Source-linked AI summary
An Interpretable Approach to PDE Solution Discovery via Structural Experience Distillation
Yunpeng Gong, Huolong Wu, Can Yang, Min Jiang
TL;DR
Physics-informed symbolic search must satisfy both observed data and PDE constraints, yet aggregate terminal scores give little information about individual substructures. SED-MCTS distills local structural evidence from evaluated expressions and reuses it to guide search; across the reported evaluations, it improves recovery accuracy, efficiency, and robustness.
Problem
Physics-informed symbolic search must fit observed fields and satisfy governing PDE constraints, but aggregate evaluations provide little information about which substructures contribute to either objective.
Source: §1 · pp. 1–2Method
SED-MCTS estimates subtree contributions through counterfactual interventions, routes reliable evidence to responsible grammar edges, and preserves useful components in a refined structural archive.
Source: §1 · pp. 2–3Results
SED-MCTS improves recovery accuracy, search efficiency, and robustness across PDE benchmarks, including coupled systems and noisy or scarce observations.
Source: §1 · p. 3, §5 · p. 10Takeaways & Limitations
Results on coupled multiphysics systems support SED-MCTS’s ability to capture within-field and cross-field structural effects.
Source: §5 · p. 10The authors identify reducing intervention costs and extending evaluation to larger, more strongly coupled PDE systems as future work.
Source: §5 · p. 10
Abstract
from arXiv · showhide
PDE solution discovery aims to identify explicit symbolic expressions for unknown physical fields from observations under known physical constraints. Existing methods, however, collapse data fidelity and physical consistency into a single terminal score used as the sole feedback signal, providing little information about which subexpressions are responsible for a candidate's final performance. This opaque terminal feedback severely limits the interpretability of the search process itself, offering no insight into why a candidate succeeds or fails. Consequently, reusable structures in otherwise suboptimal candidates are often discarded, whereas incidental syntax along successful search trajectories may be repeatedly reinforced. We propose SED-MCTS, a Monte Carlo tree search approach that distills structural experience from evaluated expressions and reuses it to guide subsequent symbolic solution search. Through counterfactual subtree interventions, SED-MCTS estimates local structural contributions, routes reliable evidence to the responsible construction edges, and preserves useful components in a refined structural archive. The approach naturally extends to coupled multiphysics systems. Across a diverse suite of PDE benchmarks, SED-MCTS achieves strong performance under a fixed evaluation budget and improves search efficiency and robustness under noisy or scarce observations.
1 Introduction
Physics-informed symbolic discovery must fit observations while satisfying governing PDEs, but aggregate terminal scores do not reveal which subexpressions help or hurt. SED-MCTS estimates local structural contributions and reuses reliable evidence to guide subsequent search and structural editing.
1 Introduction
- PDE solution discovery must balance observed-field accuracy with consistency under governing differential operators and any imposed initial or boundary conditions.A candidate can fit available observations yet retain nonzero PDE residuals at unobserved collocation points.Source: §1 · p. 1
- Aggregate terminal scores obscure how individual subexpressions affect data accuracy and physical consistency, especially when one field’s subtree changes other fields’ residuals in coupled systems.A subtree can improve fitting while worsening physical consistency, or be essential for reducing PDE residuals despite limited effect on observed values.Source: §1 · p. 2
- Useful components may occur in poorly ranked expressions, but existing symbolic-regression methods generally retain only a few top complete expressions rather than systematically reusing such structures.Candidate building blocks can include temporal decay, spatial oscillation, or cross-coupling terms.Source: §1 · p. 2
- Standard MCTS backs up delayed scalar returns along whole construction paths, so it does not distinguish useful construction steps or accumulate local structural knowledge.MCTS builds grammar-valid prefixes and maintains node and edge statistics, but conventional feedback does not exploit these structures for subtree credit assignment.Source: §1 · p. 2
- SED-MCTS uses counterfactual subtree interventions to estimate local contributions, route reliable evidence to responsible edges, and preserve useful components in an elite-diverse archive.The resulting edge statistics guide MCTS selection and editing; the framework extends to coupled multiphysics through conditional attribution.Source: §1 · p. 2, Figure 1 · p. 3
- Across single-field, coupled, noisy, and scarce-observation evaluations, the paper reports improved accuracy, search efficiency under a fixed terminal-expression budget, and robustness.These are the three evaluation scopes and outcomes identified among the paper’s contributions.Source: §1 · p. 3
2 Related Work
Related work strengthens expression generation, physics-informed evaluation, and component reuse, but typically lacks persistent construction-level credit grounded in physical attribution. SED-MCTS connects structural assessment to reusable search experience.
2 Related Work
- Physics-informed symbolic methods evaluate expressions using differential-equation residuals and applicable initial or boundary constraints, but generally retain aggregate candidate-level feedback.Sparse equation discovery instead selects terms from a predefined library, whereas symbolic solution discovery searches for closed-form functions.Source: §2 · pp. 3–4
- Prior genetic-programming approaches reuse components through recombination, libraries, archives, and transfer, but their utility criteria generally do not provide persistent responsibility-aware physical attribution.RSSP uses physics-informed local sensitivity to guide pruning and variation, primarily modifying the evaluated candidate rather than accumulating cross-candidate construction credit.Source: §2 · p. 4
- Existing MCTS methods use learned or heuristic proposals, modified value estimates, or promising-state transitions, yet typically back up returns across whole trajectories and lack direct reuse of components from lower-ranked candidates.SED-MCTS addresses this gap by converting evaluated candidates into structure-aware experience for later tree construction and compatible component variation.Source: §2 · p. 4
3 Method
SED-MCTS converts terminal evaluations into local structural evidence, routes that evidence to the grammar edges responsible for useful subtrees, and reuses it through an archive and structural editing. Its method combines physics-informed subtree interventions with MCTS search and refinement.
3 Method
- SED-MCTS converts each completed expression’s terminal evaluation into local structural evidence, routes it to responsible grammar edges, and stores reusable subtrees in a refined archive.Accumulated evidence guides subsequent MCTS construction and lightweight structural editing.Source: §3 · p. 4
3.1 Physics-Informed Symbolic Search
- Typed-grammar MCTS evaluates expressions under a joint data–physics objective and uses context-preserving subtree neutralization to measure local responses.For coupled fields, only the selected subtree is neutralized while other fields and fitted constants remain fixed, capturing within-field and cross-field effects.Source: §3.1 · pp. 4–6
3.2 Responsible-Edge Experience and Statistical Prior
- Node-level evidence is assigned only to grammar edges that generated the attributed subtree, while per-edge memory tracks support, weighted reward, and its second moment.An uncertainty penalty lowers scores for edges with few or highly variable observations.Source: §3.2 · p. 6
3.3 Refined Structural Archive
- The archive retains positively attributed subtrees by return type and complexity, favoring high reward-evidence products while preserving structurally novel patterns.Canonicalization merges grammar-equivalent or constant-only variants; removing an archive entry does not shrink the search space.Source: §3.3 · p. 7
3.4 Prior-Guided Structural Editing
- Structural editing diversifies high-attribution motifs while keeping their context and dependencies anchored; low-attribution regions can receive simpler replacements, and uncertain regions remain editable.Mutation reuses statistical edge knowledge, crossover inserts validated archive components, and edits are retained only under the stated reward or simplification criteria.Source: §3.4 · p. 7
4 Experiments
Across PDE benchmarks, SED-MCTS performs strongly, while responsible-edge structural experience improves search efficiency and recovery under noisy or scarce observations. The experiments also assess general symbolic regression and coupled multiphysics systems.
4.3 PDE Recovery, Robustness, and General Symbolic Recovery
- Responsible-edge routing improves recovery across noise and retained-data conditions, with the strongest gains under high noise and low data.Each setting reports recovery rates over 100 trials.Source: Figure 3 · p. 8, §4.3 · p. 10
- SED-MCTS achieves the highest exact-recovery rate on Nguyen, perfect recovery on NguyenC and Jin, and the best result on Livermore.These suites cover polynomial, trigonometric, exponential, logarithmic, and nested nonlinear expressions.Source: Figure 4 · p. 8, §4.3 · p. 10
- SED-MCTS achieves the best dense-grid MSE on five of six PDE benchmarks and ranks second to RSSP only on Poisson3D.It remains competitive or has the lowest error on elliptic Poisson benchmarks, and achieves the best average MSE.Source: Table 1 · p. 8, §4.3 · p. 10
4.1 Experimental Setup
- Methods share observations, residual points, grammar, constant optimizer, and evaluation grid; full constant-optimized expressions define the common search budget.Subtree interventions are bounded structural diagnostics, not additional candidate-expression evaluations.Source: §4.1 · p. 8
4.4 Coupled Multiphysics Systems
- Across nine coupled multiphysics problems, SED-MCTS leads on both fields in Joule Heating 1, 3, and 4 and Parabolic 3, with other methods leading several cases.Its improvements are especially pronounced for the thermal field in strongly coupled Joule-heating cases.Source: Table 2 · p. 9, §4.4 · p. 10
4.2 Comparison of MCTS Prior Sources
- Under a fixed budget of 10,000 evaluated expressions, the edge-routed contribution prior performs best on both MSE improvement and relative steps to target accuracy.Its reward distribution shifts probability toward higher-quality expressions, indicating a change in explored program-space regions.Source: §4.2 · p. 9
5 Conclusion
SED-MCTS converts terminal evaluations into reusable structural experience for symbolic PDE discovery. Experiments report improved recovery accuracy, search efficiency, and robustness, including in coupled multiphysics systems.
5 Conclusion
- SED-MCTS converts terminal evaluations into reusable structural experience for symbolic PDE solution discovery.Context-preserving subtree neutralization identifies useful components, routes evidence to responsible grammar edges, and stores components in a refined archive.Source: §5 · p. 10
- Experiments demonstrate improved recovery accuracy, search efficiency, and robustness under noisy and scarce observations.Results on coupled multiphysics systems also confirm the framework's ability to capture within-field and cross-field structural effects.Source: §5 · p. 10
- Future work will reduce intervention costs and extend the framework to larger, more strongly coupled PDE systems.Source: §5 · p. 10