Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search
Authors: Fangzheng Sun
Yang Liu
Jian-Xun Wang
Hao Sun
Venue: ICLR 2023 notable top 5% (2023)
Keywords: symbolic regression
Monte Carlo tree search
governing equations
nonlinear dynamics

Abstract: Nonlinear dynamics is ubiquitous in nature and commonly seen in various science and engineering disciplines. Distilling analytical expressions that govern nonlinear dynamics from limited data remains vital but challenging. To tackle this fundamental issue, we propose a novel Symbolic Physics Learner (SPL) machine to discover the mathematical structure of nonlinear dynamics. The key concept is to interpret mathematical operations and system state variables by computational rules and symbols, establish symbolic reasoning of mathematical formulas via expression trees, and employ a Monte Carlo tree search (MCTS) agent to explore optimal expression trees based on measurement data. The MCTS agent obtains an optimistic selection policy through the traversal of expression trees, featuring the one that maps to the arithmetic expression of underlying physics. Salient features of the proposed framework include search flexibility and enforcement of parsimony for discovered equations. The efficacy and superiority of the SPL machine are demonstrated by numerical examples, compared with state-of-the-art baselines.

Published as a conference paper at ICLR 2023
SYMBOLIC PHYSICS LEARNER: DISCOVERING GOV- ERNING EQUATIONS VIA MONTE CARLO TREE SEARCH
Fangzheng Sun', Yang Liu?, Jian--Xun Xun Wang', Hao Sun
*Northeastern University, Boston, MA, USA; University of Chinese Academy of Sciences, Beijing, China; University of Notre Dame, Notre Dame, IN, USA; "Renmin University of China, Beijing. China.
Emails: sun,fa fat.ff/onoohhatemmeeet 22uyang22 @ wwws.ac.cmt jwang (e.e33ennddddd. nd.edu; haosun @ ruc.edu.cn
ABSTRACT
Nonlinear dynamics ubiquitous in nature and commonly seen in various science and engineering disciplines. Distilling analytical expressions that govern nonlinear dynamics from limited data remains vital but challenging. To tackle this funda- mental issue, we propose novel Symbolic Physics Learner (SPL. machine to discover the mathematical structure of nonlinear dynamics. The key concept is to interpret mathematical operations and system state variables by computational rules and symbols, establish symbolic reasoning of mathematical formulas via expression trees, and employ a Monte Carlo tree search (MCTS) agent to explore optimal expression trees based on measurement data. The MCTS agent obtains an optimistic selection policy through the traversal of expression trees, featuring the one that maps to the arithmetic expression of underlying physics. Salient features of the proposed framework include search flexibility and enforcement of parsimony for discovered equations. The efficacy and superiority of the SPL machine are demonstrated by numerical examples, compared with ofate-of-the-art baselines.
INTRODUCTION
We usually learn the behavior of nonlinear dynamical system through its nonlinear governing differential equations. These equations can be formulated as y(t) dy dy/dt dt = F(y(t)), where yt) = {i (2())......... (t)} E IR Ix,, denotes the system state at time t, FF-) a nonlinear function set defining the state motions and n, the system dimension. The explicit form of F(.) for some nonlinear dynamics remains underexplored. For example, in a mounted double pendulum system, the mathematical description of the underlying physics might be unclear due to unknown viscous and frictional damping forms. These uncertainties yield critical demands for the discovery of nonlinear dynamics given observational data. Nevertheless, distilling the analytical form of governing equations from limited noisy data, commonly seen practice, an intractable challenge.
Ever since the early work on the data-driven discovery of nonlinear dynamics (Džeroski & Todorovski 1993, Dzeroski Todorovski] 1995), many scientists have stepped into this field of study. During the recent decade, the escalating advances in machine learning, data science, and computing power have enabled several milestone efforts of unearthing the governing equations for nonlinear dynamical systems. Notably, breakthrough model named SINDy (Sparse Identification of Nonlinear Dynamics) (Brunton et al. 2016) has shed light on tackling this achallenge. SINDy was invented to determine the sparse solution among pre-defined basis function library recursively through sequential threshold ridge regression (STRidge) algorithm. SINDy quickly became one of the state- of- art methods and kindled significant enthusiasm in this field of study (Rudy al. [017 Long al.) 2018 Champion et al., 2019] Chen et al. 2021: Sun et al., 2021 Rao et al.] 2022). However, the success of this pparsity-promotin approach relies on a properly defined candidate function library that requires good prior knowledge of the system. It is also restricted by the fact that a linear combination of candidate functions might be insufficient recover complicated mathematical expressions. Moreover, when the library size is massive, it empirically fails to hold the sparsity constraint.
At the same time, attempts have been made to tackle the nonlinear dynamics discovery problems by introducing neural networks with activation functions replaced by commonly seen mathematical
"ooresponding author

---

Published as a conference paper at ICLR 2023
operators (Martius & Lampert 2017 Sahoo et al. 2018 Kim et al. 2019 Long et al. 2019). The
intricate formulas are obtained via symbolic expansion of the well-trained network. This interpretation of physical laws results in larger candidate pools compared with the abbary-based representation of physics employed by SINDy. Nevertheless, since the sparsity of discovered expressions is primarily achieved by empirical pruning of the network weights, this framework exhibits sensitivity user- defined thresholds and may fall short to produce parsimonious equations for noisy and scarce data.
Alternatively, another inspiring work (Bongard & Lipson| 2007; Schmidtt & Lipson 2009) re- envisioned the data-driven nonlinear dynamics discovery tasks by casting them into symbolic regres- sion problems which have been profoundly resolved by the genetic programming (GP) approach (Koza & Koza 1992, Billard & Diday 2003). Under this framework, symbolic regressor is es- tablished to identify the governing equations that best describe the underlying physics through free combination of mathematical operators and symbols, leading to great flexibility in model selection. One essential weakness of this early methodology is that, driven exclusively by the goal of empirically seeking the best-fitting expression (e.g. minimizing the mean-square error) in genetic expansion process, the GP-based model usually over-fits the target system with numerous false-positive terms under data noise, even sometimes at subtle level, causing huge instability and uncertainty. However, this ingenious idea has inspired a series of subsequent endeavors (Cornforth & Lipson [0122] Gaucel et al. 2014 Ly & Lipson| 2012: Quade al. 20066: Vaddireddy et al. 22200) In a more recent work, Deep Symbolic Regression (DSR) (Petersen et al.,
et al. (022), a reinforcement learning-based model was established and generally outperformed the GP based models including the
commercial Eureqa software (Langdon & Gustafson 2010). Additionally, the Al-Feynman methods
(Udrescu & Tegmark 2020; Udrescu et al. 2020 Udrescu & Tegmark| (20TT) ameliorated symbolic regression for distilling physics laws from data by combining neural network fitting with a suite of maysics-inspired techniques. This approach is also highlighted by a recursive decomposition of a complicated mathematical expression into different parts on tree-based graph, which disentangles the original problem and speeds up the discovery. It outperformed Eureqa in the uncovering Feynman physics equations (Feynman et al., [9655.. However, this approach is built upon ad-hoc steps and, to some extent, lacks flexible automation in equation discovery.
The popularity of adopting the tree-based symbolic reasoning of mathematical formulas (Lample) & Charton 2019) has been rising recently to discover unknown mathematical expressions with a reinforcement learning agent (Kubalík et al. 2019 Petersen et al., [021] Mundhenk al.] (0211)
However, some former work attempting to apply the Monte Carlo tree search (MCTS) algorithm as an alternative to GP for symbolic regression (Cazenave 2013: White al. 2015 Islam et al. 2018
et al. (02T) failed to leverage the full flexibility of this algorithm, resulting in the similar shortage that Ga-based symbolic regressors possess as discussed earlier. Despite these outcomes, we are conscious of the strengths of the MCTS algorithm in equation discovery: it enables the flexible representation of search space with customized computational grammars to guide the search tree expansion. A sound mathematical underpinning for the trade-off between exploration and exploitation is remarkably advantageous as well. These features make it possible to inform the MCTS agent by our prior physics knowledge in nonlinear dynamics discovery rather than randomly searching in large spaces.
Contribution. We propose a promising model named Symbolic Physics Learner (SPL) machine, empowered by MCTS, for discovery of nonlinear dynamics. This architecture relies on a grammar composed of (i) computational rules and symbols to guide the search tree spanning and (ii) a composite objective rewarding function simultancously evaluate the generated equations with observational data and enforce the sparsity of the expression. Moreover, we design multiple adjustments to the conventional MCTS by: (1) replacing the expected reward in UCT score with maximum reward to better fit the equation discovery objective, (2) employing an adaptive scaling in policy evaluation which would eliminate the uncertainty of the reward value range owing to the unknown error of the system state derivatives, and (3) transplanting modules with high returns to the subsequent search as a single leaf node. With these adjustments, the SPL machine is capable of efficiently uncovering the best path to formulate the complex governing equations of the target dynamical system.
2 BACKGROUND
In this section, we expand and explain the background concepts brought up in the introduction to the SPL architecture, including the expression tree (parse tree) and the MCTS algorithm.
2

---

Published as a conference paper at ICLR 2023
Expression tree. Any mathematical expression be represented by combinatorial set of symbols and mathematical operations, and further expressed by a parse tree structure (Hopcroft et al. 2006 Kusner et al. 2017) empowered by a context-free grammar (CFG). A CFG is a formal grammar characterized by a tuple comprised of 4 elements, namely, 9 = (V, x,, R. S), where V denotes finite set of non-terminal nodes, E finite set of terminal nodes, R finite set of production rules, each interpreted as mapping from a single non-terminal symbol in V to one or multiple ferminal/non-termins node(s) in (V U where represents the Kleene star operation, and S single merminll node standing for a start symbol. In our work, equations are symbolized into parse trees: we define the start node as equation symbol f, terminal symbols (leaf nodes) corresponding to the independent variables formulating the equation (e.g, I. 2/, and a placeholder symbol C' for identifiable constant coefficients that stick to specifie production rules. The non-terminal nodes between root and leaf nodes are represented by some symbols distinct from the start and terminal nodes (i.e.. ). The production rules denote the commonly seen mathematical operators: unary rules (one non-terminal node mapping to one node) for operators like cos(-). exp(.), log(1 [), and binary rules (one non-terminal node mapping to two nodes) for operators such as + -, x, + A parse tree is then generated via a pre-order traversal of production rules rooted f and terminates when all leaf nodes are entirely filled with terminal symbols. Each mathematical expression can be represented by such a traversal set of production rules.
Monte Carlo tree search. Monte Carlo tree search (MCTS) (Coulom 2006) is an algorithm for searching optimal decisions in large combinatorial spaces represented by search trees. This technique complies with the best-first search principle based on the evaluations of stochastic simulations. It has already been widely employed in and proved the spectacular success by various gaming artificial intelligence systems, including the famous AlphaGoo and AlphaZero (Silver et al.) 2017) for computer Go game. A basic MCTS algorithm composed of an iterative process with four steps:
1. Selection. The MCTS agent, starting from the root node, moves through the visited nodes of the search tree and selects the next node according to a given selection policy until it reaches an expandable node or a leaf node.
2. Expansion. At an expandable node, the MCTS agent expands the search tree by selecting one of its unvisited children.
3. Simulation. After expansion. if the current node is non non-terminal, the agent performs one or multiple independent simulations starting from the current node until reaching the terminal state. In this process, actions are randomly selected.
4. Backpropagation. Statistics of nodes along the path from the current node to the root are updated with respect to search results (scores evaluated from the terminate states reached).
To maintain proper balance between the less-tested paths and the best policy identified so far, the MCTS agent sticks to a trade-off between exploration and exploitation by taking action that maximizes the Upper Confidence Bounds applied for Trees (UCT), formulated as (Kocsis & Szepesvári (0066::
UCT(s, a) = Q(s, + ev (n[[N(s))NNN(...
where Q(s,a) is the average result/reward of playing action a in state in the simulations performed in the history, encouraging the exploitation of current best child node; N(s) is number of times state s visited, N(s, a) the number of times action a has been selected at state s, and (s)[Ns) consequently encourages exploration of less- child nodes. Constant controls the balance between exploration and exploitation, empirically defined upon the specific problem. Theoretical analysis of UCT-based MCTS (e.g., convergence, guarantees) is referred to Shah al. (0199)
METHODS
Existing studies show that the MCTS agent continuously gains knowledge of specified tasks via the expansion of the search tree and, based on the backpropagation of evaluation results (i.e., rewards and number of visits), render a proper selection policy on visited states to guide the upcoming searching (Silver et al. (017).. In the proposed SPL machine, such a process is integrated with the symbolic reasoning of mathematical expressions to reproduce and evaluate valid mathematical expressions of the physical laws in nonlinear dynamics step-by-step, and then obtain favorable selection policy
3

---

Published as a conference paper at ICLR 2023
Repeat
Selection
Expansion
Simulation
Propagation
--4
0.35
0:25
0.5
0.25
M UCT
randomiy pick
unvisited parse
Preprocess
10 simulations rewordd
10 sirullations
Difforentitte
In hwwpppomm
RMSE
Figure I: Schematic architecture of the SPL machine for nonlinear dynamics discovery. The graph explains the MCTS phases of one learning episode with an illustrative example.
pointing to the best solution. This algorithm is depicted in Figure T with illustrative example and its overall training scheme shown in Algorithm Discussion of the hyperparameter setting for this algorithm is given in Appendix Section
Rewarding. To evaluate the mathematical expression f projected from a parse tree, we define numerical reward E R C IR based on this expression and input data D = (Y; YY), serving as the search result of the current expansion or simulation. It is formulated as
17"
(2)
1 V MI f(Y
where (y1. RE the m dimensional state variables of a dynamical system, Y; E RIIN the numerically estimated state derivative for ith dimension, and N the number of measurement data points. 7) denotes discount factor, assigned slightly smaller than I; is empirically defined as the total number of production rules in the parse tree. This numerator arrangement is designated to penalize non non-parsimonious solutions. This rewarding formulation outputs reasonable assessment to the distilled equations and encourages parsimonious solution by discounting the reward of non-parsio non-parsimonious one. The rooted mean square error (RMSH) in denominator evaluates the goodness-of- fit of the discovered equation w.r.t. the measurement data.
Training scheme. A grammar G (V, Σ, R, is defined with appropriate nodes and production rules to cover all possible forms of equations. To keep track of non-terminal nodes of the parsing tree, we apply last-in-firstoout (LIFO) strategy and denote the non-terminal node placed last on the stack NT as the current node. We define the action space A = R and the state space S as all possible traversals of complete/incomplete parse trees (i.c., production rules selected) in ordered sequences. At the current state s, = [a,, Q2 ...Gr] where t E N is the discrete traversal step-index of the upcoming production rule, the MCTS agent masks out the invalid production rules for current non-terminal node and on that basis selects a valid rule as action a++1 (i.e, the left-hand side of valid production rule is the current non-terminal symbol). Consequently, the parse tree gains a new terminal/nonnterminn branch in accordance with 0 meanwhile the agent finds itself in new state $t+1 = [a1, 42, .,, Q1+1]. The agent subsequently pops off the current non-terminal symbol from NT and pushes the non-terminal nodes, if there are any, the right- right-hand side of the selected rule onto the stack. Once the agent attains an unvisited node, a certain amount of simulations are performed, where the agent starts randomly select the next node until the parse tree is completed.
4

---

Published as a conference paper at ICLR 2023
Algorithm 1: Training SPL for discovering the 8h governing equation 1, ..., m)
Input: Grammar (V, Σ, RR,), S), measurement data [Y;Y):; Parameters: discountrregularizatonn factor 7), exploration rate c, tmazi
r controls equation parsimony;
Output: Optimal governing equation f";
for each episode do
Selection: Initialize Su 0, 0, NT (S);
while s; expandable and < tmaz do
Choose dt+1 arg max (ST a); Take action dc+1, observe s' NT;
8141 +- note as visited, L 4- & + 1;
end
11
Expansion: Randomly take an unvisited path with action a, observe s', NT;
12
S141 < note visited, t < 1 1;
13
i' = then
14
Project Backpropagate r++1 and visited count and finish episode;
15
end
16
Simulation: Fix the starting point 8t, NT; for each simulation de while sı non-terminal and < Imaz do Randomly take an action a, observe s NT:
17
19
8++1- +- w
end 21
22
if NT = then
Project f and calculate Tr+15
24
end
end
26
Backpropagate simulation results;
27 end
The reward is calculated or the maximal size exceeded, resulting in a zero reward. The best result from the attempts counts the reward of the current simulation phase and backpropagates from the current unvisited node all the way to the root node.
Greedy search. Different from the MCTS- gaming AIs where the agents inclined to pick the action with a high expected reward (average returns), the SPL machine seeks the unique optimal solution. In the proposed training framework, we apply greedy search heuristic to encourage the agent to explore the branch which yields the best solution in the past: (s,, a) is defined as the maximum reward of the state-action pair, and its value is backpropagated from the highest reward in the simulations upon the selection of the pair. Mcanwhile, to overcome the local minima problems due to this greedy approach policy search, we enforce a certain level of randomness by empirically adopting the c-greedy algorithm, a commonly seen approach in reinforcement learning models.
Adaptive-scaled rewarding. Owing to the unknown level of error from the numerically estimated state derivatives, the range of the RMSE in the SPL reward function is unpredictable. This uncertainty affects the scale of rewarding values thus the balance between exploration and exploitation presented in Eq. ( Besides adding "I" to the denominator of Eq. (2) to avoid dramatically large numerical rewards, we also apply an adaptive scale of the reward, given by
r" (s,
Q(s,a) max Q(s',a) a')
where r* denotes the maximum reward of the state-action pair. It is scaled by the current maximum reward among all states S and actions A reach an equilibrium that the Q-values stretched to the scale [0, 1] at any time. This self-adaptive fashion yields well-scaled calculation of UCT under different value ranges of rewards throughout the training.
Module transplantation. A function can be decomposed into smaller modules where each is simpler than the original one (Ddrescu al. 2020). This modularity feature, as shown Figure 2 helps us develop a sovide-anddconquee heuristic for distilling some complicated functions: after every certain amount of MCTS iterations, the discovered parse trees with high rewards are picked out and thereupon
5

---

Published as a conference paper at ICLR 2023
reckoned individual pro- duction rules and appended to the set of production rules R; accordingly, these trees are "transplanted" the future ones as their mod- ules (i.e, the leaves). To avoid early overfitting prob- lem, we incrementally en- large the sizes of such mod- ules from a baseline length to the maximum allowed size of the parse tree throughout the iterations. The augmentation part of R is refreshed whenever new production rules are created, keeping only the ones engendering high rewards. This approach accelerates the policy search by capturing and locking some modules that likely contribute to, or appear as part of the optimal solution, especially in the cases of the mathe- matical expression containing "deep" operations (e.g., high-order polynomials) whose structures difficult for the MCTS agent to repeatedly obtain during the selection and expansion.
mi-f
mi:f
COS
sin
exp
Grammars
8i
Figure 2: A module transplantation process: A complete parse serves as a single production rule and is appended to the grammar pool.
SYMBOLIC REGRESSION: FINDING MATHEMATICAL FORMULAS
GP
SPL
4.1
DATA NOISE SCARCITY
2%
10%
5%
15
100%
Data scarcity and noise are commonly seen in measurement data and become one of the bottleneck issues for discovering the govern- ing equations of nonlinear dynamics. Tackling the challenges in bigh-level data scarcity and noise situations is traditionally regarded as an essential robustness indicator for a nonlinear dynamics discovery model. To this end, we present an examination of the proposed SPL machine by equation discovery task in the presence of multiple levels of data noise and volume, comparing with GP-based symbolic regressor (implemented with gplearn python package [ The target equation is (xx) = 0. + 0.5z² + 2x, and the independent variable X is uniformly sampled in the given range [-10, 10). Gaussian white noise is added to the dependent variable Y with the noise level defined as the te--mean--quaal ratio between the noise and the exact values. For discovery, the two models are fed with equivalent search space: {+, -, X, +, cost, x } as candidate mathematical operations and symbols. The hyperparameters of the SPI. machine are set as 1) = 0.99, Imaz 50, and 10,000 episodes of training is regarded as one trail. For the GP-based symbolic regressor, the population of programs is set as 2,000, the number of generations as 20. The range of constant coefficient values is [-10, 10). For 16 different data noise and scarcity levels, each model was performed 100 independent trails. The recovery rates are displayed as 4 x 4 mesh grid w.r.t. different noise/scarcity levels in Figure 3 It is observed that the SPL machine outperforms the GP-based symbolic regressor in all the cases. A T-test also proves that the recovery rate of the SPL machine is significantly higher than that of GP (e.g.. p-value = 1.06 x 10-7).
100
0%
84%
91%
18%
100%
10056
100%
100%
data volume
75%
1000
94%
94%
97%
95%
100%
100%
200%
100%
50%
10000
92%
91%
100%
25%
100000
98%
99%
94%
93%
100%
10055
200%
100%
level
noise level
Figure 3: The effect of data noise/scarcity recov- ery rate. The heatmaps demonstrate the recovery rate of GP and the SPL machine under different data conditions, summarized over 100 independent trials.
4.2 NGUYEN'S SYMBOLIC REGRESSION BENCHMARK
Nguyen's symbolic regression benchmark task (Uy et al. 2011) is widely used to test the model's s robustness in symbolic regression problems. Given a set of allowed operators, a target equation, and data generated by the specified equation (see Table for example), the tested model is supposed to distill the mathematical expression that is identical to the target equation, or equivalent to it (e.g., N equuttinn can be recovered log(2² +22 + 1), Nguyen- 10 equation can be recovered as sin(z + y). and Nguyen-11 equation can be recovered as exp(y log(z))). Some variants of Nguyen's benchmark equations also considered in this experiment. Their discoveries require numerical
All simulations are performed standard workstation with NVIDIA GeForce RTX 2080Ti GPU.

---

Published as a conference paper at ICLR 2023
Recovery rate of three algorithms in Nguyen's benchmark symbolic regression problems. The SPL machine outperforms the other two models in average recovery rate.
Table I:
Benchmark
Expression
SPL.
NGGP
GP
N
2
+ x² +x
100%
100%
99%
Nguyen-2
+ + xx + xx + +x* +
+ x.
100%
100%
90%
Nguyen-3
100%
100%
34%
Nguyen-4
6 + x + x² + + +
99%
100%
54%
N
sin(x*) cos
95%
80%
12%
6
sin(x") sin(x + +x²)
100%
100%
11%
Nguyen-7
In(r + 1) + In(r? + 1) Ve (n) + sin(y²) 2 2sin(r) cos (1)
100%
100%
17%
N
100%
100%
100% 1
Nguyen-9
100%
100%
76%
N0
100%
100%
86%
N1
100%
100%
13%
Nguyen-12 12
x² x² 1/2
28%
4%
0%
Nguyen-1
3. + 2.r22² + 1.78x
100%
100%
0%
N
0.48x 3.39I + 2.12 1.78r
94%
100%
0%
Nguyen-5
sin(x*) ) cos(æ)) 0.75
95%
98%
1%
Nguyen-8
v1.23r
100%
100%
56%
Nguyen-9
11.() + sin (0.5y²)
96%
90%
0%
Average
94.5%
92.4%
38.2%
estimation of the constant coefficient values. Each equation generates two datasets: one for training and another for testing. The discovered equation that perfectly fits the testing data is regarded as successful discovery (i.e., the discovered equation should be identical or equivalent to the target one). The recovery rate is calculated based on 100 independent tests for each task. In these benchmark tasks, three algorithms are tested: GP-based symbolic regressor, the neural- -uideed GP (NGGP) (Mundhenk)
et al. 2021 ,, and the SPI. machine. Note that NGGP is an improved approach over DSR (Petersen
2021 They are given the same set of candidate operations: (+,- - x, + +xx(())x)))))..) cos( ()), sin(.)} for all benchmarks and (V,,In()) are added to the 7, 8, 11 benchmarks. The hyperparameters of the GP-based symbolic regressor are the same as those mentioned in Section[4.1] configurations of the NGGP models are obtained from its source code?) detailed setting of the benchmark tasks and the SPL model is described in Appendix Section B The success rates are shown in Table I is observed that the SPL machine and the NGGP model both produce reliable results in Nguyen's 's benchmark problems and the SPL. machine slightly outperforms NGGP. This experiment betokens the capacity of the SPL machine in discovery of equations with divergent forms.
al.
(a) the adaptive scaling in reward calculation, (b) the discount factor 17" that drives equation parsimony in Eq. (2), (c) module transplan- tation in tree generation, and (d) all of the above. The four models were tested on the first 12 Nguyen equations (see Appendix Section C). Results show the average recovery rates for these models are all smaller than that produced by SPL (see Appendix Table[C.I]. where the module transplantation brings the largest effect. Hence, these modules are critical to guarantee the proposed model efficacy.
Ablation Study:
We consider four ablation studies by removing:
PHYSICAL LAW DISCOVERY: FREE FALLING BALLS WITH AIR RESISTANCE
is well known that, in 1589-1592, Galileo dropped two objects of unequal mass from the Leaning Tower of Pisa and drew a con- clusion that their velocities were not affected by the mass. This has been well recognized globally the "textbook" physical law for the vertical motion of free-falling object:
Table 2: Baseline models (c; unknown constants).
Physies Model
Derived model expression
Model-1 Model-2 Model-3
H(t) co + Cit est² + cat H(()) = co + cjt + cat cat ([) co + ci log(cosh(c2t))
NGGP source code: wwwpg//www..oooooooooooeeeeeeeee
7

---

Published as a conference paper at ICLR 2023
the height of the object is formulated as (t) = ho + tyt - g where hy denotes initial height, to the initial velocity, and g the gravitational acceleration. However, this ideal situation rarely reached in our daily life because air resistance serves as significant damping factor that prevents the above physical law from occurring in real-life cases.
Many efforts have been made to uncover the effect of the air re- sistance and derive mathematical models to describe the free-falling objects with air resistance (Clancy 1975; Lindemuth, 1971] Green wood et al., 1986). This sec- tion provides data-driven discov- ery of the physical laws of rela- tionships between height and time in the cases of free-falling objects with air resistance based multi- ple experimental ball-drop datasets (de Silva et al. 2020), which con- tain the records of 11 different types of balls dropped from a bridge (see Appendix Figure D.I). For discovery, each dataset is split into a training set (records from the first 2 seconds) and testing set (records after 2 seconds). Three mathematically derived physics models are selected from the literature as baseline m for this experiment (see Table 2). and the unknown constant coefficient values are estimated by Powell's conjugate direction method (Powell 1964). Based on our prior knowledge of the physical law that may appear in this case, we use (+. +, exp())) cosh(.), (,, log(-)) as the candidate grammars for the SPL discovery, with terminal nodes {t, const). ). The hyperparameters are set 17 0.9999, tmax 20, and one single discovery is built upon 6,000 episodes of training. The physical laws distilled by SPL from training data are applied to the test data and compared with the ground truth. Their prediction errors, in terms of MSE, are presented Table [3] (the SPL-discovered equations are shown in Appendix Table D.I The full results can be found in Appendix Section [ It can be concluded that the data-driven discovery of physical laws leads to a better approximation of the free-falling objects with air resistance.
Table 3: Mean square error (MSE) between ball motion predic- tion with the measurements in the test set. The SPL machine reaches the best prediction results in most (9 out of 11) cases.
Type
SPL
Model-1
Model-2
Model-3
baseball
0.3
2.798
94.589
3.507
blue basketball
0.457
0.513
69.209
2.227
green basketball
0.088
0.1
85,435
1.604
volleyball
0.111
0.574
80,965
0.76
bowling ball
0.003
0.33
87.02
3.167
golf ball
0.009
0.214
86.093
1.684
tennis ball
0.091
0.246
72.278
0.161
whiffle ball 1
1.58
1.619
65.426
0.21
whiffle ball 2
0.099
0.628
58.533
0.966
yellow whiffle ball
0.428
17.341
44.984
2.57
orange whiffle ball
0.745
0.379
36.765
3.257
6 CHAOTIC DYNAMICS DISCOVERY: THE LORENZ SYSTEM
The first nonlinear dynamics discovery example is 3 3-dimensional Lorenz system (Lorenz, 1963) whose dynamical behavior (Ir, y, 2) is governed by à a
Table 4: Summary of the discovered gov- erning equations for Lonrez system. Each cell concludes if target physics terms are distilled (if yes, number of false positive terms in uncovered expression).
.),ú = x(p - 2) y, 2 = ry B2 with parameters == 10, 8 8/3, and p 28. The Lorenz attractor has two lobes and the system, starting from anywhere, makes cycles around one lobe before switching to the other and iterates repeatedly, exhibiting strong chaos. The synthetic system states (x., y, 2) are generated by solving the nonlinear differential equations using the Matlab ode 113 (Shampine, 1975 Shampine Re- ichelt 1997) function. 5% Gaussian white noise is
Model
à
Furoqa
Yes (1)
Yes (3)
Yes (1)
PySINDY
Yes (1)
No (N/A)
Yes (2)
NGGP Yes (10)
Yes (8)
Yes (16)
SPL
Yes (0) Yes (0)
Yes (0)
added to the clean data to generate noisy measurement.
The derivatives of the system states (æ, ÿ. 2) are unmeasured but estimated by central difference and
smoothed by the Siritzkk-Gollay filter (Savitzky & Golay 1964) in order to reduce the noise effect.
In this experiment, the proposed SPL machine is compared with three benchmark methods: Eureqa, PySINDy and NGGP. For Eureqa, NGGP, and the SPL machine, {+,- x are used as candidate operations; the upper bound of complexity is set to be 50; for pySINDy, the candidate function library includes all polynomial basis of (r. y, 2) from degree 1 to degree 4. Appendix Table E.
"Model I: http:://anndayyhhyssss.uoooooooo hww.://www..tt..aaaaaaaaaaaaeena attinto//rrraalll//caalllllmm
"Model 2:
Model 3: http:://a..wkiiiddii.og/////eeee fall
8

---

Published as a conference paper at ICLR 2023
presents the distilled governing equations by each approach and Table summarizes these results: the SPL machine uncovers the explicit form of equations accurately in the context of active terms, whereas Eureqa, pySINDy and NGGP yield several false-positive terms in the governing equations. In particular, although Eureqa and NGGP are capable of uncovering the correct terms, their performance is very sensitive the measurement noise indicated by the redundant terms (despite with small coefficients) shown in Appendix Table E.I] Overall, the baseline methods fail to handle the large noise effect, essentially limiting their applicability in nonlinear dynamics discovery. It is evident that the SPL machine is capable of distilling the concise symbolic combination of operators and variables correctly formulate parsimonious mathematical expressions that govern the Lorenz system, outperforming the baseline methods of Eureqa, pySINDy and NGGP.
7 EXPERIMENTAL DYNAMICS DISCOVERY: DOUBLE PENDULUM
This section shows SPL-based discovery of a chaotic double pendulum system with experimental
data (Asseman et al. 2018) as shown in Appendix Figure E.3 The governing equations are given by:
W1 = و1 cos( A0) + c2w3 sin (A)) + c3 sin + R1 (61, 02 01. 62),
W ا cos A0) + c sin A)) + cg (02) R2(01, 02, 0,, 02)
where 01, 0g denote the angular displacements; W/ 01, W2 02 the velocities; W1, We the accelerations; R, and R2 denote the unknown damping forces. Note that AO 0 02.
The data source contains multiple camera-sensed datasets. Here, 5,000 denoised random sub- sub-sampee from 5 datasets are used for training, 2,000 random sub-samples from another 2 datasets for validation, and dataset for testing. The derivatives of the system states are numerically estimated by the same approach discussed in the Lorenz case. Some prior physics knowledge is employed to guide the discovery: (1) the terms روy cos( A0) for W1 and Wن cos(At) (A)) for Wن are assumed to be part of the governing equations based on the Lagrange derivation; (2) the angles (01, 82, Ao) are under the trigonometric functions cos(.) and sin(-); (3) directions velocities/relative velocities may appear in damping. Production rules fulfilling the above prior knowledge are exhibited in Appendix Section F.2 The hyperparameters are set as 1) = 1, bmax 20, and 40,000 episodes of training are regarded as one trail. 5 independent trials are performed and the equations with the highest validation scores are selected as the final results. The uncovered equations are given as follows:
0.099142 cos (A)) 0.103 sin(A0) A0) 69.2744 sin (0)) + 0.515 cos(O),
(5)
و = -1 1.3684 cos A0) + 1.363 sin( A0) sn. + 0.032uh,
where the explicit expression of physics in an ideal double pendulum system, displayed in Eq. (5 are successfully distilled and damping terms are estimated. This set of equation is validated through interpolation the testing set and compared with the smoothed derivatives, as shown in Appendix Figure E.4 The solution appears felicitous as the governing equations of the testing responses.
CONCLUSION AND DISCUSSION
This paper introduces Symbolic Physics Learner (SPL) machine to tackle the challenge of distill- ing the mathematical structure of equations for physical systems (e.g.. nonlinear dynamics) with scarce/noisy data. This framework is built upon the expression tree interpretation of mathematical operations and variables and an MCTS agent that searches for the optimistic policy to reconstruct the target mathematical formula. With some remarkable adjustments to the MCTS algorithms, the SPL model straightforwardly accepts our prior or domain knowledge, or any sort of constraints of the tasks in the grammar design while leveraging great flexibility expression formulation. The robustness of the proposed SPL machine for complex target expression discovery within a large search space indicated in the Nguyen's symbolic regression benchmark problems, where the SPL. machine outperforms mate-of-theear symbolic regression methods. Moreover, encouraging results are obtained in the tasks of discovering physical laws and nonlinear dynamics, based on synthetic or experimental datasets. While the proposed SPL machine shows huge potential in both symbolic regression and physical law discovery tasks, there are still some imperfections that can be improved: (i) the computational cost is high for constant coefficient value estimation due to repeated calls for an optimization process, (ii) graph modularity is underexamined. and (iii) robustness against extreme data noise and scarcity is not optimal. These limitations are further explained in Appendix Section F

---

Published as a conference paper at ICLR 2023
ACKNOWLEDGMENTS
The work is supported by the National Natural Science Foundation China (No. 92270118) and the Beijing Outstanding Young Scientist Program (No. BJJW2YJH012011110000009999
REFERENCES
Alexis Asseman, Tomasz Kornuta, and Ahmet Ozcan. Learning beyond simulated physics, In Modeling and Decision-making the Spatiotemporal Domain Workshop-NIPS 2018.
Billard and Diday. From the statistics of data to the statistics of knowledge: symbolic data analysis. Journal of the American Statistical Association, 98(462):470-487, 2003.
Josh Bongard and Hod Lipson. Automated reverse engineering of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 104(24):9433-944, 2007.
Steven L Brunton, Joshua L Proctor, and Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 1 113(15):3952-3937, 2016.
Tristan Cazenave. Monte-carlo expression discovery. International Journal on Artificial Intelligence Tools, 22(01): 2013.
Kathleen Champion, Bethany Lusch, Nathan Kutz, and Steven L Brunton. Data-driven discovery of coordinates and governing equations. Proceedings of the National Academy of Sciences, 1 116(45); 22445-22451, 2019.
Zhao Chen, Yang Liu, and Hao Sun. Physics-informer learning of governing equations from scarce data. Nature Communications, 12:6136, 2021.
Laurence Joseph Clancy, Aerodynamics. John Wiley Sons, 1975.
Theodore Cornforth and Hod Lipson. Symbolic regression of multiple-ting -ime--calee dynamical systems. In Proceedings of the 14th annual conference Genetic and evolutionary compulation, pp. 735-742, 2012.
Rémi Coulom. Efficient selectivity and backup operators monte-carlo tree search. International conference on computers and games, pp. 72 83. Springer, 2006.
Brian M de Silva, David M Higdon, Steven L. Brunton, and J Nathan Kutz. Discovery of physics from data: universal laws and discrepancies. Frontiers in artificial intelligence, 3:25, 2020.
Saso Dzeroski and Ljupco Todorovski. Discovering dynamics: from inductive logic programming to machine discovery. Journal of Intelligent Information Systems, 4(1):89-108, 1995.
Saão Džeroski and L.juééo Todorovski. Discovering dynamics. In Proc. tenth international conference on machine learning, pp. 97-103, 1993.
Richard P Feynman, Robert B Leighton, and Matthew Sands. The feynman lectures physies; vol. American Journal of Physics, 33(9):750-777 -52, 1965.
Sébastien Gaucel, Maarten Keijzer, Evelyne Lutton, and Alberto Tonda. Learning dynamical systems using standard symbolic regression. In European Conference Genetic Programming, pp. 25-36.
Springer, 2014.
Margaret Stautberg Greenwood, Charles Hanna, and Rev John W Milton. Air resistance acting on sphere: Numerical analysis, strobe photographs, and videotapes. The Physics Teacher, 24(3): 153-159, 1986.
John E Hopcroft, Rajeev Motwani, and Jeffrey D Ullman. Automata theory, languages, and computa- tion. International Edition, 24(2), 2006.
Mohiul Islam, Nawwaf N Kharma, and Peter Grogono. Expansion: novel mutation operator for genetic programming. In IJCCI, pp. 55-66, 2018.
10

---

Published as a conference paper at ICLR 2023
Samuel Kim, Peter Lu, Srijon Mukherjee, Michael Gilbert, Li Jing, Vladimir Ceperic, and Marin Soljacic. Integration neural network-basee symbolic regression deep learning for scientific discovery. arXiv preprint arXiv:1912.0488: 2019.
Levente Kocsis and Csaba Szepesvári. Bandit based monte-carlo planning. In European conference on machine learning, pp. 282-293. Springer, 2006.
John Koza and John R Koza. Genetic programming: on the programming of computers by means of natural selection, volume 1. MIT press, 1992.
Jifí Kubalik, Jan Žegklitz, Erik Derner, and Robert Babuška. Symbolic regression methods for reinforcement learning. arXiv preprint vrXXiv1903.00000 2019.
Matt J Kusner, Brooks Paige, and José Miguel Hernández-Lobtu Grammar variational autoencoder.
In Proceedings of the 34th International Conference on Machine Learning, pp. 1945-1954. JMLR. org, 2017.
Guillaume Lample and François Charton. Deep learning for symbolic mathematics. arXiv preprint 1912vv:912.01411 2019.
William B Langdon and Steven M Gustafson. Genetic programming and evolvable machines: ten years of reviews. Genetic Programming and Evolvable Machines, 11(3):321 -338, 2010.
Jeffrey Lindemuth. The effect of air resistance falling balls. American Journal of Physics, 39(7): 757-759, 1971.
Zichao Long, Yiping Lu, Xianzhong Ma, and Bin Dong. Pde-net: Learning pdes from data. In International Conference Machine Learning, pp. 3208-3216. PMLR, 2018.
Zichao Long, Yiping Lu, and Bin Dong. Pde-net 2.0: Learning pdes from data with numeric- symbolic hybrid deep network. Journal of Computational Physics, 399: 2019.
Edward N Lorenz. Deterministic nonperiodic flow. Journal of the atmospheric sciences, 20(2): 130-141, 1963.
Qiang Lu, Fan Tao, Shuo Zhou, and Zhiguang Wang. Incorporating actor-critic in monte carlo tree search for symbolic regression. Neural Computing and Applications, pp. 1-17, 2021.
Daniel L Ly and Hod Lipson. Learning symbolic representations of hybrid dynamical systems. The Journal of Machine Learning Research, 13(1):3585-366 3618, 2012.
Georg S Martius and Christoph Lampert. Extrapolation and learning equations. 5th International Conference on Learning Representations, ICLR 2017-Workshop Track Proceedings, 2017.
T Nathan Mundhenk, Mikel Landajuela, Ruben Glatt, Claudio P Santiago, Daniel M Faissol, and Brenden K Petersen. Symbolic regression via neural-guided genetic programming population 2021.
seeding. arXiv preprint
Brenden K Petersen, Mikel Landajuela Larma, Terrell N Mundhenk, Claudio Prata Santiago, Soo Kyung Kim, and Joanne Taery Kim. Deep symbolic regression: Recovering mathemat- ical expressions from data via risk-seeking policy gradients. In International Conference on Learning Representations, 2021.
Michael JD Powell. An efficient method for finding the minimum function of several variables without calculating derivatives. The computer journal, 155) 1964.
Markus Quade, Markus Abel, Kamran Shafi, Robert K Niven, and Bernd R Noack. Prediction of dynamical systems by symbolic regression. Physical Review E, 94(1):012214, 2016.
Chengping Rao, Pu Ren, Yang Liu, and Hao Sun. Discovering nonlinear PDEs from searce data with physics-encoded learning. In International Conference Learning Representations, pp. 1-19, 2022.
Samuel H Rudy, Steven L Brunton, Joshua Proctor, and Nathan Kutz. Data-driven discovery of partial differential equations. Science Advances, 3(4)xe1602614, 2017.
11

---

Published as a conference paper at ICLR 2023
Subham Sahoo, Christoph Lampert, and Georg Martius. Learning equations for extrapolation and control. In International Conference on Machine Learning, pp. 4442-4450, 2018.
Abraham Savitzky and Marcel JE Golay. Smoothing and differentiation of data by simplified least squares procedures. Analytical chemistry, 36(8):1627-1639, 1964.
Michael Schmidt and Hod Lipson. Distilling free-form natural laws from experimental data, science, 324(5923):81-85, 2009.
Devavrat Shah, Qiaomin Xie, and Zhi Xu. Non-asymptotic analysis of monte carlo tree search. arXiv preprint arXiv:1902.05213, 2019.
Lawrence F Shampine. Computer solution of ordinary differential equations. The initial value problem, 1975.
Lawrence F Shampine and Mark W Reichelt. The matlab ode suite. SIAM journal on scientific computing, 18(1):1-22, 1997.
David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. Nature, 550(7676)3354 359, 2017.
Fangzheng Sun, Yang Liu, and Hao Sun. Physics-informed spline learning for nonlinear dynamics discovery. In Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, pp. 2054-2061, 2021.
Silviu-Marian Udrescu and Max Tegmark. Ai feynman: physics-inspired method for symbolic regression. Science Advances, 6(16):eaay6631, 2020.
Silviu-Marian Udrescu and Max Tegmark. Symbolic pregression: discovering physical laws from distorted video. Physical Review E, 103(4):043307, 2021.
Silviu-Marian Udrescu, Andrew Tan, Jiahai Feng, Orisvaldo Neto, Tailin Wu, and Max Tegmark. Ai feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity. arXiv preprini arXiv:2006. 10782, 2020.
Nguyen Quang Uy. Nguyen Xuan Hoai, Michael O'Neill, Robert I McKay, and Edgar Galván- López. Semantically- crossover in genetic programming: application to real-valued symbolic regression. Genetic Programming and Evolvable Machines, 12(2):91-119, 2011.
Harsha Vaddireddy, Adil Rasheed, Anne E Staples, and Omer San. Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensor observation data. Physics of Fluids, 32(1):015113, 2020.
David White, Shin Yoo, and Jeremy Singer. The programming game: evaluating mets as an alternative to gp for symbolic regression. In Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation, pp. 1521-1522, 2015.
12

---

Published as a conference paper at ICLR 2023
APPENDIX
A HYPERPARAMETER SETTING
We perform a parametric study on the value of discount factor based on Nguyen's benchmark problems without measurement noise. Empirically, setting 1/ 0.9999 ensures the scores of ground truth equations stand out and successfully enforces the sparsity in all experiments. For the discovery of very chaotic dynamical systems based on measurement data, we expect some physies terms from the governing equations to have weak impact on the state variables (c.g.. in the chaotic double pendulum system experiments, the effects from physics terms sin(01) and sin(02) are hard to be captured), and the effect of data noise is unknown. Hence, we set 1j 1 to leverage the full strength of data fitting to enable the detection of physics terms that are offset or overwhelmed by data noise but are pivotal to the systems. Nevertheless, empirical, which depends on our desire for the degree of parsimony of the target equation(s).
must acknowledge that this selection process is
As for the hyperparameters in the training schema (i.e., maximum module transplantation, episodes, maximum tree size, maximum augmented grammars), we have conducted parametric convergence tests for each experiment to ensure the learning curves (i.c., maximum scores in the history) converge. shows the setting of these hyperparameters.
For example, as discussed in Section Table B.2
B NGUYEN'S BENCHMARK PROBLEMS
This section provides more detailed experiment settings for the Nguyen's benchmark tasks that described Section 4.2 of the main text, where the training hyperparameters for the SPI. machine in these equation discovery experiments operations allowed for three tested models and Table B.2 displays training hyperparameters for the SPL machine in the Nguyen's benchmark tasks.
also listed. Table B.1 presents the candidate mathematical
Moreover, the utilization of CFG in the SPL machine facilitates the flexibility of applying some prior knowledge including the universal mathematical rules and constraints. This feature empirically turns out to be an accessible and scalable approach for avoiding meaningless mathematical expressions. In the Nguyen's 's benchmark tasks, one multiple mathematical constraints are given the SPL machine. These constraints include
1. Only variables and constant values are allowed in trigonometric functions.
Table B.1: Candidate operators for each Nguyen's 's benchmark task. const denotes constant values.
Benchmark
Candidate Operations
N
+, +,-, x, sin) exp
Nguyen-2
+, +, 4, (inn)) exp
Nguyen-3
+,,, xx,+, cos(
Nguyen-4
+, +,-,x, -, x,+, cos(
s).iin)))...... (,sin()),epp exp
Nguyen-5
+, -, X, x, cos
(-i)),)())))))))) (.in()))expl
Nguyen-6
+. -, x, +, cos(
cos()),sin()))ess
Nguyen-7
+, -, +
(.),,in())),
(-)p((),
log(-), v-
Nguyen-8
+, -, X, + cos
(.),sin()))))))))
log(.). (-), V-
Nguyen-9
+, X, + cos(
(,ni())),, exxp
N0
+, -, X, + cos
11
+, -,-
cos
expp
lloo())) V
12
+, -, X, + co8
),sin()),
exxpp
1
+, -, X, cos
sinnn)))
e,exp( (,,i())
const
2g
+, -, x, C.0s
s,nin))),
exxpp(
const
5
+, +,-,X, x, +, cos
sinin (onn())))))))).c)
const
N
+, x, cos
(.sinn)),epp exp
l,g(()), V-, const
Ng
+, X, + cos
sin exp const
13

---

Published as a conference paper at ICLR 2023
Table B.2: Training Hyperparameter settings for the SPL. model in Nguyen's benchmark tasks.
Benchmark
Maximum Module
Episodes Between
Maximum
Maximum Augmented
Transplantation
Module Trunsplantation
Tree Size
Grammars
N
20
10,000
50
5
Nguyen-2
20
10,000
50
5
N
20
100,000
50
5
Nguyen-4
20
100,000
50
5
Nguyen-5
20
100,000
50
5
Nguyen-6
20
10,000
50
5
Nguyen-7
20
5,000
50
5
Nguyen-8
20
5,000
50
N
20
10,000
50
5
Nguyen-10 10
20
10,000
50
Nguyen-11
20
10,000
50
5
Nguyen-12 12
20
100,000
50
5
Nguyen-1
20
2,000
50
5
Nguyen-2"
20
10,000
50
5
Nguyen-5e
20
10,000
50
5
N
20
2,000
50
5
Nguyen-9
20
1,000
50
5
Table B.3: Mathematical constraints for the SPL machine in each Nguyen's benchmark problem.
Benchmark
Constraints Utilization
N 1
(1,3)
Nguyen-2
[1.3]
Nguyen-3
[1,3)
Nguyen-4
[1,3]
Nguyen-5
(2,3)
Nguyen-6
[2,3)
N
[2,3)
Nguyen-8
[]
Nguyen-9
(2,3,4)
Nguyen-10
(2,3.4)
Nguyen-11
[2,3)
Nguyen-12
(1,3,4)
Nguyen-1
[1,3]
Nguyen-2
(1.3)
5
[2.3]
Nguyen-8
Nguyen-9c 32)3,4)
2. Variables in trigonometric functions, logarithms and roots are up to the polynomial of 3.
3. Trigonometric functions, logarithms and roots are not allowed to form unreasonable com- posite functions with each other, such as (io...)(...)).
4. Some small integers (e.g. 1, 2) used directly leaves.
They can be easily implemented into the SPL machine by defining or adjusting non-terminal nodes or production rules the customized CFG. Constraints adopted by each task are shown in Table B.3
C RESULTS OF ABLATION STUDY
14

---

Published as a conference paper at ICLR 2023
We consider four ablation studies by removing the following:
(a) the adaptive scaling in reward calculation,
the discount factor 17" that drives equation parsimony in Eq. (2)
(c) module transplantation tree generation,
(d) all of the above three.
The resulting models are denoted with Model A, Model B, Model C, and Model D (note that Model D equivalent to the vanilla MCTS). The ablation study is performed on the 12 classic Nguyen's benchmark problems, where the recovery rate of each model is calculated. The results of the ablation study are summarized Table C.T It clear that module transplantation brings the largest gain in recovery rate. All the ablated models fail to uncover the Nguyen- 12 equation (the most difficult case), where the coefficient 1/2 needs to be represented by mathematical operators and symbols, c.g.. x/(e + z), // (y + v). etc.
D FREE FALLING BALLS WITH AIR RESISTANCE
This appendix section reveals more details on discovering the physical laws, in the context of relationships between height and time, for the cases of free-falling objects with air resistance based on multiple experimental ball-drop datasets (de Silva et al. 2020). The datasets contain the records 11 different types of balls, as shown in Figure dropped from a bridge, collected at a 30 Hz sampling rate. The time between dropping and landing varies in each case due to the fact that air resistance has different effects on these froe-dropping balls and induces divergent physical laws. Consequently, we consider each ball as an individual experiment and discover the physical law for each them. The measurement dataset of free-drom dreppiro ball is split into training set (records from the first seconds, 60 measurements) and a testing set (records after 2 seconds).
The physical laws of the three baseline models and distilled by the SPL machine from training data are exhibited in Table D.I These discovered physical laws are then applied to forecast the height of the balls the time slots the testing dataset. These predictions, in comparison with the ground truth trajectory recorded, are shown in Figure D.2 The prediction error is shown in the Table 3 the main text.
E NONLINEAR DYNAMICS DISCOVERY
This appendix section elaborates more details on the two governing equation discovery experiments presented in the main text, including the datasets and full results.
Table C.I: Summary of the ablation study results.
Benchmark
Expression
SPL
Model A
Model
Model C
Model D
Nguyen-1
2
100%
100%
100%
14%
12%
Nguyen-2
a rd + x² x
100%
100%
100%
0%
0%
3
+ +
x² +x
100%
100%
100%
0%
0%
Nguyen-4
6 + 5 r + x² +
99%
96%
92%
0%
0%
S
sin(x*) cos((r)
95%
95%
92%
92%
88%
Nguyen-6
sin(x²) (x +x²)
100%
100%
96%
100%
100%
Nguyen-7
In(x + In(x" 2
100%
100%
100%
100%
100%
Nguyen-8
100%
100%
100%
100%
100%
Nguyen-9
sin(x) ) sin(y²)
100%
100%
100%
100%
100%
10
2 sin x: cos(1)
100%
100%
100%
0%
0%
Nguyen-11
x
100%
96%
96%
92%
87%
Nguyen-12
r
3
+ 1/4
28%
0%
0%
0%
0%
Average
93.5 %
90.58%
89.67%
49.83%
48.92%
15

---

Published as a conference paper at ICLR 2023
Table D.I: Uncovered physics from the motions of free-falling balls by the SPL machine and three baseline models. Note that these formulas the raw equations produced by SRL. Further simplification helps better parsimony of the formulas.
Type
Model
Expression
baseball
SPL
((t) 47.8042 0.6253r 4.5383:"
Model-1
H(t) 47.082 1.456r 5.629r 0.3766
Model-2
((t) = 45.089
8. 8.156t + 5. 5.448 exp(
Model-3
H(E) 48.051
183.467 log (o0_
blue
SPL
((t) 46.4726
5.105t + 0.251
basketball
Model-1
H(t) 16.513
0.493t
3.9126 2 0.03c 7.963
Model-2
((t) = 43.522
5 5.306 exp
Model-3
((t) 46.402
84.791
(0g(cosh(0.3319))
green
SPL
((t)) 45.9087
4.1465:² log(cos((1))
baskethall
Model-1
((t) 46.438
0.34r
3.882r 0.055
Model-2
((t) = 43.512
8.043: + 5.346 cxp
Model-3
((t)) 46.391
124.424
volleyhall
SPL.
((t)) 48.0744
3.7772:²
Model-1 ((t) 48.046 08.046
0.362r
4.352r 0.2188
0.218
Model-2 ((t) = 15.32
7.317( + 5.037.3377 6.037
Model-3
((t)) 48. 124
107.8161og(roeh
107.816
bowling
SPL
((t) 46.1329
3.81732
0.28466
4.14 1 (xpp exp(-12. 12.453888)
ball
Model-1
H(t) 16.139
0.091r
- 3.504t
0.431e
Model-2
((t) = 13. 336
8.525t
5.676
Model-3
H(E) 46.342
247.371
Log (og(
golf ball
SPL
H(t) 49.5087
4.9633/ log(cos((t))
Model-1
((t)) 49.413
0.532t 0..10
Model-2
H(t) = 46. 356
8.918t + 5. exp1
Model-3
((t) 49.585 585
178.47
log(ccsh(0.233))
tennis
SPL
H(t) 47.8577
4.0374:
ball
Model-1
((t)) 47.738
0.658
0.325
Model-2
H(t) = 45.016
7.717t 5.212exx
Model-3
((t)) 47.874
114.19)og(ccmh(0022888
114.19
whiffle
SPL
H(t) 4.1563t
ball
Model-1
((C) 46. 969
0.574t
4..05
Model-2
((t)) = 44. 259
6.373t
4.639exx(01
Model-3
H(t) = 17. 17.062 34.083
94.083log(coehh(0-6622))
whiffle
SPL
H(t) 18. 6063 65.85333ss(
ball 2
Model-1
((t)) = 47.215
- 4.39
0.
Model-2
((t)) = 44. 113
-.73
7411
Model-3
H(t) = 17.255
38.299lo(ccoh((0.4777))
38.
yellow
SPL
H(t) 148. 9111991////
14.5828t ((og(ccos((()) -1455225//0oo((cco((())) 3.065)
whiffle ball
+48. 6092 (cg((ccccsad))///cgoooo//////
3.065)
Model-1
H(t) = 48. 613 0.0171 4.936 0.8226
1.936/
Model-2
H(E) 45.443 6.789 4.97
Model-3
H(t)
12.49log(coah((8886))
12.49
orange
SPL.
m(t) 1.6626t 77.8622xxpp-000061155
whiffle
Model-1
H(t) = 47.836
1.397t
3.822: 0.422:
ball
Model-2
((E)) 44.389
7.358t + (xp
Model-3
47.577
log(rosh(0.8954)
16

---

Published as a conference paper at ICLR 2023
DUKE 1AL
Figure D.1: The experimental balls that were dropped from the bridge (de Silva al. 2020). From left to right: golf ball, tennis ball, whiffle ball 1, whiffle ball 2, baseball, yellow whiffle ball, orange whiffle ball, green basketball, and blue basketball. Volleyball is not shown here.
baseball
blue basketball
green basketball
volleyball
25
20
20
bowling ball
golf ball
tennis ball
whiffle ball
23
L3
20
35
30
whiffle ball 2
yellow whiffle ball
orange whiffle ball
35.0
39
223
339
27.5
SPL discovered
253
Model-1 discovered
22.
Model-2 discovered
253
Model-3 discovered
173
True measurement
15.
2.
33
23
344
25
Figure D.2: Trajectories after 2 seconds predicted by uncovered physical laws.
E.
LORENZ SYSTEM
The 3-dimensional Lorenz system is governed by
o (yy xx) ý = z(p 2) y
1 xy Bz
with parameters o = 10, B = 8/3, and p = 28, under which the Lorenz attractor has two lobes and the system, starting from anywhere, makes cycles around one lobe before switching to the
17

---

Published as a conference paper at ICLR 2023
Noisy Measurement
Estimated Derivatives
150
50
-100
300
wwm Mmn
X
30
10
time
Figure E.1: Lorenz system for the experiment. Noisy measurement data and numerically estimated derivatives smoothed by Savitzky-Gobb Golay filter.
Table E.I: Discovered governing equations for the Lorenz system.
Model
Discovered Governing Equations
Eureqa
0.56 - 9.02x + 9.01y
= -0.047 + 18.79x + 1.86y 0.046xy 0.74xz
-3.04 2.23= + 0.882y
PySINDy
è -0.46 - 9. 9.17y
22.32x 0.15y 0.85zz
& 6.04 2.832 + 0.15.² 0.8lay
NGGP
-000 0.37x/ 0000://==0.0077::: 0.00074.²y
0.0047 - 10.02n 1 +
1001: 2yz 0.00011y
0.0000633
6.59c
26.36x - 1.5y 0.83z=
7
13.200/2211.00//2222111 13.08//2 4.526 -5,5 0.0038x
-1125x/ 5.00000..77000000022-----------0
-0.64 0.036y -
1 0.000211: 0.001ly= 0.00021x/2
8.01e 8.01e~yy/x +.00022y00 -.000
8.01e
0.000212yz + 0.00021
0.001y -0.0022100000 -.001
3.89c "22" 6.66e 2²²
SPL
9.966x + 9.961y
27.764r 0.942y 0.994xz
= 2.655z 0.996zy
True
10x + 10y
28x y 22
= 2 Zy
other and iterates repeatedly. The measurement data of the Lorenz system states in this experiment contains a clean signal with 5% Gaussian white noise. The derivatives of Lorenz's state variables are unmeasured but numerically estimated and smoothed by the Savitzky-Golay filter. The noisy synthetic measurement data and numerically obtained derivatives are shown in Figure
Table presents the distilled governing equations for the Lorenz system by the SPL machine compared with baseline models. It is observed that the SPL machine uncovers the explicit form of equations accurately in the context of active terms, whereas Eureqa, PySINDy and NGGP yield several false-positive terms the underlying governing equations. The predicted system responses (starting from a different initial condition) simulated from these uncovered equations are shown in Figure E.2 Although is hard to reproduce the most accurate coefficients due to the tremendous emrors induced by numerical differentiation of noisy measurement data depicted in Figure E. the SPL machine is still capable of distilling the most concise symbolic combination operators and variables to correctly formulate the parsimonious mathematical expressions that govern the Lorenz system dynamics. The predicted responses for the governing equations unearthed by the SPL machine simulate the system in decent manner.
18

---

Published as a conference paper at ICLR 2023
Eureqa
PYSINDY
SPL
S6
50
75
x
25 2
30
y
Figure E.2: Response prediction for seconds by identified governing equations (dashed plots) under a different validation IC of Lorenz system, in comparison with the ground truth trajectory (grey),
A
B
c
h
l2
0
time
Figure E.3: Double Pendulum system experiment and measurement data (Asseman et al. 2018): A. experiment setup. B. displacements of the moving masses. C. model the system with 0, and 02. D. angles of two masses transformed from displacements.
E.2 MOUNTED DOUBLE PENDULUM SYSTEM
The second nonlinear dynamics discovery experiment is a chaotic double pendulum system (Asseman) et al. 2018). The measured data, in form of videos, represents the chaotic motion of double pendulum on the device shown in Figure E.3/A filmed with high-speed camera. The positional data converted into angular form based on the geometry information (see the model shown in Figure
E.3C). The governing equations can be derived using the Euler Buler-Lagrange method, given by
(m1 + 12 + m2l202 cos(0, 02) + mzl2w3 sin 02) + (m1 + m2)g (0,) F1,
mzl202 + m cos(0; 02) m2lw? sin 02) m2g sin si2) F2.
which, by denoting w as the velocity, can be converted to the following state-space form:
ò W1,
d W2,
(8)
W1 = C1 cos( (0) + 62w3 sin( + ca sin(O1) + Ri 01,02,
W2 c cos A0) + czws (1) )) + c3 sin( R2(0,, 0z 0, 02)
where A = 0, 22 and R, (0,, 02 0, 82) and R 02 0, 0z) denote the damping terms for the last two differential equations.
The data source contains multiple video datasets. For this discovery, 5,000 denoised random sub- samples from 5 datasets used for training purposes, and 2,000 random sub-sample saob-samles from another 2 datasets for validation, and dataset for testing. Some prior knowledge guiding this discovery includes:
19

---

Published as a conference paper at ICLR 2023
Table E.2: Discovered governing equations of the mounted double pendulum by the SPL model.
Phase
Expression
W1
0.099143 cos A0)
69..774 0.515 - -1.36841 cos( A0) + 1.363w sin (A0) - -9.913 sin(O2) 0.032w1
0.103w3 sin A0)
W2
1. the two terms We cos( A0) for W1 and (os( for W2, which can be easily derived based on our prior knowledge on the system, are assumed known in the two governing equations. However, their coefficients are unknown and need be estimated.
2. the remaining of the formulas are potentially comprised of the free combination of W1, W2 w1, W2, as well as the angles (f), 82. A)) under the trigonometric functions cos(.) and sin(.).
3. velocities and relative velocities of two masses, well their directions (sign function) might contribute to the damping.
Based on the above information, the candidate production rules for ان and We equations are shown below, where non-terminal nodes are = {A, W, T, S} and C' denotes the placeholder symbol for the constant coefficient values.
A + A + A, A AX A.A A C. A A+ A
W + W x W, W + wh, W + w2, W + + W + W2,
+T T.T 01, T 02. + wh. - W2, + W1, S W2 - اn cos(01 - 02), A 42 cos 02).
- cos (T), (in)
- (S
Note that our prior knowledge can be casily incorporated in the proposed SPL machine to improve the discovery performance, rather than relying on the free combination of mathematical operators and symbols. The hyperparameters are set regarded one trail. independent trials are performed and the equations with the highest validation scores are selected as the final result. The uncovered equations are shown in Table E.2 They are validated through interpolation on the testing set and compared with smoothed derivatives, as shown in Figure E.4 The solution appears felicitous as the governing equations of the testing responses.
= 1, faaz 20, and 40,000 episodes of training are
F DISCUSSION AND FUTURE DIRECTIONS
The effectiveness the proposed SPL. machine is empowered by the following elements: (1) The use of MCTS enables the flexible representation of search space with customized computational grammars, composed of finite set of mathematical operators and symbols, to guide the search tree expansion. (2) The iitooratioo-expliitattin trade-off nature of MCTS is remarkably useful for searching the optimal mathematical expression tree. (3) The key adjustments, including the greedy search, the adaptive-scaled rewarding, the reward regularizer, and the expression module transplantation. make it possible to efficiently uncover the best path to formulate complex equations. (4) The SPL machine straighfoowwarly accepts our prior domain knowledge, any of constraints of the tasks in the grammar design while leveraging great flexibility in expression formulation.
While SPL shows huge potential in both symbolic regression and governing equation discovery tasks, there are still some imperfections to be improved. In this appendix section, few bottlenecks and their potential solutions are presented:
1. Computational cost. Computational cost for this framework is one of the major issues, especially when constant coefficient estimation is required. Evaluating the solution in the simulation phase happens very frequently for the MCTS algorithm where the policy selection relies heavily on large number of historical rewards. However, the constant coefficient value estimation, which requires repeated calls for optimization process and can be slow,
20

---

Published as a conference paper at ICLR 2023
ground truth
SPL discovered
200
MM mm
8, [rad/s4]
M
-200
8, [add//
500
Mm M mmm mmm/ M
-500
2.0
4.0
6.0
8.0
Time [Sec]
M
8, [Iad/s
-60
130
8, [rad/s-]
-130
30.0
32.0
34.0
36.0
Time [Sec]
Figure E.4: Discovered governing equations of the mounted double pendulum system on a different dataset in different time sections: in 2-8 seconds the masses in chaotic motions while in 30-36 seconds the masses tend to move periodically due to accumulative damping. 0₃ and 02 are obtained through smoothed numerical differentiation and predicted from the discovered governing equations.
Table E.3: Average training time (in seconds) of SPL and NGGP in the Nguyen's benchmark problems
Benchmark
SPL [s]
NGGP [s]
Nguyen-1
8.776
2.734
Nguyen-2
7.296
3,296
Nguyen-3
81.287
3.945
Nguyen-4
567.061
5.764
Nguyen-5
431.228
77.627
Nguyen-6
64.651
104.588
Nguyen-7
14.995
3.024
Nguyen-8
5.59
2.896
Nguyen-9
5.743
13,229
Nguyen-10
53.245
86.497
Nguyen-11
10.163
44.399
Nguyen-12
187,9
334.757
N
452.734
362.075
2
295.769
1188.215
N
2178.891
1365.777
Nguyen-8e
77.892
129.349
Nguyen-9e
2001.402
3066.41
21

---

Published as a conference paper at ICLR 2023
is needed for evaluation purposes. In particular, the constant coefficient value is estimated via concurrently solving an optimization problem, e.g., by Powell's conjugate direction method (Powelll 1964). For example, when the tree structure changes is updated, the optimization the constant coefficients should be re-performed simultaneously. The SPL. machine not the only symbolic regressor suffering from the computational cost in constant coefficient value estimation. In fact, the state-of-the-art symbolic regression model, the neural-guided GP (NGGP) (Mundhenk al. 202T). becomes much slower the Nguyen's ' benchmark variant tasks (see Table E.3). The current implementation of the SPL machine tries empirically avoid this issue by limiting the number of placeholders in a discovered expression and simplifying the expression before evaluation, but still cannot reach great efficiency. This bottleneck might mitigated if parallel computing is introduced to MCTS simulation phase.
2. Graph modularity underexamined. The current design of the SPL training scheme does not leverage the full graph modularity: modules are reached by transforming a complete parse tree into grammar. However, in some cases, there might be some influential modules appearing frequently as part of the tree. This type of graph modularity is described in the Al- Feynman method (Udrescu et al. 2020). Deploying more comprehensive graph modularity into the SPL machine will boost its efficacy the complex equation and nonlinear dynamics discovery tasks.
3. Robustness against extreme data noise and scarcity. Although it is observed that this reinforcement learning-based method able to unearth the parsimonious solution to the governing equations from synthetic or measurement data with a moderate level of noise and scarcity. is not effective when the data condition is extreme, or if there are missing values that make it challenging to numerically calculate the state derivatives. It is reasonable to investigate the integration between the SPL framework with a differentiable surrogate model built upon neural networks (Long al., 2018 Chen al. 2021) spline learning (Sun) et al. 2021) for further robustness in non nonlinear dynamics discovery tasks.
22


Reviews:
[{"review_id": "fclVQVOjtbT", "reviewer": "Reviewer_GVk6", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "The paper comprehensively reviews the different families of methods to tackle the data-driven discovery of nonlinear dynamics by comparing their advantages and disadvantages. Based on these discussions, the papers propose a new model, the Symbolic Physics Learner (SPL) machine, empowered by MCTS, to discover the mathematical structure of nonlinear dynamics. The paper proposes three changes: achieving a new expected reward, using adaptive scaling in policy evaluation, and transplanting modules with high returns. SPL shows its effectiveness in a wide range of tasks for discovering the mathematical structure of nonlinear dynamics.\n", "summary_of_the_review": "The paper is attractive to me since there are various strengths in terms of problem formulation, modeling, experiments, and writing.", "strength_and_weaknesses": "**Strengths**\n\n**1. The paper is well-organized and easy to follow.**\n\nThe organization and the writing of the whole paper are very attractive to me. The introduction and the background sections go over the different lines of existing work and the background one needs before diving into the proposed method in this paper. Figure 1 and Algorithm 1 clearly illustrate the high-level architecture and the underlying details of the method, respectively. Sections 4 to 7 discuss how the method could be applied to different benchmarks and tasks.\n\n**2. The proposed method adopts three useful adjustments to the conventional MCTS.**\n\n The proposed Symbolic Physics Learner (SPL) machine carefully incorporates the three adjustments to the conventional MCTS, which largely improve the prediction performance and stability.\n\n\n\n**3. The experimental sections are impressive to me.** \n\nThe experiments are conducted on four separate cases. The results show a consistent improvement on a bunch of tasks for SPL.\n\n\n**Weaknesses**\n\n**1. The model novelty.** \n\nI am a little bit concerned about the novelty of the proposed method. It looks like an incremental model that makes several modifications on top of a popular model.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "MXUoGlmMzF", "reviewer": "Reviewer_X2Dz", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "This paper proposes Symbolic Physics Learner (SPL), a method for symbolic regression and discovery of dynamics.\nThe authors formalize mathematical operations using expression trees and context-free grammars,\nand employ Monte Carlo Tree Search (MCTS) to explore the space and select\noptimal solutions.\nFurthermore, they propose adjustments to conventional MCTS, including adaptive\nscaling and module transplantation.\nThe authors demonstrate the efficacy of the proposed method in a number of\nsettings and achieve superior performance compared to other\nstate-of-the-art methods.\n", "summary_of_the_review": "I recommend that this paper is accepted. It is a well written paper with important technical contributions and strong quantitative results. A comparison with related works on MCTS for symbolic regression could help the readers appreciate its contributions more. More thorough ablation studies on the introduced components could further clarify their efficacy.", "strength_and_weaknesses": "The paper is well written and easy to follow.\nThe authors perform a multitude of experiments on symbolic regression and\ndiscovery of nonlinear dynamics.\nComparison with state-of-the-art related works clearly demonstrates the\nshortcomings of related works and the efficacy of the proposed method.\n\nI am missing a more detailed description and direct comparison with related works on MCTS for symbolic regression.\nHow is your work different from theirs?\nSimilarly, I would expect a quantitative comparison with existing MCTS methods for symbolic regression.\n\nThere is one ablation study on the importance of adjustments to MCTS (sec. 4.2),\nyet the results are barely mentioned, and I miss some analysis on them.\nYou could have included the results (and the analysis) in the appendix.\nIt is surprising that removing each adjustment separately results in a performance lower than 60%.\nCan you provide more detailed results, and perhaps explain why that might be the case?\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "8SfryJbtkJ", "reviewer": "Reviewer_k15E", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "This manuscript proposes a novel symbolic physical learner (SPL) that extracts analytical formulas governing nonlinear dynamics from limited data. The SPL uses a Monte Carlo tree search (MCTS) agent to search for optimal expression trees based on data.\nThe specific differences of the SPL with respect to previous studies are the following three points. (1) replacing the expected reward in UCT scores with a maximum reward to better fit the equation discovery objective, (2) employing an adaptive scaling in policy evaluation which would eliminate the uncertainty of the reward value range owing to the unknown error of the system state derivatives, and (3) transplanting modules with high returns to the subsequent search as a single leaf node. \nThe effectiveness and superiority of the SPL machine are demonstrated by numerical experiments comparing it to a baseline of state-of-the-art symbolic regression algorithms.", "summary_of_the_review": "Symbolic regression is an important technique for future physics research using machine learning. In addition, equation tree search using reinforcement learning, which was improved in this study, is an important technique that has shown its usefulness in other fields as well. As such, this research is important. On the other hand, the description and organization of the paper and the setting of numerical experiments are still insufficient, and I think that the paper should be accepted after these points are improved.", "strength_and_weaknesses": "strengths:\nThe strength of the research is its stable symbolic regression compared to prior methods.\n\nweaknesses:\nThe following three points are drawbacks\n1 There is no discussion or theoretical analysis of why the proposed method is effective. Therefore, I cannot eliminate the suspicion that the effectiveness of the proposed method over the prior methods depends on the superiority of the hyperparameter tuning.\nI think it is good to show what kind of \"prior physics knowledge in nonlinear dynamics\" can be implemented by the proposed algorithm, and provide numerical experiments to demonstrate its effectiveness.\n\n2 Where the stability of the estimation results due to hyperparameter settings is unclear.\nPlease explain the optimization method for hyperparameters and the stability of the estimation results with respect to variations in hyperparameters.\n\n3 Where it is unclear whether the hyperparameter tuning of the prior method is appropriate.\nIt is same as above point. In particular, we would like to know why the results in Table 1 do not match the results of similar experiments presented in the paper proposing the prior method. (Table 1 in [Mundhenk et al., 2021])\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "3iS_lBYr2HU", "reviewer": "Reviewer_uNr2", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "The authors propose a Monte Carlo tree search (MCTS) algorithm to generate optimal expression trees based on measurement data. The method is validated using Nguyen’s symbolic regression benchmark task (Uy et al., 2011), along with several other benchmarks.\n\n\n\n", "summary_of_the_review": "This paper presents a novel approach to symbolic regression, and shows good improvement over existing methods. Overall, a strong contribution.", "strength_and_weaknesses": "Strengths:\n\nThis is a strong paper, and has clear novelty in the area of symbolic regression.\nThe paper has extensive comparison to competing methods, with good results.\n\n\nWeaknesses:\n\nThe theoretical aspects of the paper could be improved.\n\nThe theoretical basis of this approach should be highlighted in comparison to existing methods of symbolic regression. In addition, since it is based on stochastic methods for which guarantees have been given for other applications, this is a clear lack. Questions include: \nWhat type of sampling distributions are used, and can these be modified during search to improve the returned function?\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: notable-top-5%