PhysPDE: Rethinking PDE Discovery and a Physical Hypothesis Selection Benchmark
Authors: Mingquan Feng
Yixin Huang
Yizhou Liu
Bofang Jiang
Junchi Yan
Venue: ICLR 2025 Poster (2025)
Keywords: AI4Science
Physics
PDEs
PDE Discovery
Primary area: datasets and benchmarks

Abstract: Despite extensive research, recovering PDE expressions from experimental observations often involves symbolic regression. This method generally lacks the incorporation of meaningful physical insights, resulting in outcomes lacking clear physical interpretations. Recognizing that the primary interest of Machine Learning for Science (ML4Sci) often lies in understanding the underlying physical mechanisms or even discovering new physical laws rather than simply obtaining mathematical expressions, this paper introduces a novel ML4Sci task paradigm. This paradigm focuses on interpreting experimental data within the framework of prior physical hypotheses and theories, thereby guiding and constraining the discovery of PDE expressions. We have formulated this approach as a nonlinear mixed-integer programming (MIP) problem, addressed through an efficient search scheme developed for this purpose. Our experiments on newly designed Fluid Mechanics and Laser Fusion datasets demonstrate the interpretability and feasibility of this method.

Published as a conference paper at ICLR 2025
PHYSPDE: RETHINKING PDE DISCOVERY AND A PHYSICAL HYPOTHESIS SELECTION BENCHMARK
Mingquan Feng', Yixin Huang', Yizhou Liu', Bofang Jiang' Junchi Yan
'Shh of Computer Science & Sch. of Artificial Intelligence, Shanghai Jiao Tong University
Sch. of Physics and Astronomy, Shanghai Jiao Tong University
(ggggmingquan, cindyh1103, jiangbofang, yanjunchi @ajjtu. edu.err https: / githut..hhhhco. com/ Foam/Fende Fwwgg..igguannnjssuueeeuu
ABSTRACT
Existing works on recovering PDE expressions from experimental observations often involve symbolic regression. This method generally lacks the explicit incor- poration of physical insights, which weaken the interpretations and effectiveness, especially in the presence of large noises. Recognizing that the primary interest of Machine Learning for Science (ML4Sci) often lies in understanding the underlying physical mechanisms or even discovering new physical laws rather than simply obtaining mathematical expressions, this paper introduces novel ML4Sci task paradigm. focuses on interpreting experimental data within the framework of prior physical hypotheses and theories, thereby guiding and constraining the dis- covery of PDE expressions. Technically, the approach is formulated as a nonlinear mixed-intege programming (MIP) problem, addressed through an efficient search scheme developed for this purpose. The experimental results on our newly designed Fluid Mechanies and Laser Fusion datasets demonstrate the interpretability and feasibility of our method. Source code and benchmarks are publicly available.
INTRODUCTION
Partial Differential Equations (PDEs) serve as powerful and pervasive tools for comprehending and describing intricate systems across physics, engineering, applied mathematics, and various other domains. Instead of solving PDEs, there has been growing interest in the task of recovering unknown mathematical equations from observed data, especially within the machine learning community. This endeavor is particularly valuable when the observed data is noisy and imperfect, mirroring real-world conditions. However, existing techniques ranging from reinforcement learning (Du et al., 2023), genetic programming (Chen et al., 2022), and sparse regression (Wentz & Doostan, 2023) exhibit heightened sensitivity data noise. It is crucial to note that previous research has simply focused on producing a combination of mathematical symbols and operations while disregarding their fundamental physical implications. In practice, users of PDE recovery tools typically seek to grasp the fundamental physical mechanism rather than merely fitting mathematical models, let alone their fragility to noise. Additionally, note that the field has yet to uncover any new physical laws or PDE systems through the existing literature on PDE discovery. Based on our understanding, physicists typically do not require 're-discoerring' expressions, as they already possess well-established expressions in most scenarios. Furthermore, even when they identify new expressions through purely data-driven methods, interpreting these findings within a physics framework remains essential. Ultimately, their fundamental pursuit lies in the physical interpretation of observed data.
To bridge the gap between mathematical expressions and physical interpretability for data-driven PDE recovery, we propose innovative strategy. It leverages physical hypotheses and laws to interpret observational data to uncover new expressions that enhance our understanding of observed systems. These task paradigms and algorithms are compared in Figure 1. Take fluid dynamics for example, our PhysPDE paradigm is based on the conservation of mass, momentum, and energy, and candidate hypotheses such constant viscosity (Newtonian fluid). The output of our paradigm is PDEs along
"Correspondence author who is also affiliated with Shanghai Innovation Institute. This work was partly supported by NSFC (62222607, 623B1009).

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Published as a conference paper at ICLR 2025
PDE Solving
Observational Data
Solver
A/
PDE Solution
du
PDE Discovery
Observational
Symbolic Regression
PDE Expression
ac
PDE Interpretation
Symbolic Regression
Du Dr
PDE Expression
Observational
Integrated approach (proposed)
Hypothesis Selection
Laws & -Conservation Law
Interpretation
-Neratorian flud Hypothesis, Flow Regime
Non-Newrtniion
Caminer Flow
du
Pipelined
Observational Data
Symbolic Regression
Expression
approach
(baseline)
Laws & Hypothesis
Large Language Model
Interpretation
Figure I: Comparison of different PDE learning paradigms. This paper focuses on the PDE Interpre- tation setting in the bottom box, which, our best knowledge, is in contrast to existing literature devoted to the other two settings. Although a pipeline of existing algorithms can interpret PDE data to some extent, propose stronger integrated approach directly solving the combinatorial problem.
with selected hypotheses. This paradigm shift seeks to align mathematical models more closely with their underlying physical realities, facilitating deeper comprehension of complex phenomena.
The possible applications include inferring key factors affecting the efficiency of heat exchangers (e.g.. turbulence) (Wang et al., 2023) and aiding in the diagnosis of physical phenomena in nuclear fusion that cannot be directly measured (e.g., magnetic fields) (Peebles et al., 2022). In these applications, existing mathematical formulas and hypotheses are relatively comprehensive and do not require "rediscovery." What scientists need is to use this knowledge to explain experimental results. The highlights of this paper include:
1) New Task Formulation:
Going beyond the prevalent regression of PDE expressions from observations in existing literature, in this paper we introduce a new ML4Sci task, which standing upon the shoulders of suite of basic physical laws and hypotheses, employing learning to identify the hypothesis that best aligns with the data.
2) New Algorithms: Besides a simple pipeline of existing models, we design a new integrated algorithm for the task. We formulate the task above as bi-level mixed-integer optimization problem, consisting of two layers: hypothesis selection at the outer level and expression recovery at the inner.
3) New Datasets and Experiments Design: Following the proposed paradigm, we present two new datasets, Fluid Mechanics and Laser Fusion. Experiments across multiple scenarios demonstrate the feasibility and effectiveness of our approach.
2 RELATED WORKS
PDE Solving. Solving PDEs that describe complex system dynamics is crucial in scientific disciplines like computational physics. The goal is to find a solution function that satisfies the equations given initial and boundary conditions. Representative datasets include PDEBench (Takamoto et al., 2022), PDEArena (Gupta & Brandstetter, 2022). and CFDBench (Luo al., 2023). Classic methodologies, including finite difference (Strikwerda, 2004), finite volume (Eymard et al., 2000), and finite element (Zienkiewicz et al., 2005), concentrate devising numerical schemes that simplify the problem into linear equation systems. Deep-learning-bars based solvers such as PINN (Raissi et al., 2019) and DeepBSDE (Han al., 2018) have been emerging, extending the capability of numerical solvers.
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Physies Theory
Hypotheses Selection
PDE System
Observational Data
Mechanio
Rud Macharic
defined System
2M
V) I-pl+
A2 V (p()))
U
Selected hypotheses
Compressible Flow,
Constitutive
0.01(1
mus(y)
Expression
log(y))
Symbolc Regression
(Optional)
h Newtonian E (0.1).
Hypothesis
Compressible (0.1).
Selection
eeq-Threshold Optimization
is New Constitutiv Eq E [0.1)...]
Tree Search
Operator [sin, Ing paw 1)31
Decision Forest/ Tree
Natroriant
MCTS
Initializztion
Search Node
seeking Obj
minn MX FiF) Al1
Compuss
Neg MSR Reward
Convergence
Threshold
Equetio
Nenaraa /quation/
Figure 2: Overview of the proposed hypothesis selection framework, selecting hypothesis and constructing PDE system from both physics theory and observational data. The constructed PDE system consists of equations from selected physics hypotheses and newly discovered expressions.
Table 1: Comparisons with related PDE solving, discovery, and interpretation tasks in physics
(Takamoto al., 2022; Raissi al., 2019; Bhamidipaty al., 2024; Rudy al., 2017). To our
knowledge, our work provides the first dataset and algorithm for the PDE interpretation task.
Task
Physies Sub- Sub-discipline
Dataset/Tool
Algorithm
PDE Solving
Computational Physics
PDEBench DynaDojo PhysPDE
PINN
PDE Discovery
Phenomenology
PDE-FIND
PDE Interpretation (ours)
Theoretical Physies
HSTS
PDE Discovery. In the domain of physics phenomenology, the pivotal problem is deducing the mathematical symbolic expressions (i.e. symbolic regression) of PDE systems from observed data. There is currently no specifically designed PDE Discovery Benchmark. However, DynaDojo (Bhamidipaty et al., 2024), as general-purpose PDE Benchmark, can be used to some extent for this purpose. Data-driven methodologies, including reinforcement learning (Du et al., 2023), genetic programming (Xu et al., 2020; Chen et al., 2022), and sparse regression (Rudy et al., 2017; Messenger & Bortz, 2021; Wentz & Doostan, 2023), have been prominently applied to this end. These approaches adeptly identify the mathematical constructs of PDEs but frequently overlook the integration of physical principles.
PDE Interpretation. critical examination existing data-driven methodologies for PDE discovery underscores a pervasive limitation: their insufficient focus ensuring interpretability within concrete scientific theory framework (e.g. physical theorems). This oversight, as highlighted by Zhang al. (2023) and Faroughi al. (2023), manifests pronounced divide between mathematical precision and physical interpretability. Further exploration by Cornelio et al. (2023) and Cory- Wright et al. (2023) accentuates this concern, advocating for deeper comprehension of the physical underpinnings of equations beyond their mere mathematical representation. Despite these insights, to the best of our knowledge, universally recognized methodology for the systematic interpretation of nonlinear PDE system data remains elusive.
PDE INTERPRETATION TASK FORMULATION
This study introduces PDE interpretation, a novel task aimed at the recovery of physics hypotheses as well as expressions from observational data. The cornerstone named Hypothesis Selection, distin-
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guished by its capacity to conceptualize systems of PDEs as amalgamations of physies hypotheses and theories, effectively narrows the chasm between physical understanding and mathematical depiction. The subsequent section delineates the technical aspects of our problem formulation and method.
Our approach grounded in two fundamental physics principles: (i) physics formulas, like PDEs/ODES, stem from physics laws, and (ii) physics laws are derived from hypotheses and basic the- ories (Carcassi & Aidala, 2022). We aim to recover valid PDEs that adhere to underlying hypotheses, moving beyond mere data fitting. We define our task as selecting the best-fit set of hypotheses from "hypothesis universe" to fit a system of PDEs, conceptualized as a two-level min-min optimization:
min min 1.0% m.8m
f(X, m. om.)
s.t. C(h, m. Pn. 0m) 2
Outer level (hypothesis selection): The outer optimization targets the decision variable and physics parameter On h is binary vector encoding the selection of physies hypotheses, while O,, is a vector comprising undetermined coefficients and constants of the system.
Inner level (symbolic regression, optional): Selecting the "discover new partial expressions" hypothesis triggers the inner optimization level, aimed at discovering new expressions within a PDE expression segment. The algorithm can modify the equation directly but confined only to the specific part of the equation that the selected hypothesis allows. At this level, m. being an integer vector, denotes selections from library of basis functions or operations. These choices instantiate a mathematical expression, characterized by coefficients represented by the real vector Om.
The combined decisions h. m and parameters 0,, 0m, informed by physics prior knowledge, collec- tively define the PDE system Eq and its constraints C. The abstract loss function is:
((x, Eq) = MSR(X, Eg) + Reg(Eg)
We combine the mean squared residual (MSR) between observation and recovered system and regularization (Reg) to enforce simplicity per Occam's 's razor principle (Walsh, 1979). For brevity, the arguments of both F.q and C' are omitted. The constraints C' encode the physical validity constraints of variables, for example, the mutual exclusiveness of hypotheses, and bounds of parameters. This problem framework is versatile, accommodating various solvers. We next provide a detailed account of our specific definition and algorithmic approach.
Decision Forest. The hypotheses exhibit complex dntrr-dpenndecces, e.g. mutual exclusiveness and hierarchy. We opt represent these dependencies via decision forest rather than as constraints in the optimization problem Eq. 1, aiming to improve readability and streamline the search.
We define decision as set of choices among a group of mutually exclusive hypotheses. For instance, encompasses two hypotheses, Bewtonian__Fa FluFFluio and
the decision labeled 1es_Newtonii Nen_Newt onian Fluid. Subsequent decisions are defined as those influenced by the choices made in the preceding decisions. For instance, the decision Syyp__No__Neete oniiinniii an is induced by the second choice made in the Ts_Newtonial decision. Once the non-Newtonian hypothesis is selected, all descendant searches are non-Newtonian. Based on the above definitions, we organize decisions hicrarchically as a forest, with decisions being nodes. Subsequent decisions are marked as child nodes of the preceding decisions. Nodes lacking parent nodes are designated as root nodes.
Under this framework, a closed subset of physics hypotheses constitutes a tree, while the aggregate of these subsets forms forest. A root-to-leaf path in any tree represents partial decision instance, and the ensemble of such paths across all trees defines complete decision instance. Given the forest, there is one-to-one mapping between the full decision instance and valid hypothesis vector h.
complete decision, combined with existing physics theorems, can uniquely derive a system of PDEs. We automatically implement such derivation via the symbolic computing tool (Meurer et al., 2017). This encoding scheme allows for a comprehensive representation of physical concepts and their mathematical counterparts. Figure illustrates an example, showeasing the decomposition of compressible Savier-Stoker equations into two hypotheses: Compressibility and Newtonian fluid. Figure visualizes the details of PDE construction from hypotheses.
The decision forest is a simplified model postulating that each decision node has a single parent and that hypotheses across decisions are mutually inclusive. In fluid mechanics as considered in the paper
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1. Laws of Physics
Physical Quantities
0p
Mass
+ V. (pu) =0
2
Consorvation
0t
228
uVV u
3
Momentum
P + + = V- VlaaVV-V---ppll
F
4 = v28 S
0T
Energy
pCpl +u VT) V (AVT) + Qp + Qui
Vu (Vu)*)
Conservation
0t
3. Selected Hypotheses
4. PDE System
Compressille-aabee
0p/Ot Vp
=u=
Newtonian=Falise
Non-fiewtonian
New Eq
(STOP Alg.)
# 0.01 cos(4)/4 log(
Type=2 (Unknown)
Body False
F = =0
P 0t + p(u V)u = p-pp
du
Turbulent-Falie
Q2 Q ud =
bothermal- True
0T/00 0, VT 0
Figure 3: Given observational data and physics theories including 1) laws of physics, 2) physical quantities, and a search space of hypotheses forest (not shown here), our algorithm can provide best-fit set of 3) selected hypotheses and 4) PDE system, possibly with newly discovered expressions. We start from the full PDE system of the unchangeable physics theorem (e.g. conservation laws in fluid mechanics). On top of that, the derivation is done by plugging the expression of the selected hypothesis into the PDE of conservation laws. The example PDE is an incompressible version of our S2 Fluid Mechanics dataset. brief introduction to fluid mechanies and the construction of these equations is provided in Appendix C.
for example, exceptions to these hypotheses are addressed by an ad-hoe filter function for simplicity. Modeling these exceptions accurately may require more sophisticated structure, e.g. a decision graph, which we leave for future work.
Loss Function. The loss function is defined with:
(Pqh (X.)
MSR(X, Eq)
(3)
size(X)
1.j
Reg( Eq) = XE size + An size(h) + Ao, |0 llo + Am||mmlll
where denotes hyperparameters. between the left L and right Rt sides of each equation Eq', among all collocation points X,. Average pooling Pool a simple weak-form (Messenger & Bortz, 2021), serves as a de-noising technique. The regularization term Reg(Eg incorporates the equation system's 's complexity of hypothesis selection and new expression discovery. The former is quantified by weighted sum of the total equation size, decision vector size, and count of active physies parameters while the latter measured by the sparsity of the chosen basis functions or operators.
calculates the residual as the averaged squared differences
Both the outer hypothesis selection and inner symbolic regression problems have exponentially large search spaces, necessitating specially designed baseline algorithms. We developed a Monte Carlo Tree Search (MCTS)-inspires algorithm called Hypothesis Selection Tree Search (HSTS) for the outer problem, and a Sequential Threshold OPtimization (STOP) algorithm for the inner problem. Due to space limitations, detailed designs of these algorithms are provided in Appendix F.
FLUID MECHANICS DATASET AND EXPERIMENTS
4.1 EXPERIMENT SETUP
Physics Hypotheses. In research, while the proposed methodological framework is adaptable across multiple physics domains, we specifically target fluid mechanics, a focal area within learning- based PDE discovery studies. We design 16 decision points across 38 hypotheses based on textbook
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(Batchelor, 1967), encapsulating fundamental physics concepts such as flow regimes, compressibility. constitutive equations, and non-isothermal flows. The validity of our design is also acknowledged by the mathematician expert in this area. We also explore hypotheses concerning new expressions of constitutive equations. Details on the complete set of hypotheses are provided in Appendix A.
Scenarios. We design three different scenarios constructed from three representative sets of hypothe- ses: 1) S1: Newtonian Fluid, 2) S2: Non-Newtonian Fluid with Unknown Constitutive Equation (Figure 3), 3) S3: Non-isothermal Flow
Scenario S1 elaborates on the derivation of the Navier-Stokes equations, which are foundational to fluid mechanics (Constantin & Foias, 1988). The flow is set to be compressible, isothermal, non-turbulent, and non- non-gravity. The viscosity is constant (Newtonian fluid assumption) μr = 0.01. Scenario S2, a variation of SI, evaluates the model's capability of discovering new expressions by substituting the constitutive equation μ 0.01 in S1 with an unknown new expression 0.01 + 0.01 cos(7)/7 + 0.01 log(7), where y is the shear rate. S3 deviates from S1 by switching to the non-isothermal hypothesis. It leads to thermal-hydraulee equations, which are widely applied in nuclear technology (Zhang et al.. 2018) and geophysics (Bächler & Kohl, 2005). The derived PDEs for these scenarios are solved via COMSOL Multiphysics commercial simulation software. Observable physics variables include velocity, density, pressure, and temperature, each modeled as two-dimensional in both space and time. For each scenario, datasets for training and testing are generated under varying boundary conditions, ensuring identical sizes.
Performance Metrics. We use the recall and precision metric. The recall is defined as the percentage of the number of correctly learned decision over the true decision, R = blearn @ birue b and precision is P |lutaann berue In/ //ellrrar where element-wist product. Vectors b are binary vectors denoting the selection of hypotheses or PDE terms. Accuracy is measured by MSR on dataset splits and efficiency by training wall-time.
4.2 DATA GENERATION AND SAMPLING
S1 and S2. The study investigates fluid dynamics within rectangular domain of dimensions
1 0.3 m, with the fluid initially set to density of 1 kg/m², x-velocity sin(#x), y-velocity sin (, and pressure at zero. Boundary conditions are implemented as no-slip walls, with the bottom wall moving left at m/s, and the top wall moving right at 1 m/s for Scenario I (SI) and 2 m/s for Scenario 2 (S2), with pressure fixed zero at the bottom-left corner. The simulation employs finite-element method on a mesh resulting in approximately 30,000 degrees of freedom, with results sampled on a 100 x 100 grid over a temporal range(O, 0.004, 1).
S3. This scenarios is adapted from COMSOL Application Gallery?. 2 The study models fluid dynamics within 0.005 m 0.04 m rectangular domain, initially set with a density of 997 kg/m², velocity of zero, pressure pg (0.04 - y). and temperature 293.15 15 K. A half-circle heater of radius 0.0025 m, positioned at (0, 0.015), heats the fluid. Boundary conditions are periodic on the left and right, an inlet at the bottom with 0.005 m/s inflow velocity and 293.15 K temperature, and outlet at the top designed to suppress backflow. Heater temperatures are set to 303. K in Scenario (S) I) and 323. 15 K in Scenario 2 (S2). The numerical solution employs a finite-element method focusing the upper half of the rectangular domain, encompassing approximately 25,000 degrees of freedom. Results are sampled over 50 x 100 spatial grid within temporal range (1, 0.005,
4.3 HYPER- PAREMETRRSE SETTING
Loss The pooling size is 5. The coefficients are λ = 10-7, No, = 10-5, An = 10 5 Am 10-5
HSTS For S1 and S2, the maximum number of rollout is 30, the e of risk-seeking object is 0.5; while in the S3, the the maximum number of rollout is 60 and e = 0.05. The bound Imin = 0.0001, maxx = 1. The exploration weight in the UCT is 1. The 10-core mrotisproc is applied at node level, using the virtual loss technique.
STOP We set the maximum iteration depth at 3, the l2 regularization λem 0.01, and the weight tolerance T = 0.005. The parameters Om are scaled to j0.05, 0.05) for numerical stability.
www. comsol 1aw.olccoomm ccom
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PDE Parameter Estimation The optimization algorithm is SLSQP implemented in SciPy (Virtanen et al., 2020), with a timeout of 5 minutes for S1 and S2, and 15 minutes for S3. The parameters are first initialized from normal distribution (0,, 1). For parameters with different orders of magnitude, e.g. the heat capacity at constant pressure Gp E [4100, 4300), we further add biases to the initialization, e.g. bias of 4200 in Cp case. The biases and parameter ranges informed by physics knowledge.
Experiment Protocols. We perform 5-fold cross- tross-validation on the training set for performance estimation, highlighting the top three decisions with the highest average validation loss. The hyper- parameters of the search algorithm are chosen without tuning, and variable initialization 0 follows normal distribution. Computations are executed on a remote server using 10 CPU cores con- currently. For the STOP algorithm, the function library comprises second-orde polynomials of {sin, cos, log, (-)(- To restrict the wall-time of ES, we limit its maximum wall-time to 24 hours, only include the new expression hypothesis in the search space in S2, and incorporate STOP for internal expression discovery. Other numerical settings, including rrper-parameter configurations, are detailed in Appendix 4.3.
Baselines. As there is no baseline directly available for our hypothesis selection problem, we designed an Exhaustive Search (ES) baseline that randomly enumerates the search space to demonstrate the efficiency of the proposed search algorithm. A variant of ES is ES-TL (Exhaustive Search with Time Limit), which terminates the ES under the same runtime our method. Also, the performance of the Ground Truth (GT) hypothesis is reported to evaluate the accuracy of parameter estimation. For the PDE symbolic regression baseline, we adopt the popular PDE-FIND (Rudy al., 2017), which is still in active maintenance by the open-source community. We also tried other more recent peer methods like genetics algorithm based SGA-PDE (Chen et al., 2022) and reinforcement learning based DISCOVER (Du et al., 2023) yet it remains in vain and these methods perform less competitive than PDE-FIND in our challenging setting. We further apply GPT-40 (Achiam et al., 2023), a Large Language Model (LLM), to extract physics properties from the underlying system. In our pre-experiment, GPT-4o alone cannot infer the physics hypothesis directly from raw data, possibly due to the large input size and combinatorial nature of the task. Thus, we fed PDE expressions (discovered by SINDy or configured manually) and physics hypothesis priors into GPT-4o to infer the correct physics hypothesis, corresponding to the pipelined approach in the Figure 1.
4.4 RESULTS AND DISCUSSION
We evaluate our framework by answering the following research questions (RQs):
RQI: Does HSTS select the hypotheses correctly and efficiently?
RQ2: Is HSTS able to recover known PDEs and to discover unknown expressions?
RQ3: Do the PDEs recovered by HSTS have acceptable interpolation and extrapolation accuracy?
RQ4: How does the pipelined baseline compare with the HSTS (integrated approach) in physics interpretation?
(RQ1) Hypotheses Selection. Table 2,3, and 4 resent the hypothesis selection performance compari- son results of HSTS with the ES baseline. With 3500 valid decision combinations the search space, HSTS successfully identifies nearly all hypotheses across all scenarios, achieving the same top-1 recall and precision as ES, demonstrating its superior capability in terms of physics interpretability. Notably, HSTS's training wall-time is significantly lower than ES's, by an order of magnitude, highlighting the efficiency of our search scheme. In Table 2. HSTS outperforms ES in top-2 and top-3 performance metrics, which is attributed HSTS's strategy of extending the search space locally, resulting in clustering of visited nodes and similar top results. For instance, in scenario S1, HSTS's top-1 result the correct solution, with its top-2 and top-3 variations differing by only one or two hypotheses. In contrast, ES's global search approach tends yield more diverse top results. Detailed listings of the selected hypotheses for each scenario are provided in Appendix D.
(RQ2) PDE Recovery and Discovery. Table 9 showcases the recovered PDE terms and parameters, while Table 5 compares the performance of the HSTS (top-1) with that of PDE-FINI in identifying PDE terms. Our results demonstrate that HSTS accurately recovers the majority of terms and parameters across all scenarios, including successfully identifying new expressions in S2. Conversely, PDE-FIND tends overfit the data with overly complex expressions, resulting in low recall and
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Table 2: RQ1 on S1: Hypothesis Selection.
Model
Recall
Precision
Wall-time(h)
Top-1
1.00
1.00
Table 5: RQ2 on all Scenarios.
Ours
Top-2
0.86
0.86
0.14
Model
Scenarin
Recall
Precision
4. Complexity
Top-3
0.71
0.71
$1
1.00
1.00
109
Top-1
0.86
0.55
ES-TL
Top-2
0.71
0.45
0.28
S3
1.00
0.93
180
Top-3
0.57
0.31
SI
0.88
0.04
1423
Top-1
1.00
1,00
PDE-FIND
0.20
0.02
1377
ES
Top-2
0.86
0.55
16.68
S3
0.22
0.01
2176
Top-3
0.86
0.55
Table 6: RQ3 on S1: Mean Squared Residuals.
Table 3: RQ1 on S2: Hypothesis Selection.
Model
Train
Valid
Test
Model
t Recull
Precision
Wall-time(h)
Ours
x10-4
5.22
x10-
x10-4
GT
2.89 x10
5.09
x10
4.50 10
Top-1
1.00
1.00
PDE FIND
1.58 x 10
2.79 10-4
1.48 x10-1
Ours
Top-2
0.86
0.86
4.10
DISCOVER
0.123
0.132
1,231
Top-3
0.86
0.55
SGA-PDE
24.43
Top-1
0.71
0.56
ES-TL+
STOP
Top-2
0.57
0.31
4.36
Top-3
0.57
0.31
Table 7: RQ3 on S2: Mean Squared Residuals.
ES+
Top-1
1.00
1,00
24.00 (Timcout)
Model
Train
Valid
Test
STOP
Top-2
Top-3
0.86
0.86
0.55
0.55
Ours
3.24 x 10-3
4.49 x10-3
4.75
x10-3
GT
3.35 x10-3
10-3
4.29 x 10-a
PDE FIND
7.21 x10-a
8.44 x10-
1.66 x10-1
DISCOVER
0,144
0.158
1.232
Table RQI on S3: Hypothesis Selection.
SGA PDE
Model
Recall
Precision
Wall-time(h)
Top-1
0.91
0,91
Table 8: RQ3 on S3: Mean Squared Residuals.
Ours
Top-2
1.00
3.12
Top-3
0.82
0.82
Model
Train
Valid
Test
Top-1
0.82
Ours
9.86 x 10-3
9.38 x 10-3
1.74 x 10-2
ES-TL
Top-2
0.73
0.62
3,30
GT
1.03x10 2
9.83x
1.76x10
Top-3
0.82
0.82
FIND DISCOVER
1.93 x10
2,06
x10-1
1.675
1.682
2.312
Top-1
0.91
0.91
ES
Top-2
1.00
1.00
21.11
Top-3
0.82
0.82
extremely low precision. Despite careful tuning of PDE-FIND's parameters and the application of denoising techniques (Messenger & Bortz, 2021), performance improvement remains elusive. We attribute PDE-FIND's limitations to its lack of embedded physics knowledge, leading to significantly larger search space compared to HSTS and, consequently, heightened difficulty in problem-solving.
(RQ3) Interpolation and Extrapolation. To assess the generalization ability of selected hypotheses and the induced PDEs, we conduct interpolation and extrapolation tests on both training and test datasets, which are generated from the same governing equations with different boundary conditions. Table 6, 7, and 8 show the MSR for all data splits. Our findings affirm that HSTS- recovered PDEs consistently show high accuracy in both interpolation and extrapolation, closely aligning with the ground truth PDEs. In contrast, PDE-FIN PDE-FIND-reee FI-D-rrecoeeeed PDEs excel in interpolation but exhibit significantly diminished extrapolation capabilities. characterized by orders of magnitude higher MSR. Furthermore, it's noteworthy that even the true PDEs themselves exhibit non- oon-eeligiible residuals, likely stemming from systematic errors in data generation, sampling, and finite difference derivative approximations. This observation, coupled with the promising results of HSTS in RQI and RQ2, underscores the robustness of HSTS in the presence of data noise. This property makes the prevalent.
algorithm well-suited for real-world scenarios where data imperfections
(RQ4) Integrated vs Pipelined. Table 10 compares the results of the physics hypothesis selection of both integrated and pipelined models. Our integrated HSTS algorithm get almost everything correct, while the SINDy+GPT pipeline isn't able to find physics properties effectively. However, ground truth PDE expressions are provided, GPT can infer physics better. We conjecture that the
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Table 9: Examples of identified PDEs and Estimated Parameters (notations defined in Appendix B).
PDE-FIND
Ours
GT
S1
dºp 160393
yedy ME
+ 68788y
Du V (-pl+)
Dr
Du V (-pI++)
Buy
12777
11125
m
Dp
+ V (pu) =0
V (pu) =0
By
0p
Up 16339n
18572
0.01000
# 0.01
0t
S2
Oux
Bup
01
dp 2204597
15580
130897
Bxdy de
Wyy Up
8'p p_
132342dy
60459ay
48085
ay
80p
dp
by
JA 0.01004 + 0.01003 (0.010071og(7))
P Du = (-
3p + V (pu) =
0.01 cos(7) + +lg((1))
Du = V (-pI+T)
V (pu) 0
Bg
S3
Sus Bug = 0.75u,p
Bug -0.001 Up 0.003a
21
By
0.5p
dbzz
Du Dr
p De
de
Φt DT = V (AVT))
+ V (pu) =0
V (-p1
pCp Dr = (IVT) + Qud
Du V (-pT + +) + Dr
8p V (pu) =
By
tQp Qud
8P
1.6u,
1.2
My
0.00992, g 9,8000
0T
26. 7ug Myy 0r0y
0² uz
17. 17.17
Or
8y
k 0.58704, Cp 4200.3
k 0.6, C₂ 4186
1 0.01, == -9.8
0.15747
pipelined approach inferior due to the limitation of information propagation. Specifically, the PDE discovery step is inaccurate due to the lack of hypothesis selection priors or feedback, and in turn, such inaccuracy bottlenecks the performance the latter hypothesis selection step.
More details about physics interpretation and the meaning of each decision are shown in appendix D.
LASER FUSION DATASET AND EXPERIMENTS
To demonstrate the application of PhysPDE in real physics research, experimented in the field of Fast Ignition (FI) of inertial confinement fusion (ICF) (Tabak et al., 1994), guided and assisted by experts in the domain. In the Fl experiments, high-density deuterium-tritium plasma is heated by relativistic electron beams (REB) over a time scale of approximately 10 ps, raising the temperature by about 10 ke (approximately 120 million Kelvin). During this process, some physical quantities (such as magnetic fields, REB divergence angles, plasma dopant types and plasma degeneracies, etc.) are crucial for evaluating the effectiveness of the Fl experiments and optimizing the design of future experiments. However, due to the extremely rapid evolution of the physical system and the difficulty of diagnosing certain physical processes occurring in high-density plasma through optical methods, these physical quantities are typically challenging to measure directly and precisely (Abu- bbu-Shawareb et al., 2022; Peebles et al., 2022). Therefore, we aim to indirectly derive these physical quantities through the evolution of directly observable quantities, such as plasma temperature. Researchers can provide prior knowledge in the form of theoretical analyses (hypotheses) about how these physical quantities influence the system's governing equations (PDEs). In this context, PhysPDE aids researchers in automating and efficiently conducting diagnostic analyses.
Leveraging domain knowledge, we developed a hypothesis decision forest (Table.15), comprising 5 decisions and 32 valid combinations. The detailed definitions and results are provided in Appendix H. In this work, we first use a numerical simulation program simulate the Fl experiment and its diagnostic process to construct dataset. In the future, the algorithm developed in this work is expected be applied to the ongoing double-cone collision ignition FI experiment (Zhang et al., 2020). The data were simulated using the specialized solver HEETS (Xu et al., 2019), with 80 x 80 x 80 spatial grids and 26 temporal grids. HEETS is particle-fluid hybrid simulation program specifically developed for simulating charged particle transport, nuclear reactions, and the evolution of fields in high-density plasma in FL. In this program, the motion of the REB is described using relativistic stochastic partial differential equations (Robinson et al., 2015), while background electron-ion plasmas are described by a reduced two-fluids model (Gardiner et al., 1985). We constructed the dataset by performing Fl simulations over physical time of 10 ps under various conditions, including different applied magnetic fields, REB divergence angles, target doping, and initial plasma degeneracy. Given the problem's 's complexity and dimensionality, each simulation required approximately 0.72h

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Table 10: Results the physies property extracted from data and expressions. Number marked red means a wrong decision is made on this property. Term colored red represents a non-existing property in the original system. Abbreviations: GT+GPT (ground truth equations are fed to ChatGPT-4o)
S1
S2
S3
is scomprrsssbbb::
__sotheermal:: 0
compressible: 1
is compressible: 1
is newtonian:
is isothermal:
is ja_isothermal:
pressure__yy wook: 1
newtonian:
newtonian:
thermal conductive: 1
Ours
is turbulent:
is L_turbullet:: O
is *s_turbulent: O
type_ pre_body_fooe force:
yyp_body_for force:
is viscosiit_diiffusiini didfffsffio::
type_non non nononeentnnn: 3
type_non, ____non__eeeee non__eetoniann: 2
type body force:
rre__on_turbullnt O
sype_non_turbulee 0
type_mu tre__m_teette muu_eeperatture
type_ type_newtonian:
type_non_turiane jes_non_turbbuett:: O
is_ c_oopreesibll::
is isothermal: 0
is_ s__compressblle: is jsjoohehhmal::
is_ ss_compressible:
is_ is_newtonian:
SINDY +GPT
is isothermal:
is pressure_ work: 1
newtonian:
is s_neettniia: is j__turbulent:
thermal conductive: 1
(with
is dttturuulen::
is $s_turbulent: is viscosity_diffuuss difffusion:
PhysPDE
upe_body__u force:
sipe_ooy__ffe force:
prompt)
yype_nom tei_non__nee non__eetonii:: 3
type_non ____no___eeee coo__newtoiian: 2
type body ffrce:
ttt_tuubulent:
aype_turbulent 1
type_mu_teet ____mmmppraturre
type_newtonian: 2
type_ ape_ttubulent: 0
is is_compressible: 1
is i_isothermal:: O
ss_compressbbl:: 1 is js_isothermal:
is_ s__ooprressbbe: 1 is is_isothermal:
is is_newtonian:
GT+ GPT (with
js_pressure_yy work: 1
newtonian: 0
is ne_netnnin:: 0
thermal conductive: I
is turbulent:
is L.ruuullettt type_ _____odd_____e force: syp_non__ewtonian 2
is ss_turbulent:
PhysPDE
type_ body force: 0
is viscosity diffusion:
prompt)
type_non_newtomian: 3
sype_body__eee forree
eype_no__tubulent O
lype_non_turbulntt
hype_mu_leet mmu_eemperature:
type_ newtonian: 1
lype_non_turbulent 0
of wall-time (parallelized across 128 CPU cores). We randomly selected one decision combination as the ground truth and divided the dataset along the temporal axis into training, validation, and test sets. To simulate data noise, we applied a factor of + 0.01 * *N 1) to the training and validation datasets. The results of the hypothesis selection (RQ1) are shown in Table. 16. According to the table,
our method identifies the correct hypotheses in 37% of the wall-time required by Exhaustive Search (ES), whereas ES-TL fails to produce the correct result.
Despite the problem's small search space, the simulation high cost makes exhaustive search impracti- cal. Previously, searches were manually conducted, relying expert experience. Our method offers an efficient and automated way to explore the search space, facilitating scientific discovery.
CONCLUSION
In this paper, have proposed PhysPDE, ML4Sci task, with two datasets and baseline algorithms. offers unified framework for discovering and interpreting PDEs, bridging the gap between theoretical understanding and practice in scientific research.
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Table 11: Fluid Mechanics Hypothesis Decision Forest
Decision ID
Decision Name
Value
Hypothesis Name
Child Decision Name
0
type non type_ _pp__turulentt
sype_non turbulent
is turbulent
turbulent
0
laminar
type_non non turbulent
creeping
0
k-epsilon
2
type_ turbuuent
1
realizable k-epsilon
0
non_newtonian
type_non, _nwttneann
3
is newtonian
newtonian
yype_newtonian
powerlaw(mu_app)
is_dilatant
4
aype_non_newtomee
1
carreau
poly_ordes
new_non_newtonian_
Fourier_order
0
poly_order
5
poly_order
1
poly_order__
poly_order_2
0
Fourier_order, 0
6
courier_order
1
Fourier_order
2
Fourier_order_2
0
pseudoplastic
is dilatant
dilatant
inviscid newtonian
type nep_onentnnann
1
type mu temperature is t-tthhmmall conduccive is therthelaaaccm ccnemal_conductive is ss_pressure__wa
nonisothermal
9
is isothermal
viscosity_diiffuee difffssonn
isothermal
0
mu powerlaw(mu_TT
10
type_mu__empeerauue
2
Sutherland
Andrade
0
not not_thermal Iiternal_conductivy Fourier
11
is coorutal_conductive
0
no wwrppserreeeeee work
12
is pressure_ewweee work
pressure work
0
no viscosity diffusion
13
is viscosity diffusion -iffositydiifuuson
viscosity_diiftssion
0
incompressible
14
is compressible
compressible
0
no b__body_ffoe force
15
type_ hedybody__o bory_force force
gravity
A FLUID MECHANICS HYPOTHESIS DEFINITION
In this paper, we focus on fluid mechanics. The full decisions, hypotheses, and parent-child relations listed in Table 11.
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B
FLUID MECHANICS NOTATION DEFINITION
DJ
0f Dt 0t
Material Derivative
+ u Vf
Op
Pressure Work
Qp = 3T(
+ u Vp)
Viscosity Diffusion
Qud = T
Body Force
= (0, - pp)) )
u(uu+ + (Vu)²) 2
Deviatoric Stress
ju(V - (.((vu)) u)1
Shear Rate
:
Strain Rate
S 5 (Vu+ (Vu)?)
C FLUID MECHANICS BASIC INTRODUCTION
Fluid dynamics is governed by the principles of conservation of mass, momentum, and energy, which are mathematically represented by the continuity equation, the Navier-Stokes equations, and energy equation, respectively.
C.I CONTINUITY EQUATION (CONSERVATION OF MASS)
This equation ensures that mass neither created nor destroyed within a fluid system. For compress- ible fluid, is expressed
0p Ot + V.V. V. (pu) =
(5)
where is the fluid velocity vector, and p density. An incompressible fluid has constant density, and Vp- = 0, and the continuity equation reduces
thus dp
u-u 0
(6)
C.2 MAVIER-STOKE EQUATION (CONSERVATION OF MOMENTUM)
These equations describe the motion of fluid substances by accounting for forces acting the fluid, including pressure, viscous forces, and external forces. They are formulated as:
du + p(u Vju= V +) Tt
where p the pressure, F represents external body forces (e.g. gravity). the deviatoric stress tensor:
2 (Vu+ (Vu) -V u)I.
uJI.
The μ is the dynamic viscosity. For Newtonian fluids, the dynamic viscosity is constant. The most commonly encountered fluids Newtonian, such water, gases, and ethanol. However, there are a number of fluids that do not have constant viscosity, known as non-Newtonian. Examples of non-Newtonian fluids include blood, paint, and cosmetic products.
C.3 ENERGY EQUATION (CONSERVATION OF ENERGY)
This equation accounts for the energy (temperature) changes within the fluid due to conduction, convection, and internal energy variations. It is given by:
OT +u. VT) = v (EVT) + Up + Qud
pCp( 0t
(9)
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Table 12: Decisions in S1 (incorrect decisions colored red)
Top-1
Top-2
Top3
is_ compressible:
is compressibbe:: 0
is compressible:
isotheemall:
is_ s_iisothermal:
isothermal:
is newtonian:
is newtonian: 1
is newtoniam: is turbuuent:
Ours
is turbulent:
is turbulent: 0
ape_body_ force: 0
upp_body__aee force: O
bype_body_aag force: 0
type neeeneetnnin:
are__eetoniia: type_non_ coon_turbulent: 0
type_non, aape_non_aeee non__eewoniinn
type_non
non turbulent:
type_non_tuue supe_non_turbullee turbulent:
is_ ""cmppressbblee
is compressible:
is isothermall: 0
is jsothermal: is "s_newtonian: is "s_pressure__we _wor:: 0
is compressible:
*s_newtonian: 1
is isothermal: is_ newtonian::
is pwwpsrrreeewwoeee work: O
thertheraaa........ t__thermal_cooddctivee _conductiee:: O
is is Lttuuulentt 0
ES
is turbulent:
is turbulent: 0
type_ body_force,
is s_sviiooitty_doe difffusion:
is viscosity diffuusion = 0
sype_newtonial 1
see_body_foor 0
type_body une_body_force force: 0
type_non non turbulent: tree_oontturbulnt:
type_mn mu nnu_teepprattre:
type_mm wt__m_mmmmeamuuuee 0
lype_newtonian:
type_newtonians
type_non_turbulent O
sype_non_turbulen
where Cp is the specific heat capacity at constant pressure, T' is the temperature, is the thermal conductivity, Qp: Qud represents pressure work and viscosity diffusion turbulence. For isothermal fluid, the temperature is assumed constant, thus the above equation is constantly zero.
These fundamental equations collectively describe fluid behavior under various conditions and essential for analyzing and predicting fluid flow phenomena.
D FLUID MECHANICS HYPOTHESIS SELECTION RESULTS
The learned decisions and values of all scenarios are listed in Table 12-14.
E ALGORITHM PRELIMINARIES
Our search algorithm is inspired by two established algorithms MCTS and Sequential Threshold Ridge Regression (STRidge)
MCTS The MCTS algorithm (Coulom, 2006) stands out robust approach frequently utilized solving optimal decision-making problems. This method employs tree search mechanism that adeptly balances the dual aspects of exploration and exploitation. At its core, the MCTS algorithm unfolds through cyclical iteration four fundamental steps.
1) Selection. The MCTS agent, starting from the root node, selects successive child nodes according to given policy until an expandable node or a leaf node is encountered.
2) Expansion. The search tree is then expanded by selecting unvisited child an expandable node.
3) Simulation. Following expansion, the agent performs independent simulations from the selected child node, exccuting random actions. The simulation halts the agent reaches a terminal state or predefined time constraint.
4) Backpropagation. Statistics of nodes along the random path are updated according to the ultimate search result.
To balance exploration and exploitation, the Upper Confidence Bounds applied for Trees (UCT) algorithm is integrated (Koesis & Szepesvári, 2006). During the selection phase, child node with the
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Table 13: Decisions in S2 (incorrect decisions colored red)
Top-1
Top-2
Top3
is compressible:
is j_jiotthermal:
is compressible:
is compressible:
is newtoninn::: 0
is isothermal:
is isothermal: 1
is pressure work: 0
newtonian:: is turbulent:
is __nettonia:: 0
thermal conduuttvv: is turbulent: is viscosity diffusion: =isooittydiiffuiio:: 0
Ours
is turbulent: 0
type_ se__bod__foreee force: 0
type_body mody_forces, sype_non_newtoniae 3
type_non_ net.noonnwwtoninn 3
type_body su__bodd__foor force: O
sype_non_turbulnt 0
sype_mu_temppeeeeeee __m__eemmmratture: 0
type_non newtonian: 3
type
'ype_non_ non turbulent:
is_ compressible:
is compressible:
is jsothermal: 0
isothermal: is_newtonian:
is compressible:: isothermal:
is ww.ettnnia:: 0
is pwwsspreeeweeeeeeeee work:
is_ p.pressure work:
is newtonian:
is thermal conduuctioede 1
is thermal conductive: thermal__ondductive;
ES
is turbulent:
is A_turbulent: O
is is_turbulent:
syp__body_foree force: 0
is viscosity diffusion: viscosity_diffusion:
is viscosity_diffusiio:: 0
type_non_newweneeee nenneetttnn:::
type_body h___bbod_foore 0
type_body_ forcee
type_mm upe_noo___
pre__oon_turbulee turbulent:
type_mm _e__mm______ nnu_temerrature
type_mu ape__m_ttee imu_temperaturee 0
type ____oon__eeee non_newtonian:
type_ www__nnnnnnntnn non__eewoniinn 3
type_non mon_turbuleen: 0
type_nop aepe_non_turie ton_uuubulent:
Table 14: Decisions in S3 (incorrect decisions colored red)
Top-1
Top-2
Top3
is compressible:
is compressible:
is compressible::
is jssisottermal::
is jsjisothermal:: 0
is s__isttheemal:
is "s_newtonian:
is is_newtonian: 1
is "s_newtonian:
is pwwssurreeeeee work:
is ___pressuu___ve work: 0
is
is ss_pressure_wwar sooreeewwooooo 0
is iitermal_conduccin
1
is
Ours
is turbulent:
is turbulent: 0
is turbulent: 0
is viscosity diffusion:
is_ viscosity diffusion:
is viscosity diffusion: Disfositydiiffusion: 0
type_hooe _____odd__ooo ffrcffooee 1
srp__boy_ffee
typ__oody_fee force::ooe
type_mu__t s____mutmmerattrree 0
type_mu nnu_teeperature:
type_mu d____u_temmpp anu_temperature: 3
sype_newtonian: 1
type_newtonian:
sspe_newtonian:
type_non_turbulens
type_non_turbulent: 0
type rre__oo_tturulentt
is ss_coprreesible:
is __compreessblle:
is sccmpressssble:
is jiiiottherma::
is_isothermal: 0
is jsisotthermal: 0
is_newtonian:
is_newtonian: 1
is_newtonian:
is pressure work: 0
is "s_pressure work: 0
is pressure work:
is thermal conductive: thermal_connductive: is .utuuulentt is Suscosity_diffusionn O
thermall_onnductive: is. thermal conductive: 1
is thermal conductive: is .urbullnt: is_ viscosity diffusion: 0
ES
is t_tturulenn: O
is viscosity_diffusion: _iiffusionn:
discosityddiffuusinn
rype_body_foree
type_body_ be__boody_fore force:
jype_body_foot force
type_mu_tempell t_m__eeeprrrrtrre: 0
rre_mu_tempprattee 0
type_mu_tente rre_mu_temperature: 0
type_ vype_newtonian:
type_ sype_newtonian:
type_ srpe_newtonian:
type_non turbulent:
type_non peee_oon_tuute -oo_uuett 0
type_non turbulent:
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maximal UCT value is selected:
((s (t0(sa)))t////Msss////......
(10)
where Q(s, a) the average win ratio playing action a in state in previous simulations; (s) is the number of times state has been visited; [(s, a) is the number of times action gets selected at state s. Constant с controls the balance between exploration and exploitation, which is represented by [>))NN (s, a) and Q(s, a) respectively. The convergence and efficiency of this algorithm have been rigorously validated as demonstrated by Shah et al. (2020).
STRidge. STRidge (Rudy et al., 2017) is an advanced algorithm that combines concepts from both ridge regression and sparsity-promoting techniques. It composed of an iterative process of ridge regression and thresholding. As of the ridge regression step. penalty terms are incorporated into the objective function limit the number predictor variables considered relevant:
f = arg min [@(U)- U. + cllsl
(11)
where denotes the library of candidate terms, U data, and regularization coefficient.
After the initial ridge regression step. STRidge applies thresholding mechanism, eliminating terms with coefficients below a threshold value. The idea to eliminate predictors that have a negligible effect on the target variable, thereby promoting sparsity in the model. This process continues until it converges stable set of predictors or until a specified number of iterations is reached. STRidge poses stricter regularization and penalties for overfitting. therefore achieving outstanding performance in PDE discovery tasks.
F HSTS AND STOP ALGORITHM
The main algorithm is summarized Alg. 1, and we will elaborate on its details below.
Algorithm 1: HSTS+STOP Algorithm
Input :Decision Forest, Operator Library, Observed Data
Param :HSTS) loss bound fmxx, risk fraction max episode ME, number of rollout NR, exploration weight
Param :STOPP) Regulariazation A., threshold T,, max iteration Output:Optimal decision h, coefficients 0,,
Dutput::Optionnn:
new expression m, coefficients 6.
Initialize h as empty vector, initialize an empty cache; for each episode < ME do h has no child then
L break; for each rollout < NR do
Selection: ho -- UCT selection of object J(Eq. 14) among descendants of h; Repeatedly select child of ho until child available; Simulation: Calculate reward(Eq.1l 12) of ho by solving Eq.15 via STOP; Backpropagation Propagate and cache the reward:
Expansion: ho <-
10
Action: h - the child maximizing (Eq. 14);
u Return best solution in the cache;
F.1 HYPOTHESES SELECTION TREE SEARCH (HSTS)
This section addresses solving the outer-level decision vector h. Although the decision forest effectively represents various physics models and constraints, the search space is vast, making naive enumeration impractical. With n nodes in the forest and a minimum of 2 choices per node, the total possible decision instances are O(2"), excluding interdependencies. To tackle this challenge, employ the Monte Carlo Tree Search (MCTS) algorithm to efficiently explore the search space for hypothesis selection task.
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Repeat/ Move
Selection
Expansion
Simulation
Propagation
DOOT
UCT
Tival
Dinoonlly
aln 17 1-1-st RR]
Unvisied Decis Irotance
Ecoos
K HoyelPte (Pu)) (0
Tects
y ( Pow.
5-n [T (tw/)
MSE
repalive
Paax 0.00EL 1ogl99)
Fisslations
Simulations
Figure 4: Overview of the proposed Hypothesis Selection Tree Search (HSTS) algorithm.
MCTS is heuristic search algorithm renowned for simplicity and effectiveness in complex decision- making scenarios (Silver et al., 2017). navigates the search space by expanding selected search nodes and back- peo-ppropggtting simulation results through the tree structure. In this paper, the MCTS algorithm is adapted to align with the unique requirements of Hypothesis Selection. The adaptation involves customizing the search node, reward, and action objective of MCTS handle the decision forest representation and evaluation of PDEs within the context Hypothesis Selection. The resulting algorithm is thus named HSTS. Figure 4 demonstrates illustrative example.
Search Node. MCTS, a search node represents an (in)complete decision instance from the decision forest, essentially set of tree paths from roots. The search begins with empty decision node and progresses to a terminal node representing a complete decision. A child node is generated by extending the parent's i decision instance with the next decision, adhering the predefined sequential order decisions within the forest.
Reward Function. Once valid decision h is sampled, construct a system of PDEs and evaluate the system with a reward function R on observation data X. To simplify notation, we denote the minimal loss of decision h as f(h):
{ (h) min min ((X, EEq, C 0.
(12)
O, m00m
While straightforward reward definition could the negative of ((),, the unpredictable numerical error level may lead to unbounded reward range. This unpredictability can potentially disrupt the balance between exploration and exploitation in the UCT selection. Thus, we normalize the loss and constrain it within the range [0, 1]:
R(h) = 1 log E(h) - log Liin
(13)
log mmax log Lmin
where the hyper-parameters 6nax and Emin serve as approximate bounds for the loss.
Risk-seeking Objective. After every N back-propagation iteration, MCTS either terminates or selects action for execution, maximizing the current state's objective. Actions correspond moving to child search nodes, while the state encompasses the current search node and all visiting and rewarding histories. In traditional Monte Carlo Tree Search (MCTS), the objective for action at state s is the expected reward under policy denoted as J(s, a) Bn--((e..) (R(h.) This approach, however, does not align with the priorities Hypothesis Selection, which emphasizes the performance of top outcomes over average-case scenarios. Similar challenges in policy-gradien methods led to the adoption of risk-seeking objectives, concentrating on the highest rewards within a batch. Inspired by such methods (Petersen et al., 2019; Tamar et al., 2014), propose e-parameterized risk-seeking objective for MCTS.
(()) Eppp((e. (R(h)) R ()(s,a)))
(14)
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This objective averages the rewards of the top fraction of samples from the distribution p(s, a). disregarding all other samples falling beneath this performance threshold.
F.2 SEQUENTIAL THRESHOLD OPTIMIZATION (STOP)
Here address the challenge of determining the boolean vector m as specified on the right-hand- side of Eq. 12. The objective is to select an optimal combination of functions from a predefined function library for constructing a new expression. where each element of vector m denotes the inclusion (or exclusion) of specific function. The loss function indicates that it is an Co regularized sparse optimization problem, which proved be NP-hard (Virgolin Pissis, 2022). An intuitive idea of this complexity is the exponential growth of the search space with the size of the function library, n, leading 2 possible function combinations.
Inspired by the sequential threshold ridge regression technique utilized in PDE discovery (Rudy et al., 2017), we propose a Sequential Threshold Optimization (STOP) method. adopts an iterative backward selection strategy, initially setting all elements of m to one. In each iteration, it tackles the following constrained optimization problem:
0% O = argmiin On
e(X, + Apm. 10 1
(15)
s.t. 0m - m) 0, C20,
where λen. serves hyperparameter to modulate the regularization strength. Subsequently, is updated to deactivate elements where 0x, falls below a predefined threshold T, with updated rule m 1 T). This iterative refinement continues until
m stabilizes, effectively reducing the search space by excluding insignificant function choices.
The STOP terminates within most (m) iterations since each iteration either diminishes the count of non-zero elements of m or terminates the algorithm. This approach trades optimality for efficiency, with STOP's output not assured to be the optimal solution due its inherent greedy selection mechanism. However, as demonstrated by our empirical findings Section 4.4, STOP capable of accurately identifying the correct expression with an appropriate configuration of hyperparameters.
F.3 NUMERICAL SOLVER OF PDE INVERSE PROBLEM
To solve the PDE parameter estimation problem Eq. or inverse problem (Isakov, 2006), we employ a two-step approach: 1) discretization of PDE derivatives leveraging the centered finite difference method (Strikwerda, 2004), transforming it into an ordinary constrained optimization problem, and 2) application of sequential least squares quadratic programming solver for optimization (Stoer, 1985).
G RESULT VISUALIZATION
The performance is visualized in Figure. 5-11.
H LASER FUSION HYPOTHESIS DEFINITION AND RESULTS
DISCUSSION
traditional benchmark should be comprehensive evaluation of existing methods. We acknowledge that PhysPDE may not fully align with the traditional definition of a benchmark paper. However, in broader sense, we believe that benchmark typically includes the following components:
1) dataset (e.g., Fluid Mechanics dataset);
2) task setting (e.g., Physics Interpretation task);
3) workflow for evaluation (e.g., integrated/pippllined approaches);
4) evaluation results.
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RQ1 Hypothesis Selection
Top2
Top3
1--
1.71
0.4
06
06
02
0.2
0.4
0.-
00
80
P%:
50-
30
Figure 5: RQ1-S1
RQ1 on 52: Hypothesis Selection
1.00
1.30
94-
0.0-
Figure 6: RQ1-S2
RO1 Hypothesis Selection
Tapt
Topa
Topa
65%
0mm
65
0.59
65
1.0
Out
1.0
Ouri
1.0
0.02
0.1
14.11
14.35
30-
5
30
Oum
05.7%
85
Omm
C2.7
65
0.m
C5.7
65
Figure 7: RQ1-S3
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on all scenarios
S1
62
Oum
FDC 7RD
Oun
PDC FND
O.m
FDC TWO
1.0
1.0
Dui
1.0
10:0
10:00
500
50300
530
Our
FDR
Oum
PCC
Oum
FDC TWC
Figure 8: RQ2
RQ3 Mean Residues
Train
Velid
Test
D600WES
Aleds
Figure 9: RQ3-S1
RQ3 on $2: Mean Square Residues
12'
PAL
Decobo
Fou
Figure 10: RQ3-S2
Square Residues
Train
Valld
Test
121-
DECOWES
nst mNO2
Figure 11: RQ3-S3
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Table 15: Laser Fusion Hypothesis Decision Forest
Decision ID
Decision Name
Value
Hypothesis Name
Child Decision Name
0
magnetic field
is magnetic field
1
magnetic seenetic_field
O
divergence_angle<<44
range_ beeesddvegenneeeeee angle
1
givergence_angleee
0
Iemperature<<T__
2
uige_temperaturs
1
temperature22_1
0
no dopant dopant
3
is is_dopant
1
uye_dopant
0
dopant Au
type_dopant
1
dopant_BB Br
2
dopant_ Cl
Table 16: RQI
Laser Fusion: Hypothesis Selection.
Model
Recall
t Precision
+ Wall-time(h)
Top-1 1 Top-2
1.00
1.00
Ours
0.40
0.40
8.67
Top-3
0.80
0.80
Top-1
0.80
0.80
ES-TL
Top-2
0.60
0.60
8.70
Top-3
0.80
0.80
Top-1 0.60
0.60
ES
Top-2 0.80
0.80
23.21
Top-3 1.00
1.00
Our paper's primary contributions lie in components 1, 2, and (the algorithm implemented for) 3. The 4-th point relatively weak since not enough baselines are available for our new task. However, have demonstrated that the combination of the existing baseline (i.e. the SINDy+ pipeline) can be applied. In this sense, PhysPDE will provide foundation for assessing and improving more ML4Sci algorithms in the future.
LIMITATIONS
important acknowledge the current limitations, including scalability exceedingly large datasets and the need for further refinement when dealing with highly complex PDE scenarios. Future endeavors will be directed toward addressing these challenges and exploring the application of PhysPDE to real-world data, enhancing its utility. A potential negative societal impact could be reducing the need for human expertise in certain scientific fields, leading to job losses.
23


Reviews:
[{"review_id": "SqTfGyxKck", "reviewer": "Reviewer_QC1C", "summary": "The authors introduce a new task of explaining observed data using existing partial differential equations in physics and other domains. Previous approaches to discover new partial differential equations via symbolic regressions and randomized algorithms starts from scratch. However, PDEs are inspired from domain knowledge and available theories. The authors propose a method HSTS and STOP algorithm that can find/construct appropriate and interpretable PDEs, incorporating available domain knowledge. Experiments show that the authors proposed method provide better performance compared to baseline methods.", "questions": "Although the author's motivation is to generate interpretable PDE models on top of existing domain knowledge, is this work limited to discovering only known theories? For example, given the knowledge of Newtonian fluid mechanics, can this method discover PDEs for non-newtonian fluids?\nAnother thing to mention here is there are previous approaches for empirical law discovery such as BACON. What benefit does the author's method provide compared to such previous works?\nAdditionally, how do the authors see the dataset being used in future works? Given these datasets, is there a metric that can measure the interpretability of PDEs discovered by different algorithms, or do we still need domain expert opinion on that?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The motivation of the work is described well in the introduction.", "weaknesses": "The method proposed by the authors needs to be explained in more detail, using an example if possible. Figure 2 caption needs to include more details, for example difference between learned PDE and discovered new expression is not clear. Figure 3 caption needs more details, are these learned from the author's algorithm or available from domain knowledge?", "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": "", "extra_scores": {}}, {"review_id": "XddgT2K8ft", "reviewer": "Reviewer_QbPL", "summary": "This work addresses limitations in existing methods for discovering Partial Differential Equations (PDEs) from experimental data, particularly the lack of physical interpretability in machine-learning-driven approaches. This approach integrates machine learning with physical hypotheses and theories to guide the discovery of PDEs, prioritizing physical interpretability over purely mathematical accuracy.\n\nMain contributions of this work are:\n\n- The proposal of a new task, termed \"PDE Interpretation\", which involves selecting physical hypotheses (laws) that align with observed data rather than simply generating symbolic expressions, addressing the gap between mathematical representations and physical meaning.\n\n- The authors, then, recast the discovery/solution task as a mixed-integer programming (MIP)-based approach formulated as a bi-level optimization problem. The outer layer selects the best-fitting physical hypotheses, while the inner layer focuses on expression recovery, involving symbolic regression when new expressions are necessary. They introduce a Monte Carlo Tree Search-inspired Hypothesis Selection Tree Search (HSTS) and a Sequential Threshold Optimization (STOP) algorithm for efficient problem-solving.\n\n- Finally, the study provides two new datasets in fluid mechanics and laser fusion, designed to test the proposed framework's effectiveness in interpreting complex physical phenomena, with scenarios tailored for both Newtonian and non-Newtonian fluid behaviors, as well as thermodynamics in nuclear fusion.\n\nThrough experiments, the authors demonstrate that PhysPDE not only accurately recovers existing PDE terms but also extrapolates effectively to new scenarios, outperforming baseline models in terms of recall, precision, and computational efficiency.", "questions": "Following up on the precious remarks, the following questions arise:\n\n   -  How does PhysPDE compare specifically to recent frameworks that incorporate physical interpretability into PDE discovery, such as those seeking to discovery interpretable models from sparse data Champion et al. (2019), or Chen, Liu, Sun et al (2021)?\n- How does PhysPDE handle the computational demands of hypothesis selection when scaling to high-dimensional systems, where the hypothesis space grows exponentially? Could the current approach handle systems with higher-dimensional PDEs or more complex, multi-parameter hypothesis spaces, such as those in climate or geophysical modeling?\n- How robust is PhysPDE to real-world data noise, which may not follow Gaussian distributions or may include systematic measurement errors? Would the authors consider augmenting their experimental setup with various types of noise, or could they propose methods to make PhysPDE more robust to noise common in observational datasets?\n-  Could you elaborate on how the hypothesis decision forest is constructed, specifically the criteria for selecting hypotheses and organizing them hierarchically? Consider sharing any automation or semi-supervised methods used in building this structure, as this could significantly impact PhysPDE’s ease of adaptation for other researchers.\n- How does the decision-making process in the hypothesis selection step work, especially in cases where multiple hypotheses may fit the data similarly well? Also, if multiple plausible hypotheses exist, does PhysPDE allow for any probabilistic ranking or ensemble approach to indicate uncertainty, or is it a hard-selection process? \n- Could the authors clarify how they evaluate interpretability and physical alignment of the recovered PDEs, beyond standard metrics like recall, precision, and MSR?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper does make an original contribution to the state of knowledge, as it reconceptualizes PDE discovery as a physically guided interpretation task. This work focuses on **PDE interpretation**, which is the term they use to denote the integration of prior physical knowledge into the discovery process. By framing the task as one of **hypothesis selection**—with a goal to align with established physical laws rather than simply fit data—the authors address the crucial challenge of **physical interpretability** in machine learning for scientific discovery. \n\nThe paper is well written, with explanations offered with adequate rigor, clearly articulating the mixed-integer programming formulation and providing a detailed breakdown of the two-layer optimization process for hypothesis selection and expression recovery. \n\nThe offered **new datasets**are certainly of significance to the PDE, and particularly fluid mechanics and laser fusion, communities. The experimentation protocol is robust, involving cross-validation, varying boundary conditions, and multiple baselines for comparative analysis.", "weaknesses": "While the paper introduces a promising framework for integrating physical interpretability into PDE discovery, there are areas where the work could be improved to enhance its impact, applicability, and rigor. More specifically,\n\n-  The paper offers a **comparative analysis** with traditional symbolic regression methods, such as PDE-FIND and GPT-based pipelines, however, it lacks an explicit comparison with recent interpretability-focused PDE discovery approaches, as for instance, the work by Champion et al. (2019), or Chen, Liu, Sun et al (2021), who adopt physics biases to discover interpretable models through sparse data. \n\n- While PhysPDE introduces a bi-level optimization approach, there is limited discussion on how this specific approach advances the interpretability or robustness over similar frameworks like the sparse identification of nonlinear dynamical systems (SINDy), where laws are also indirectly accounted for in the selection of the basis functions. \n\n- The paper provides an experimental evaluation of PhysPDE on relatively controlled (and wely generated) datasets, but the scalability of the framework for **high-dimensional, real-world systems** remains unclear. In practical applications such as climate modeling or biomedical engineering, PDE systems can have many more variables and parameters, leading to extremely large hypothesis spaces. The **Monte Carlo Tree Search (MCTS) and Sequential Threshold Optimization (STOP)** methods presented here may struggle with such scalability without further refinement.\n\n- Although the authors mention that PhysPDE is tested under data noise, the experiments are still based on **synthetically generated datasets**, where the noise distribution can be controlled (as is here the case). Real-world data is typically noisier and may have **systematic errors, non-Gaussian noise, or missing data points**, which could impact the framework’s robustness. In this setting, reliance on a structured hypothesis decision forest could lead to inaccuracies, as the framework might overly rely on a narrow set of physical assumptions that may not fully capture complex, real-world phenomena.\n\n- The **design of the hypothesis decision forest** is critical to PhysPDE’s interpretative power, yet the criteria for constructing this decision structure, such as the choice of hypotheses and decision hierarchy, are not discussed in detail. It is unclear how flexible or adaptable this forest structure is, which could limit the framework's usability across different scientific domains. For instance, if a user wanted to apply PhysPDE to a new domain like biomechanics, would they need to manually create a decision forest from scratch?", "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": "", "extra_scores": {}}, {"review_id": "lZY0mvly6C", "reviewer": "Reviewer_Xb5C", "summary": "The paper proposes a new ML4Sci task which focuses on identifying the hypothesis that best align with the data. Prior physics hypothesis and theories are used and the proposed paradigm focuses on identifying which ones align best with the data. The approach is introduced using a mixed-integer programming formulation. Furthermore, the paper designs new data sets for this problem, specifically from the domains of Fluid Mechanics and Laser Fusion.", "questions": "1) Are you planning to release the data sets and the code? (Are these available for review?)\n2) Could you elaborate more on the use of LLMs and in particular GPT-4o for extracting physics hypothesis from the data? Why are LLMs specifically appropriate for this task?\n\nUpdate after discussion period: I have increased my score while keeping my confidence low. The main reproducibility concern has been addressed during the review process.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Overall, the paper proposes a novel task in the ML4Sci field, which could potentially lead to new methods and approaches developed in this field.", "weaknesses": "- Unless I missed it, the data introduced data sets are not open access and aren't shared during the review process, which limits the possible impact of the paper. \n- Given the complexity of the proposed approach (Fig. 1 which includes many components) and newly generated data sets, the results of the paper are not reproducible with the current information available to the reader.", "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": "", "extra_scores": {}}, {"review_id": "4GQTGHy7XU", "reviewer": "Reviewer_A96C", "summary": "Physical insights via following existing physical theories and hypotheses are benchmarked in this paper to learn PDE. This allows for possibly good interpolation and extrapolation of PDE learning.", "questions": "I hope to see the weaknesses being addressed. It should not be hard to address them since these are clarity issues, but they are critical.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The overall idea is interesting and useful. I like the idea and think it is going in the correct direction -- to make PDE learning physically interpretable via construction of differential equations under the current physical framework, as opposed to symbolic regression only. The datasets used are in different areas, and the physical laws considered are representative.", "weaknesses": "It still remains unclear to me how your decisions (such as is_compressible, is_newtonian) directly affect the PDE system. You have a Figure 3 that shows how the PDE formulation is defined under both different conditions, such as is_compressible=yes and is_newtonian=false. However, I still do not know how this is constructed. What is the connection between these conditions and the PDE appearance? What does it mean for a PDE to have is_compressible=yes? How does that reveal in the PDE terms? I know you are using a decision tree algorithm and use data-label pairs to train a model to predict whether is_compressible=yes. However, but as a reader, understanding what does it mean for a PDE to be is_compressible=yes is important for them to know whether and how that is learnable. This is also good introduction for ML people who may be less familiar with physics.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)