An Interpretable Approach to PDE Solution Discovery via Structural Experience Distillation

2026-10-08

AI Summary

This study addresses the challenges of opaque terminal feedback and the loss of reusable structures in symbolic solution search for partial differential equations (PDEs) by proposing SED-MCTS. This method integrates Monte Carlo Tree Search with counterfactual subtree intervention and introduces a structural experience distillation mechanism. By leveraging local contribution evaluation and reliable evidence routing, it extracts and preserves effective structural components, thereby enhancing the interpretability of the search process. Experimental results across diverse PDE benchmarks demonstrate that SED-MCTS achieves superior performance under fixed computational budgets. Furthermore, it significantly improves search efficiency and robustness in scenarios characterized by noise interference and sparse observations.

Abstract

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.

Problem

• PDE solution discovery
• symbolic regression
• interpretability
• terminal feedback
• structural reuse

Innovation

• Monte Carlo Tree Search
• Structural Experience Distillation
• Counterfactual Subtree Interventions
• PDE Solution Discovery
• Symbolic Regression