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

Abstract: Humans are capable of reasoning about physical phenomena by inferring laws of physics from a very limited set of observations. The inferred laws can potentially depend on unobserved properties, such as mass, texture, charge, etc. This sample-efficient physical reasoning is considered a core domain of human common-sense knowledge and hints at the existence of a physics engine in the head. In this paper, we propose a Bayesian symbolic framework for learning sample-efficient models of physical reasoning and prediction, which are of special interests in the field of intuitive physics. In our framework, the environment is represented by a top-down generative model with a collection of entities with some known and unknown properties as latent variables to capture uncertainty. The physics engine depends on physical laws which are modeled as interpretable symbolic expressions and are assumed to be functions of the latent properties of the entities interacting under simple Newtonian physics. As such, learning the laws is then reduced to symbolic regression and Bayesian inference methods are used to obtain the distribution of unobserved properties. These inference and regression steps are performed in an iterative manner following the expectation–maximization algorithm to infer the unknown properties and use them to learn the laws from a very small set of observations. We demonstrate that on three physics learning tasks that compared to the existing methods of learning physics, our proposed framework is more data-efficient, accurate and makes joint reasoning and learning possible.

Under review as a conference paper at ICLR 2021
A BAYESIAN-SYMBOLLE APPROACH TO LEARNING AND REASONING FOR INTUITIVE PHYSICS
Anonymous authors Paper under double-blind review
ABSTRACT
Humans are capable reasoning about physical phenomena by inferring laws of physics from very limited set of observations. The inferred laws can potentially depend on unobserved properties, such as mass, texture, charge, etc. This sample- efficient physical reasoning is considered core domain of human common-senss knowledge and hints at the existence of physics engine in the head. this paper, we propose Bayesian symbolic framework for learning sample-efficient models physical reasoning and prediction, which are of special interests the field of intuitive physics. In our framework, the environment is represented by top-down generative model with a collection of entities with some known and unknown properties as latent variables to capture uncertainty. The physics engine depends on physical laws which are modeled as interpretable symbolic expressions and are assumed be functions of the latent properties of the entities interacting under simple Newtonian physics. As such, learning the laws is then reduced to symbolic regression and Bayesian inference methods are used obtain the distribution of unobserved properties. These inference and regression steps are performed in an iterative manner following the expectation- maximizziion algorithm to infer the unknown properties and use them to learn the laws from a very small set of observations. demonstrate that on three physics learning tasks that compared to the existing methods of learning physics, our proposed framework is more data- efficient, accurate and makes joint reasoning and learning possible.
INTRODUCTION
Imagine ball rolling down ramp. asked 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 in- depth 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 efficiently reason and plan action about common- sense tasks even in absence of complete information (Spelke) 20000 [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
al. 2015 Gerstenberg et 2015 Sanborn 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 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--onzzinn
et al.J 2019 Ehrhardt et al. 2017
Kipf et al. 2018 Seo al. 2019 Baradel et al. 2022)). Among these, neural-networt based learned models of physics (Breen et al.] 2019: Battaglia et al. 2016 Sanchez-Gonzales 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.

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Fixed Symbolic Force Laws
Learnable Symbolic Force Laws
Neurally Learned Interaction
Neurally Leurned Interaction
Fixed
Summation
Summation
A Aggregation
Aggregation
Deep Neural Networks
Physics
Engine
Newtonian Dynamies
Newtonian Dynamics
Hamiltonian Dynamics
Neurally Leurned Dynamics
Iniegrator
Inlegrator
Integrator
More adaptive but less sample efficient
Figure 1: From left to right rule-based to purely data-driven models of physics. Examples for
each column are (1) (Smith et al. 2019), (2) (Ullman et al. 2018), (3) BSP (Ours), (4) OGN &
HOGN (Sanalez-Gonzalez al. 2019), (5) IN (Battaglia et al. 2016) and (6) (Breen al. 2019).
Inspired by humans' highly data-efficient ability of learning and reasoning about their environment, we present sayesian-symbolic physics (BSP), the first fully Bayesian approach to symbolic intu- itive physics that, by combining symbolic learning of physical force laws and statistical learning of unobserved properties of objects, enjoys the sample efficiency symbolic methods with the ac- curacy and generalization of data-driven learned approaches. In BSP, pose the evolution of the environment dynamics over time as generative program of its objects interacting under Newtonian mechanics using forces, as shown in figure 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 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 neural network. Therefore, we also introduce grammar of Newtonian physics that leverages dimensional analysis to induce 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 sxseetation-maximizaae Carlo (MCMC) and SR, for maximum likelihood fitting of the model.
(EM) based algorithm, which combines Markov chain Monte
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 to 5 observations to learn the exact force laws usually more than 10x fewer than that of the closest neural alternatives.
RELATED WORK
At high level, the logic of physics engines can be decomposed into dynamics module and 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 [ 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 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 al. (2016). ODE graph networks (OGNs) and Hamiltonian ODE graph networks (HOGNs; (Sanhezz (Sanchez-Gonzaleeee 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-Cionzalezzee al. 2019). On
(Sancee,
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the other end of the spectrum (left) are the fully symbolic, rule-based physics models and engines (Smthh et al., 2019 Allen et al. 2019 Wu et al. ( Ullman et al. 2018). While these methods are suitable for reasoning tasks, they lack the flexibility of the data- learned models as they cannot 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 Cranmer 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 interest in intuitive physics studies.
MAYESIAN-SYMOOTE 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 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 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 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 E [1...N] by vector of properties 2", such as mass, charge, coefficient of friction, and shape, some of which may be unobserved. At each time step 4, a state vector st (P), vf) is associated with each entity i, where P E IR. and vž E Rd are position and velocity vectors respectively and is the dimensionality of the environment, typically 2 or 3. Let {T' where r' = Pi.r (p). ........... pf)). be the set of observed trajectories from an environment with N entities. Together with a prior on 2, the generative process of BSP defines a joint probability distribution p(D, F) over the observed trajectory data D and latent properties 2 given the force function F An example of the generative process of a three-body problem is shown in figure 2
The state transition of entity in Newtonian system depends not only its properties and current state but also on its interaction with other entities in the environment. Therefore, we define a pairwise interaction function F(z', s 23, ,,,5²), where i, (1 ... N): we interpret F as the force applied to i due to its interaction with j. Then, the trajectory т; of each entity is generated by transition function T that consumes the current state and all of its interactions
$ = T (s), F F(z", 2f st)),
F(z*, , 2 ,,,,,,,s)))
(1)
As forces are additive, forces on entity i can be easily summed to get the total force applied as f = Σj= sf 2 s²). Similar to Sannaee--Gonzalez et al. (2019), we use numerical inte- gration to simulate the Newtonian dynamics, updating with the acceleration obtained from fü.
As physical dynamics typically sensitive to initial states, assume the noise-free initial states are given either as part of the data D point-mass prior over the initial state, thus are omitted in the notation.
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Figure 2: The generation of an observed trajectory: a three-body example with unknown mass. Cir- cles are the learnable force function, rectangles fixed functions, rounded rectangles random variables and others are deterministic variables.
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Figure 3: A grammar of expressions of Newtonian physical laws
Specifically, we choose the Euler integrator since its update rules correspond to the basic relations between position, velocity and acceleration. With these specifications, equation Ionnationnbbbae becomes
at = /mmm,, v = at,t Pi+1 = pt + Att
where m'' is the mass the recipient of the force fÿ and At is the step size the Euler integrator. Finally, add Gaussian noise each trajectory {r'} Tē=1, that is, D [Fi] (Pi...., pr), p ~ (pi o²) and a is the noise level. For clarity, Appendix provides the complete generative process represented by a probabilistic program.
Ji=1 where F
3.2 A GRAMMAR OF NEWTONIAN PHYSICS
In order attain sample efficiency. we chose to learn ,' (²) using symbolic search, but this approach can be inefficient if the search space of possible functions is too large, inaccurate if the search space is too small. Therefore, we constrain the function F to be a members of context-free language called the grammar of Newtonian physics, G. which we now describe.
We consider the following terminal nodes in G: learnable constants C2, C3, masses m,, my, fric- tion coefficients His M4j shapes 8;, 8j, positions p;: Pj. velocities Va, V, and the contact point e for a pair of entities and the following arithmetic expressions: (-)2 (square), +, -, x, +, || (L.2- norm), normalise(-) and project(.:-) (project vector to an unit vector). However, naively support- ing all possible arithmetic expressions for any combination of terminal nodes would make SR highly inefficient and even lead to physically meaningless force laws. Therefore, in order to constrain ex- pression search further, we introduce physics-inspired production rules, along with preterminals and nonterminals nodes, shown in Figure 3 We now discuss the design choice of our grammar.
Motivated by dimensional analysis in natural sciences, in which the relations between different units of measurement are tracked (Brescia 2012), build the concept of units of measurement into the nonterminals of G. That is, mass has the unit kilogram (Kg), unit meter (Meter) for distance and meter per second (MeterSec) for speed. With this unit system in place, only allow addition
In of contact, e the middle position of the two objects, i.c. (p. P,)

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Algorithm 1: Robust inxectation-maxxiizzatioo for Bayesian-symbolic physics
initialize the force function Fo as constantly zero;
for i 1,..., m do
simulate k + k' chains from p(= DD; F(-1) using HMC:
// E-step starts
compute ESS for each chain and remove k' chains with the smallest ESS; select the last sample from each chain as {=1,...,7k]]
get current loss function L; c) = Σx L(e, c; 2, D):
// M-step starts
get candidates C = {(t ci), ..... (ft,, G)} by Algorithm 2 with Li for repetitions;
find (t", ,c* from with the best loss and set F; = getF ,,, // Update force
end
return = Fm and {z1,.... 2x) ~ p(z D; Fm)
and subtraction of symbols with the same units, therefore pruning away physically meaningless expressions, e.g. Ky - Meter. Importantly, this leads to forces laws that have the unit of Newton.
We also forbid the direct use of absolute positions Pt. Pj and с and only allow their differences This ensures that all force laws are reference-invarian that is, independent on the choice of the reference frame. To consistent with the unit system, we call vectors obtained from operations on positions vectors, meter vector (Meter Vec) and those velocities tor Vec).. These variables are all vectors thus we call them eeference-invarian vectors (RefInu Vec). When such vectors are normalised, divided by their corresponding "scalar vari- ables", they becomes unit-less vectors (Unitless Vee) that can be used to describe a direction.
meter per second vec-
The start symbol of the grammar is Force We allow forces to be summed by right- branchin or be conditioned on a Boolean expression. In order to support conditional forces, which are forces that only apply when condition is true, such as collision force and friction. Since the goal of our work is not perceptual learning, we provide perceptual primitives (collision detection as a function doesCollide and isOn check if a disc on mat) in the grammar. Collision then handled by applying an extra force to the entity when doesCollide is true; the force must still be learned.
Finally, some care is needed to ensure the grammar is unambiguous. For example, if we used a 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 expansive. Instead, we represent this rule in right-branching way by the introduction of BaseCoeff and BaseForce as nonterminals. Although the grammar puts basic constraints on plausible physical laws, still expressive. For example, there are more than 6 million possible trees of depth 6.
3.3 LEARNING ALGORITHM
Following the EM approach, our learning method (Algorithm[]) alternates between an E-step, where object properties are estimated given the current forces (Section 3.3.2), and an M-step step, where forces are learned given object properties (Section 1. See Appendix B. for hyperparameters.
3.3.1 SYMBOLIC REGRESSION WITH LEARNABLE CONSTANTS
Symbolic regression is function approximation that searches over the space of mathematical ex- pressions that are specified by user-provided context-free grammar (CFG; Koza 1994). A CFG consists a start symbol, sets of nonterminal, preterminal and terminal symbols and set of production rules. If each production rule in the grammar is specified with a probability (with the probabilities for all rules summed to 1), such grammar is called probabilistic context-free grammars (PCFGs), which effectively defines a distribution over possible expression trees. As such, one can sample from PCFG and/or evaluate the probability of given tree. In our work, we the cross-entropy method for SR. The method starts with PCFG of given grammar that has equal probabilities of all production rules. At each iteration, it samples number of trees (up-to specified depth d) from the current PCFG and evaluates their fitness by a loss function L. After this, trees with the top-k fitness are seleted and used fit PCFG, which will be used in the next iteration, via maximal likelihood.
TThis is in fact consistent with how such variables are pre-processed in the neural network approaches. Usually the mean of pair of positions are subtracted from the pair to make them reference-invariant.
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Algorithm 2: Cross-entropy method with learnable constants
initialize a PCFG Po for 9 uniformaly;
for i 11=1.... .,...,,........... m do
initialize an empty candidate set C
for = 1,... ................................ n do
sample an expression e, ~ Pi-1. 6 with a maximum depth of d;
solve oj = arg min, [(e,, c) by L-BFGS; // Lower-level optimization compute the loss of the sampled tree lj = [(e,, (3) and add (eg, (j) wC
end
if < m then
|
fit a PCFG P. on trees from C with the top-k fitness via maximum maximum-likelihood ;
end
return the best expression tree e* from C and the corresponding constant as c²
To learn the force laws, we need to find an expression e E G and a setting for the learnable constants [c1, C2 Cy] that define the force function Fe,c. The loss used by the cross-entropy method in- volves computing the log- log-liiklhoood of the generative model. As the observed trajectory is generated sequentially given initial state, the computation of the log- lilliiikiioodd term cannot be parallelized and can be computationally expensive in practice. Following the loss form of (Battaglia et al. 2016 we use leacher-forced or vectorized version of the log-likelihood
Sanchez-Gonzalez al. 2019),
T-1
Le = Σ
log log.N (P T (5) Pe,o(',,5,, = ...,, Fe,o By NN (,,, oo) (3)
where T follows equation 2 and s (p) vt) As such, we assume velocity is also available in the dataset D. Clearly, equation 3 differs from the sequential version, as the input for the integrator contains noise at each step. However, similar to previous works, we found it is not issue when learning forces by regression and allows a speed-up of 10x in terms of computing the log- bog-likelihood.
In order to favor simpler trees, add regularization term, weighted log- ogg-robabilitty under uniform PCFG prior of G, to the negative log-likelihood. our final loss per trajectory
C(e, LLL(e 2, D) + À Alog Po(e).
(4)
hyper-parameter that controls the regularization. The loss for multiple trajectories is just summation of L over individual trajectories. Optimizing equation f can be seen as maximum a posterior (MAP) estimation.
Here Po is the uniform PCFG of G and
When using the cross-entropy method for symbolic regression, the continuous constants [c1,02, 02, ca] require care as they can take any value. To handle this, we use bilevel optimization, where the upper-level is the original symbolic regression problem and the inner-level is extra optimization for constants. Specifically, we use an L-BFGS step to optimize the constants before computing the loss of each candidate tree within the cross-entropy iterations. For cases where the constants are very small, e.g. the gravitational constant G = 6.67 x 10-11, we parameterize the constants
10-9 to avoid numerical issues in the inner-level optimization. Traditionally, if such strategy is not used, constants are either randomly generated from a predefined, fixed integer set or continuous interval, or for evolutionary algorithms, they can be mutated and combined during evo- lution to produce constants that fit better; such constants are often referred ephemeral constants (Davidson al. 200T). Compared to these methods, the benefit of our formulation that the evalu- ation of each tree candidate depends on the symbolic form only as the constants are optimized-awwy making the search more efficiently. Note that although the literature has not explicitly considered way of constant learning bilevel optimization, such strategy is not new and is similar in spirit to (Cerny et al. 2008 Kommenda et al.] [2013] Quade et al. (0016). In contrast to recent use of bilevel optimization in meta- Letaa-earrnng, e.g. (Finn al. 2017). our method is simpler: As our upper-level optimization is gradient-free free, we do not need to pass gradient from the lower-level to the upper-level.
In practice, the cross-entropy method itself is sensitive random initializations, in order to bustify the M-step, we repeat it for r runs and pick the best optimizer. We provide a complete description of cross-entropy method with learnable constants in Algorithm

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300
(a) NBODY
(b) BOUNCE
(c) MAT
Figure 4: Example scenes for the datasets used for evaluation. Entities in gray are static.
(a) NBODY
(b) BOUNCE
(c) MAT
Figure 5: Predictive error (RMSE) per frame on the held-out set with a varying number of scenes for training. The line plots are mean values out of five experiments with different shuffling of the training set and the error bars are minimum and maximum values.
3.3.2 REASONING ABOUT UNKNOWN PROPERTIES
With a force function F given, reasoning of unknown properties is reduced to posterior inference on in the generative model specified in Section 3.1 Since for fixed F, the model in our framework is end-to-end, piecewise differentiable with respect to properties, we perform inference by sampling using Hamiltonian Monte Carlo (HMC; [Danne et al. 1987 Neal et al. 201T). Other particle-based alternatives like importance sampling and sequential Monte Carlo are possible but are less efficient.
In order to draw A samples from the posterior robustly in the E-step, we first run k + k' independent HMC chains by the no-U-turn sampler (NUTS; Hoffmman & Gelman] 2011) for a reasonably large number of iterations, where k also a hyper-parameter to choose. After this, we remove k' chains with the smallest effective sample size (ESS). This reduces the chance of using samples from chains that mixed poorly or got stuck in bad region due to random initialization. Finally, we pick the last sample from each chain as the samples returned by the E-step (=1,-..; 2
4 EXPERIMENTS
In this section, present battery of empirical evaluations of the symbolic M-step and demonstrate how our complete EM algorithm is capable of joint reasoning and learning.
Datasets. We consider three simulated datasets for our evaluation. These three datasets, if combined together, correspond to the dataset used in Ullman et al. (2018) for assessing physics learning and reasoning in humans. The first dataset, NBODY (n-body simulation with 4 bodies), is populated by placing a heavy body with large mass and no velocity at [0, 0J and three other bodies at random positions with random velocities such that, they would 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 within the duration of the simulation. The ground truth force learn is the gravitational force between the bodies. The second dataset, BOUNCE, is generated by simulating elastic collision between a set of dises and the box that they are contained within. The gravitational constant set small such that the gravitational force is ignorable and the ground truth force to learn is the collision resolution force. The last dataset, MAT, simulates the friction-based interaction of discs and a mat. We populate this data by rolling discs with different initial states over mats with random sizes and positions and applying friction when they come in contact. The ground truth force to learn is this friction. Each dataset consists of 100 scenes, 20 of which are held-out for testing. All scenes are simulated using a physics engine with a discretization of 0.02, for 50 frames; see figure 4 for an example of each dataset and Appendix B.2 for the ground truth force expressions under our grammar.
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320
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(a) Condition (true) (b) Condition I (learned) (c)
Condition 2 (true)
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Figure 6: Generalization of the approximate bounce law in a vertical world with downward gravity,
Baselines Being most closely related our work, we use specific instantiation of the OGN model Sanchez-Gonzalee et al., (019)) as the neural baseline. The original OGN model does not assume any particular Hamiltonian dynamics and thus outputs the partial derivatives of both position and momentum variables to integrate the dynamics. In our specific realization, we only make it to output the partial derivative of the velocity variable since under the Newtonian dynamies, the partial derivative of the position variable is analytically known as the velocity. Since OGN (and other neural methods, such as IN) is designed to learn with all properties given, we assume all the properties are fully observed and compare our symbolic force model against OGN, See Appendix B.3 for details of the neural architecture, training and parameterization setup for OGN.
4.1 DATA EFFICEEFI SYMBOLIC VS NEURAL
We now compare the symbolic M-step of BSP against the OGN-based neural baselines in terms of data-efficiency. Specifically, we check how accurate the per-frame predictions these models are on held-out data, and how their accuracies change with the amount of training data. We use a noise-free version of the trajectory in this evaluation and provide all the properties as observed data since the neural baselines cannot be trained if the properties are not fully observed. For each dataset, holding out 20 scenes for evaluation, we randomly shuffle the remaining 80 scenes for training and use the first l scenes to fit the model. Because an average human can perform such learning tasks with about 5 scenes (Ullman et al. 2018), we only vary h from 1 to 10 in our experiments. We use the root mean squared error (RMSE) per frame as the performance metric, repeat each of the experiments five times with different training set and finally report the mean, maximum and minimum performance for all the methods, as shown in figure 5 As reference, we include the performance of F 0, a zero-force baseline, corresponding to the constant velocity baseline in (Battaglia et al.) 2016). We also include the performance of the ground truth force F", which has an RMSE per frame of O. As it can be seen, the symbolic M-step is more sample-efficient than the neural baseline for all datasets within this limited data regime. For NBODY and MAT, BSP can find the ground truth force function with 1 scene and 4 scenes respectively. For BOUNCE. the neural bascline cannot reach the performance of F = 0 even after 10 scenes for training. This is a known issue with neural network approaches when learning collision as the inherently sparse nature of the collision interaction does not provide enough training signal (Battaglia et al., 2016). As BOUNCE is the only case where our method fails to find the true law within 10 scenes, we include the typical inferred law in Appendix (Ca as well the predicated trajectories of some selected scenes for inspection.
4.2 GENERALIZATION
is worth checking how the laws learned in Section 4.1 generalier to new scenes beyond the train- ing data. In cases where the true law is successfully recovered, the expression will generalize to novel scenes undoubtedly. Therefore, it is more interesting to inspect the generalization ability of an approximate law, that law which is completely equivalent to the true law but is close. The emerged law for the BOUNCE dataset is such an example as mentioned earlier. It has an expression of F c ||v 17--o Vyl12 c doesCollide(p; $t, P sy); see figure 10 in Appendix C for the actual tree. Although it is not identical to the true law, is still good approximation: it takes into accounts the velocity difference into consideration and finds the correct force direction. We now consider applying this law to a completely new scene: a vertical-vier world where the gravity is pointing in downward direction. figure 6 shows the predicted trajectory with true and the approxi- mate law with two different initial conditions. As it can be seen, the approximate law successfully generalize this novel world. For the first condition, the projection is very close the true one, while
p.-e
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Expression tree for Fa
Figure 7: Results of the EM algorithm on NBODY. figure 7a to figure 7c shows the posterior of
mass for Entity in Scene with the corresponding force function during EM. In figure 7b the
force function F 239.99
m
P.
The constant in figure 7d is c
2.04 x 10
pp. p.lla
P,
3.0
SAA
Bonsfeers
130
100
100
(a) Fo 0 (b)
pt (c)i=
F4 F
(d) Expression tree F4
Figure 8: Results of the EM algorithm on MAT. figure Sa to figure 8c shows the posterior of friction coefficient in Scene with the corresponding force function during EM. In figure 8b the force
function Ft 22.99 P4 [[vila
Vi_ isOn(p:, 8:: Pý. s,). The constant in figure 8d is
8.605.
for the second condition, the concept of bounce is also correctly transferred. The corresponding animations for these plots can also be found in the supplementary material for further inspections.
4.3 LEARNING WITH UNOBSERVED PROPERTIES
Now, we demonstrate how the joint learning and reasoning in the BSP method (Algorithm I can recover the true force law when some properties are unobserved. As the first experiment, we three scenes from the (noisy) NBODY dataset (with four entities per scene), such that the true masses are given, the M-step can successfully learn the true force law. Next, assume that the masses of the three light entities are unknown with uniform prior (00002, 9) and the mass of the heavy entity is known. We use Algorithm to fit the same generative model that simulates the data using BSP figure 7 shows the posterior distribution over mass and the force function at initializa- tion (figure 7aj. middle (figure 7b) and convergence (figure 7cp. In this run, after iterations, our algorithm successfully recovers the true force function. We repeat this experiment ten times with randomly sampled scenes and for eight of them, BSP successfully recovers the true force law. Note that because the intermediate learned force law Ft is incorrect, the variance of the posterior (in figure 7b) is larger than the one from the true force law (in figure (c). Compared the expression at convergence in figure [d with the true law, the algorithm replaces p; Pý with p; c and scaled constant. This valid as the contact point is defined as c = (P; + P,) /2 when there is no contact.
For the second experiment, that the only unknown is the friction coefficient of the mat with a truncated Gaussian prior T Truncated(A((Ho, 22). (%) 5) (truncated between 0 and 5), where (4n is the true coefficient, Note that the variance 2 is large enough to be uncertain, justifying a fair choice of the prior. Similarly, we Algorithm to fit the same generative model that simulates the data using BSP. figure 8 shows the posterior distribution over mass and the force function at initialization (8a). middle (8b) and convergence (8c) of the algorithm. Compared the expression at convergence with the true law, the algorithm learns V, instead of v, as the mat velocity is zero, i.e. V, = 0, in all scenes,
use five scenes from the (noisy) MAT dataset. We assume
5 DISCUSSION AND CONCLUSION
We present BSP, the first fully Bayesian approach to symbolic intuitive physics by combining sym- bolic learning of physical force laws and statistical learning of unobserved properties. Our work paves the way for using learnable, data-efficient IPEs in intuitive physics by providing a computa- tional framework to study how humans' iterative leasongg is mentally performed.

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Under review as a conference paper at ICLR 2021
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Frank Wood, Jan Willem Meent, and Vikash Mansinghka. A approach probabilistic pro- gramming inference. In Artificial Intelligence and Statistics, pp. 1024-1032, 2014.
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Algorithm 3: Complete generative process given force laws
// Sample latent variables
for i ,....,,.. .N do
ASSUME z' from prior for entity ;;
initial state not given then
ASSUME Pij from prior for entity i:
ASSUME vó from prior for entity i;
set S/ (p vt);
end
T for / ,,...,,. do
for 1, 1,..., N do
// Compute force and acceleration
for j ,,..,, N do
compute f F(2 s;-1, 22, s
end
compute f = L= f(
compute a: f:/m';
// Euuer''' integration update v = v a,A:;
update p = +
set s (p).
// Sample observations
OBSERVE P from ((p)
end end
A COMPLETE GENERATIVE PROCESS
Section 3 describes the top-down generative model piece by piece, to improve the clarity of the EM framework, we provide the complete generative process of the observation given force function F, which corresponds to the E-step our method, as a probabilistic program Algorithm 3 this the keyword ASSUME and OBSERVE for sampling latent varialess and observations separately, following the notations from Wood et al. (2014).
probabilistic program,
B EXPERIMENTAL DETAILS
B.1 HYPER-PARRMETTRS FOR ALGORITHM
I
For the E-step, we
3 and and the hyper-parameters for NUTS 150 adaptation steps, 150 HMC iterations, maximum tree depth of 4 and a target acceptance ratio of 0.75. For the M-step, we repeat
runs and the hyper-parameters for the cross-entropy method are: 1, 000 total populations, 500 selected populations. 4 iterations and a maximum depth of 8. The weighting parameter for the PCFG prior are for the NBODY and BOUNCE datasets and 10-4 for the MAT dataset.
B.2 GROUND TRUTH FORCES
The symbolic tress of ground truth forces that are used to generated the datasets used in Section are given in figure
B.3 TRAINING SETUP OF OGNS
For the OGN baseline, function the rectified linear unit (ReLU) for the node model and use a MLP of (50 + +50) + 50
multilayer perceptron (MLP) of d, - 50 50 where the activation
50
50 dout as the edge model. For training, the ADAM optimizer (Kingmal
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Under review as a conference paper at ICLR 2021
& Ba 2014) with learning rate of 5 10 for total number of 10,000 passes of scenes for the NBODY and MAT datasets and a total number of 20,000 passes of scenes for the BOUNCE dataset. For example, if 5 scenes used, the total number of pass of the dataset would be 10000/5, 5, for the NBODY dataset. This makes sure that the training time for all experiments are fixed.
In addition, found that need provide additional prior knowledge how forces related to the mass and acceleration by parameterzing them as F.(.) = m ag(+), where 0 is NN parameters, otherwise they fail learn. This parameterization is fact consistent with (Sanchez-Gonzalez al., 2019) in which NNs output partial derivatives of the Hamiltonian system.
C THE LEARNED BOUNCE LAW
As mentioned Section 4.1 and discussed in Section 4.2 the only case in which BSP fails to infer the true law (within 10 scenes) is of special interest for further inspection. typical approximate law learned in Section 4.1 is shown in figure [0] see Section 4.2 for discussion on how this law differs from the true one. To highlight, there are basically two mismatches between the true law and the learned law. First, there is no projection operation that correctly calculates the effect of speed. Second, the mass-based coefficient is missing. To assist inspection, we also provide some visual- izations in figure [1 using initial conditions from the training set for inspection. The corresponding animations can be found in the supplementary material.
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Under review as a conference paper at ICLR 2021
Force
BaseForee
Constant
Coeff
Unitless Vee
C1
BaseCoeff + BaseCoeff
normalize( Reffno Vec)
KySq
MeterSq
Meter Vee
1 Pt Pj
my X m2
Meter?
|| Mecll2
p:
(a) Gravitational force (depth 7); c. 8.17 10
Force
BaseForce
Bool
Constant
Coeff
Unitless Vec
docsCollide(p: 8,, Pj. 8z.
02
BaseCoeff BaseCoeff
normalized (Refaaiiieeeeeeeeeee
KgSq + Kg
Meter
MeterVec
my X m2
m2 + m2
-
project( SetterSe Vec, Unitless Vec)
V; Vj
normalize RefIne Vec)
MeterVec
p:
(b) Collision force (depth 8); c2 2/5 100 where E = 2 x 10-2 is the step size of the integrator
Force
BaseForce
Bool
Constant
Coeff
Unitless Vec
isO. 83)
C3
Cueff x BuseCoeff
normalizel Reftre Vec)
Kg
P2
MeterSec Vec
mi
Vi
(C) Friction force (depth 5): c3 = 9.8
Figure 9: Expression trees (under 9) of true force laws that generates the datasets used in Section 4
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Force
BaseForce
Bool
Constant
Coeff
Unitless Vec
docsCollidc(pr. 8, Pj,
c
BaseCoeff
normalized Reshro Vec)
Meter
MeterVea Vec
normalize ReflnwVec)
-
MeterVec
Figure 10: Approximate bounce law learned by BSP under our grammar; 130.22
300
32
32
DS
21
-30
(a) Scene (true)
(b) Scene (learned)
(c) Scene (true)
(d) Scene 2 (learned)
Decs FanW
Nec
Dec FariWa
320
320
322
(e) Scone 3 (true)
(f) Scene (learned)
gg) Scene (true)
(h) Scene 4 (learned)
Figure 11: Predicated trajectories of the true bounce law and the learned bounce law.
16


Reviews:
[{"review_id": "ai3drtlY_SY", "reviewer": "AnonReviewer3", "summary": "", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes an Bayesian-symbolic physics (BSP), an intuitive physics model that jointly infers symbolic force laws and object properties (mass, friction coefficient). The inductive bias is force summation, F=ma, and a grammar of force laws to express object interactions. The inference is done via an EM method that alternates between object property estimation (E-step) and force law induction (M-step), using techniques like symbolic regression and Hamiltonian Monte Carlo (HMC). Some preliminary experiments are shown for the method's effectiveness and data efficiency. \n\n**Strength**: \nThe paper fills an important missing position in the spectrum of intuitive physics models, as Figure 1 argues. The force law grammar, to my knowledge, is something novel in this area, and represents a reasonable inductive bias that balances expressivity and physical plausibility (with two further physical constraints: dimensional analysis and reference invariance). The grammar also helps improve data efficiency. From the inference side, the proposed EM approach is also reasonable. Symbolic regression for force law inference can be interesting for future research.\n\n**Weakness**:\nExperiments seem a bit weak for now, and I have some technical concerns about the inference. \n\n**Questions/suggestions**:\n1. Learning with unobserved properties (Figure 6) is the key experiment setting according to the paper's selling point, but is obviously lacking. It'd be great to see more scenarios (than graviton) with more diverse settings (e.g. object mass, initial position), some quantitive numbers, and comparison with some baseline (if possible). \n2. For some qualitative sense, would also be nice to show true trajectories vs. predicted trajectories in the main paper, or if the learned symbolic law force is correct (for Figure 5). Would be interesting to see if any symbolic laws are predicted wrong and how they look like.\n3. For object property inference, why use Monte Carlo methods over variational/gradient-based methods? Also, is it possible to use MCMC for formula inference? Some explanations or experiments would solidify design choices for the inference part.\n4. Is there an ambiguity when force constants and object properties are jointly inferred? How tricky is designing priors for these? (I see some tricks are used in the paper to handle small constants, for example)\n5. The paper currently only compares with one side of the spectrum, i.e. more neural approaches. Comparing BSP with more symbolic approaches like (Smith et al., 2019), the strength is supposed to be the ability to handle non-standard object interactions outside fixed physics engine but within the force grammar. Is it possible to generate some random force laws and show BSP still works while baselines from BOTH sides of the spectrum may fail? It could be the most powerful experiment to support this position in the spectrum.\n\nIn general I like the ideas and hope to see more experiments and justifications for design choices.\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "QbkjhQCriTC", "reviewer": "AnonReviewer1", "summary": "", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a fully Bayesian approach to symbolic intuitive physics that, by combining symbolic learning of physical force laws and statistical learning of unobserved properties of objects. The paper proposes an EM-based method where in E-step object properties distribution are sampled using the current best estimated force laws and in M-step symbolic regression is used to update those laws.\n\nThe paper is clearly-written with clear explanation of the proposed EM-style method. However, one of the main claim that the method \"enjoys the sample efficiency of symbolic methods with the ac- curacy and generalization of data-driven learned approaches\" is not well-supported in the experiments. While in Sec. 4.1 it shows the proposed method is more data efficient than the neural baseline, it's not clear how the method generalized to complete different scenarios. Ideally the symbolic physical laws are universal thus can be applied to all physical interactions.\n\nSome other questions / comments:\n- One of the contribution states \"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.\" This (10x more data-efficient) is hard to tell from figure 5 as it doesn't show when the neural baseline reaches the same performance.\n- The error bars in figure 5 is somewhat not very indicative of the stability of the method and some of them for the neural baseline are extremely large. It states that  the values are \"out of five experiments with different shuffling of the training set\". How many different shuffling is there?\n- Sec 3.3.1 \"In practice, as the cross-entropy method itself is sensitive to random initializations, in order to ro- bustify the M-step, we repeat it for r runs and pick the best optimizer.\" Is 'm' supposed to be r in Algorithm 2? \n- For figure 6, if it converges after iteration 3, can we show the expression tree for F3 instead F5? How much do they differ? ", "extra_scores": {}, "extra_text": {}}, {"review_id": "HploPUHwck6", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "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.\n\nThe methodology described in the paper enables prior physical knowledge to be incorporated into the statistical machine learning models in the form of a probabilistic context free grammar, which in return makes inference via EM applicable. The paper is well written in general, i.e. the main idea of the paper is easy to grasp and all the related technical concepts are explained in a very simple way. Mathematical notation is clear and consistent. Proposed methodology is novel and sound. Claims of the authors are supported by the experimental results on simulated datasets, and there seems to be no fallacy in empirical evaluations. Nevertheless, experimental section can be extended by large-scale applications or real world datasets. In other words, learning the underlying physical rules from a data set which is collected through real-world sensors would provide further evidence to the impact of such a framework. In addition, I think it’s necessary to include more results from symbolic and neural-network based learned models as a reference. Although relation to prior work is clearly addressed in the paper, authors compare their results to only OGN. \n\nIn my opinion, learning symbolic rules for physical reasoning and prediction in a sample-efficient manner will certainly be of interest to the ICLR community. I believe that the proposed methodology is clearly worth exploring, and the presented experimental results are promising. On the other hand, it is no surprise that a carefully constructed grammar for physics is sample-efficient. Authors chose to restrict the grammar with a small set of production rules, in order to narrow down the search space for deriving force terms. However, this can be also seen as a compromise from the generality of the model, since such a predefined grammar might be too restrictive and it might necessitate adding new rules for each new physical phenomena. For instance, in the fields related to optics or thermodynamics, to be able to model phenomena such as refraction-diffraction of light or heat transfer, one might need to modify the grammar manually. \n\nAnother drawback of the methodology could be enforcing position and velocity vectors to reside in the Cartesian coordinate system. Such a choice limits the expressiveness of the model, for instance allowing a general coordinate system could make it easier to model harmonic movement via angular coordinates. In short, my major concern is that enforcing too many constraints inspired by the physics rules we already know introduces a bias, which in return affects the generalization of the model. \n\nMy final remark is about the EM algorithm in the paper. I think that the details about the EM should be written more clearly and thoroughly. I recommend writing the generative model explicitly which could improve the clarity of the paper in general, as well as allowing different inference algorithms to be applicable.", "extra_scores": {}, "extra_text": {}}, {"review_id": "Flv7OHapmPZ", "reviewer": "AnonReviewer2", "summary": "", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a fully Bayesian approach to learn an intuitive physics model by combining symbolic regression and statistical learning (MCMC). \n\nPros:\n\n+ Personally, I like this paper as it approaches the learning of the intuitive physics model in the right direction, highlighted in figure 1. It is a nice combination of symbolic and learning-based approach.\n\nCons:\n\n- The related work section is too brief to see the main difference between the present work and prior work. The current related work section only focuses on the machine learning-based intuitive physics model but does not cover symbolic regression in general. Using symbolic regression has a long history, especially in material science, soft robotics, and machine learning in general. Prior work has demonstrated that SR can indeed learn the physical law in a much more complex setting [1-2].\n\n- For researchers who are opposed to the idea of intuitive physics, they would ask where the prior knowledge comes from. Some cognitive research has shown that they are innate for humans and some animals. But this rule cannot be applied for a machine: If we give them all the rules, why not just directly use the physics engine? In the experiments, the results demonstrated here are too simple so that most of the naive physics-based simulator can produce similar results at a probably faster speed. The question is, what the benefit of learning? One may argue that if I know some properties individually, I might be able to transfer this knowledge to unseen scenarios. The authors do demonstrate some capability in 3.3.2 and 4.2, but it seems to come naturally with SR based on the given prior, not something new or surprise. Such capabilities have been demonstrated in [1-2].\n\n- I would like to see if the authors would be able to demonstrate the learned physics can be generalized to scenarios beyond the training datasets. For instance, can the learned model from MAT dataset be transferred to simple scenarios created by bullet-like engine with a similar friction-based interaction, but not identical? The learned physical knowledge should be general enough. Even not as the perfectly correct Newtonian physics, it should be able to generalize to unseen scenarios.\n\n[1] Distilling free-form natural laws from experimental data, Science 2009\n[2] AI Feynman: A physics-inspired method for symbolic regression, Science Advance 2020", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a method for learning physics combining symbolic computation and learning in an interesting way, targeting sample efficiency. At the initial evaluation, it was on the fence but leaning towards acceptance, with 3 slightly positive and one slightly negative review. 

The strengths lie in the combination between symbolic reasoning and statistical ML with a formulation around the classical EM framework. On the other hand, an important issue of the paper is its quite simplistic evaluation on now very easy problems and benchmarks. While benchmarks tend to be simple in the field of learning physics, current work does address more difficult problems than the problems tackled in this paper.

Another issue discussed was the simple trade-off in injecting hand-crafted inductive bias into a system leading to increased sample efficiency, which was perceived as unsurprising by some reviewers. While this is common in ML, and even strongly more so in learning physics from data synthetically generated with known physical laws, it was perceived to be particularly unsurprising in this paper where the benchmarks are indeed very simple and the laws directly encoded.

The AC discussed this paper with the PCs, and it was judged that the weaknesses in evaluation, in particular the simplicity of the tasks, cannot compensate for the interesting hybrid symbolic/ML formulation, and decided to reject the paper.
Decision: Reject