Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networks

2026-06-18

AI Summary

Traditional approaches struggle to extract interpretable mathematical structures from numerical solutions or neural network approximations of partial differential equations (PDEs). This work proposes the Agentic Symbolic Search (ASYS) framework, which uniquely integrates agent-guided symbolic search with domain knowledge–driven inductive biases. ASYS employs differentiable symbolic programming to generate candidate analytical expressions and combines evolutionary search for structural optimization with gradient-based parameter tuning. The method overcomes limitations inherent in purely analytical, grid-based numerical, and data-driven PDE representation strategies. Evaluated on five challenging PDE problems, ASYS not only recovers known analytical solutions but also discovers novel interpretable expressions, including a two-dimensional interface formula for the Allen–Cahn equation and a nine-parameter contraction law for the Keller–Segel model—both previously unreported in the literature.

Abstract

Mathematicians understand a PDE solution through mathematical structures rather than tables of computed values. Historically, this has been the product of mathematical analysis, carried out by hand for each problem individually. Neither numerical simulation nor neural networks produce those structures directly. We propose Agentic Symbolic Search (ASYS), a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. This makes the search an automated form of inductive-bias injection rather than blind symbolic regression. For problems with known analytical forms, ASYS recovers these forms naturally; for other problems, ASYS constructs analytical approximations which can guide mathematicians toward further analysis. In our experiments, across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, in settings where no closed-form description was previously available. ASYS shows the possibility of a new paradigm for characterizing PDE solutions, beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.

Problem

• PDE
• symbolic representation
• mathematical structure
• analytical approximation
• interpretable solution

Innovation

• Agentic Symbolic Search
• symbolic regression
• inductive bias
• evolutionary search
• differentiable symbolic programs