efficiency. The idea in the paper is to use a Bayesian inference method in order to be able to determine a list of parameters and how to combined different functions to describe the physics of a particular environment.\n\nPros\n- The method does propose an interesting grammar that can be used to more efficiently explore the search space of possible physical functions to describe an environment\n- The method also shows some promising performance across a number of different environments for understanding the underlying physical properties and only a few data samples.\n- Learning the physical functions as a combination of terms is also a promising direction for this research. It could be interesting to explore this further and apply the method to systems where the physics may not be well known to discover the physical properties of new physical systems. \n\nCons\n- The assumptions for the model are still not perfectly clear and well contextualized with respect to prior work. This is referring to where and how the symbolic representation for the items in the environment comes from. It's not clear if prior methods also have access to this or similar information.\n- It would also be good to better outline the limitation on the types of physical systems this model can learn to predict over. The equations that are supported seem to indicate the most Newtonian/classical physical systems should be possible but it is not clear what are the limitations of the grammar, along with the types of functions that can be represented.\n- Well it is mentioned in section 6, it's not clear how the perceptual video data used in the real experiments or The ULLMAN experiment is processed into symbolic values that can be used by BSP. Is this mechanism easy to automate such that it's not a significant limitation of applying the method in the paper?\n\n", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Additional comments on the work:\n- It says the environment is model is a top-down generative model in which things are assumed to interact with each other, does this mean that the method requires the top-down model beforehand?\n- Figure one in the paper is challenging to understand. Is each column designed to represent the different papers that are described in the caption? If this is true then this figure is displaying a lot of information but it is difficult to understand the connections between these different papers and what is the significance of the transitions between them. It is important to discuss this well in the related work section to help contextualize how this work fits into prior methods and how it is different.\n- In the introduction, it's outlined that the paper introduces a grammar for Newtonian physics that also greatly helps the overall learning problem. It would be helpful to be able to describe this grammar more in the introduction so the reader can already get an intuition for why this grammar is helpful.\n- From the description given in section 3.1 the model is given a vector of intrinsic physical properties for each of the items in the environment. This particular information can greatly assist in being able to learn the overall problem and being able to estimate interacting bodies between them. It would be good to know if previous methods use a similar structure in this paper or if the primary contribution is being able to take advantage of this type of structure where other methods have not had access to this graph of intrinsic properties.\n- While the first feature of the grammar described in the paper makes sense such that units are preserved between operations. The second point of making sure to use absolute positions feels like something that is well known in the machine learning community as it reduces the complexity of the problem in order to learn behaviors that are more egocentric and less absolute.\n- In section 4.1 the paper reminds the reader that the average person can perform physical learning tasks similar to the ones we use with fewer than five scenes. Yet it seems that the environments described in the paper are new and have not been evaluated on people to understand how many scenes are needed for a person to be able to accurately understand and perform predictions in those physical simulations. It is important to better validate these environments to make sure that humans can easily understand the physics in them given just a few scenes. This will also provide information with respect to the motivation that people can understand these systems very quickly. On the other hand there may be environments that already exist in the community that can be used to perform these types of experiments. For example, the environments used in the Battaglia at all, 2016 paper.\n- A number of the referenced papers are citing ArXiv versions of the papers when the papers are already published. The proper published papers should be cited in the paper not the ArXiv versions. For example, Interaction networks for learning about objects, relations and physics.\n- The ablations in section 4.2 are interesting and also provide good information as to the helpfulness of the grammar that's designed in the paper. However part of the motivation for creating the grammar was to make the learning problem more tractable. Therefore it would also be helpful to include an additional comparison over the same ablations but when more time is given to the training algorithm in order to learn the problem. In this case, the method proposed in the paper should be able to learn quickly with fewer iterations of learning in order to be able to discover the proper physical settings while not using the full grammar may be able to learn with the same performance but will require a significant amount of time that is undesirable.\n- For the analysis and section 5.2 how does bsp compare to the prior neural-based methods for being able to do predictions in those real environments? It's not mentioned whether or not those real environments are impossible to solve using the neural-based methods. Some additional context here in comparison to prior work would be extremely helpful to the reader and help the reader understand how the method performs outside of the SYNTH environments that were created for this paper.\n- Well it is mentioned in section 6, it's not clear how the perceptual video data used in the real experiments or The ULLMAN experiment is processed into symbolic values that can be used by BSP. Is this mechanism easy to automate such that it's not a significant limitation of applying the method in the paper?\n\n----- Post Author Response ----\nI have updated my score thanks to the authors' responses and the supplementary material.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Well it is mentioned in section 6, it's not clear how the perceptual video data used in the real experiments or The ULLMAN experiment is processed into symbolic values that can be used by BSP. Is this mechanism easy to automate such that it's not a significant limitation of applying the method in the paper?", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "Z7AUlWgyOd7", "reviewer": "Reviewer_2v2s", "summary": "A neural-symbolic Bayesian framework for learning intuitive physical models, by making abduction from a few observations provided by the environment and domain knowledge given by grammar.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This paper proposes a neural-symbolic Bayesian framework for learning intuitive physical models, by making abduction from a few observations provided by the environment and domain knowledge given by grammar. The framework is consistent with the general methodology, i.e., *sample(hypothesis)-search-observation loop* of learning-by-explanation paradigm. The solutions, E-M algorithm, and S-R are also canonical methods for combining symbolic knowledge with statistical observation, where the former serves as the hypothesis-generator operation and the latter selects a most plausible hypothesis as the interpretation of the observation. In general, the authors formulate a significant problem and provided the community with a human-like computational approach. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "* **Novelty**: Although exploiting the E-M algorithm and S-R method for solving reasoning-by-observation problems like the ones presented in this work is valid, this method has already been an established and typical framework for explanatory learning that tries to learn a (logical and statistical) model from raw observation and ungrounded primitives. Hence, I think the proposed method is not a principle contribution for novelty. A domain-specific PCFG is a canonical workstream in the neural-symbolic community, either. The novelty of this work comes in applying the learning-by-explanation framework to induce physical rules with data efficiency. This is a principle contributor to the community, yet it goes not so far as the authors claimed;\n\n* **Scalability**: The authors exploit heuristics like checking the consistency of measurement for pruning the hypothesis space, which serves as metaknowledge. This is a good trial but make the method much too ad-hoc. It may be better if formulated in a formal logic way, making the symbols generalizable. Either, the given grammar is way overstrong, making both the framework hard to extrapolate and the task trivial. It would be better if the framework makes abstraction over a small set of primitives;\n\n* **Validity**: Since physical reasoning is not entirely a novel task for the community, it is necessary to carefully benchmark methods for the task. However, the authors only compared with four end-to-end learning methods, without consideration of neural-symbolic counterparts like DeepProblog and Abductive Learning. Since neural methods can hardly encode domain knowledge, the comparison is naturally unfair. It may make the results more solid if the authors compare their methods with neural-symbolic counterparts using identical domain knowledge. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "IC4rrb3FF8l", "reviewer": "Reviewer_6f6C", "summary": "The authors propose the Bayesian symbolic (BSP) framework for inferring both the symbolic force laws and the values of latent properties of entities.\nThey contribute a grammar of Newtonian physics over which they perform inference with the BSP framework.\nEmpirical results show higher sample efficiency than neural approaches, and performance that is between humans and chance on the ULLMAN dataset.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: on the one hand, the method can be seem as novel compared to the approaches in Figure 1. On the other hand, the methodology seems to be straightforward variant of [0] to intuitive physics domains.\n\nClarity: the writing and presentation is clear.\n\nQuality: \n\nWhile it is true that neural approaches are more sample inefficient, Figure 4 is also an unfair comparison. As the authors acknowledge in section 6, they assume that BSP has access to a grammar of Newtonian physics that neural approaches do not have. Indeed, neural approaches must both learn the grammar and how the grammar applies to a particular dataset, while BSP only needs to learn about the latter. Therefore, the results in Figure 4 do not seem to be very informative of possible advantages BSP has over neural baselines. \n\nWhat might make for a better empirical case for BSP is if the authors could dig deeper into Figure 1 and identify explicit axes along which to measure the advantages and disadvantages of BSP vs other baselines in transfer to a new domain *after* the model has been trained. This could help address the problem with the unequal starting point between BSP and neural approaches. For example, \n- suppose we first learn in a multi-task fashion on BOUNCE and MAT, and then transfer to a domain with balls bouncing around over a mat? How would this compare with IN [1] or NPE [2]?\n- the authors state that [tracking algorithm + BSP] learns Hooke's law, after having trained on this environment, what would be the sample efficiency of [tracking algorithm + BSP] be on a spring environment with a different spring constant? And how would this compare with a neural approach that has object perception built in, such as [3]?\n\nCan BSP generalize to completely different kinds of domains from that which it has been trained, such as generalizing to collisions of cars in a simulated freeway from training on the BOUNCING environment? Presumably that would be what I'd expect the strength of symbolic regression to be, it would strengthen the paper if this hypothesis were true, or alternatively if the authors could provide an explanation of why this hypothesis is false.\n\nSignificance: \n\nI am assuming that the authors propose BSP and the Newtonian grammar as a hypothesis for a computational model of intuitive physics in humans.\nIn this case it would have been enlightening to see how BSP compares to the computational model in Ullman et al. (2018) -- would the authors be able to provide this comparison, and comment on the tradeoffs between the two approaches.\nWithout a proper comparison with an alternative hypothesis (e.g. Ullman et al. 2018) as a baseline, rather than simply the random chance baseline, it is not clear how to judge the merits of BSP in modeling humans.\n\nQuestions:\n\nHow does the wall clock time of inference compare between humans, BSP, and a neural method like [1] or [2]?\n\nI was not able to find the appendix, but I think it would be very valuable to see to understand the proposed method.\n\nSummary:\nOverall, while the proposed idea is intriguing, the empirical results seem to fall slightly short of clearly elucidating the merits of the method. The comparison to neural baselines would likely need to be in a transfer setting in order to be a fair comparison, and the human experiments require a discussion with an additional baseline (e.g. Ullman et al. 2018) on the axes along which one or the other method is a better explanation for human intuitive physics induction.\n\n[0] Ellis, Kevin, Catherine Wong, Maxwell Nye, Mathias Sable-Meyer, Luc Cary, Lucas Morales, Luke Hewitt, Armando Solar-Lezama, and Joshua B. Tenenbaum. \"Dreamcoder: Growing generalizable, interpretable knowledge with wake-sleep bayesian program learning.\" arXiv preprint arXiv:2006.08381 (2020).\n[1] Battaglia, P. W., Pascanu, R., Lai, M., Rezende, D., & Kavukcuoglu, K. (2016). Interaction networks for learning about objects, relations and physics. arXiv preprint arXiv:1612.00222.\n[2] Chang, M. B., Ullman, T., Torralba, A., & Tenenbaum, J. B. (2016). A compositional object-based approach to learning physical dynamics. arXiv preprint arXiv:1612.00341.\n[3] Veerapaneni, R., Co-Reyes, J. D., Chang, M., Janner, M., Finn, C., Wu, J., ... & Levine, S. (2020, May). Entity abstraction in visual model-based reinforcement learning. In Conference on Robot Learning (pp. 1439-1456). PMLR.\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes, they were addressed.", "review_text": "", "extra_scores": {"ethical_concerns": "No issues.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1.5"}, "extra_text": {}}, {"review_id": "8kbgw3gZq9t", "reviewer": "Reviewer_9SzY", "summary": "This paper presents the Bayesian-symbolic framework (BSP), a novel framework that combines symbolic reasoning and learning to discover intuitive physics. BSP uses learning to infer object properties, and uses symbolic expression to recover physical laws. The models are learned in an EM manner, where BSP alternates between inferring the object properties with the current force, and inferring the forces given the object property distributions. The dynamics are assumed to be Newtonian. The experiments show that BSP outperforms selected baselines, and achieve relatively good results compared to humans on the SYNTH benchmark and PHYS101 dataset.\n", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "=== Strengths ===\n\n+ The paper is well written and the presentation is strong.\n\n+ The evaluations on physical forces in section 5.1 are comprehensive, as it covers both gravity, collision, and friction.\n\n+ The general idea of combining symbolic reasoning and statistical inference is nice. It also bypasses the non-differentiability issue in collisions, often encountered by pure neural network methods.\n\n+ The use of prior knowledge such as dimensional analysis to prune the symbolic search is neat, as these are also tricks that people commonly use in physics.\n\n=== Weaknesses ===\n\n- This is my opinion only, but the EM optimization method seems slightly naive here. How often would such a search get stuck in the local optimum or saddle points, for example, due to an wrongly searched expression?\n\n- I am really curious to know whether BSP can discover certain physical constants. For example, the Kepler constant K of Earth, or the gravitational constant G? This is my opinion only, but these constants are more exciting compared to the properties such as masses or the spring constants analyzed in section 5.\n\n- I am slightly concerned about the performance of BSP in table 3, as it does not outperform humans. I would imagine humans are not good at such tasks, and would expect better results from BSP, especially considering the task is more of a classification problem rather than inferring the exact physical quantity. Could the authors explain this more in detail?\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes, limitations and impact are discussed in section 6.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper presents a neural-symbolic Bayesian framework to learning physical models in a similar way as human, by making abduction from a few observations provided by the environment and domain knowledge given by grammar. It aims to address an important problem. The proposed solution is reasonable with promising results.

A few concerns have been raised: (1) evaluation: stronger baselines should be included and some results are not as good as human; (2) how the grammar was learned - ad hoc versus principled ways; (3) how general the models can be - only for some specific physical systems versus general systems. The authors were able to address these concerns to a certain extent. After discussion, it was agreed that while the paper still has some limitations, the work is interesting enough to be presented to the NeurIPS community. 
Decision: Accept (Poster)