An Interpretable Approach to PDE Solution Discovery via Structural Experience Distillation

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.