Interpretable search improves discovery of symbolic PDE solutions
An Interpretable Approach to PDE Solution Discovery via Structural Experience Distillation
Artificial Intelligence
Summary
Solving certain physics problems often means finding simple equations from data, but current methods don't explain how parts of these solutions help or hurt. This paper presents a method that learns from parts of past attempts, reusing good pieces to find better equations faster and more reliably. The new approach works well even with limited or noisy data and can handle complex systems involving multiple interacting processes. The authors show their method improves discovering understandable physical equations across many test problems.
What this means in practice
- •For computational physics teams: Discover interpretable symbolic solutions for physical systems under known constraints, improving model clarity and reliability in simulations.
- •For engineering simulation developers: Enhance efficiency and robustness of discovering PDE-based models from noisy or scarce observational data in multiphysics simulations.
Authors
Yunpeng Gong, Huolong Wu, Can Yang, Min Jiang
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.