Neuro-Symbolic AI for Analytical Solutions of Differential Equations

ICLR 2026 Conference Submission17518 Authors

Keywords: Neural-Symbolic AI, Compositional Learning, Formal Grammars, Differential Equations, Analytical Solutions

Abstract: Analytical solutions to differential equations offer exact insight but are rarely available because discovering them requires expert intuition or exhaustive search in large combinatorial spaces. We introduce SIGS, a neuro-symbolic framework that automates this process. SIGS uses a formal grammar to generate only syntactically and physically valid building blocks, embeds these expressions into a continuous latent space, and then searches this space to assemble, score, and refine candidate closed-form solutions by minimizing a physics-based residual. This design unifies symbolic reasoning with numerical optimization; the grammar constrains candidate solution blocks to be proper by construction, while the latent search makes exploration tractable and data-free. Across a range of differential equations SIGS recovers exact solutions when they exist and finds highly accurate approximations otherwise, outperforming tree-based symbolic methods, traditional solvers, and neural PDE baselines in accuracy and wall-clock efficiency. These results are a step forward integrating symbolic structure with modern ML to discover interpretable, closed-form solutions at scale.

Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)

Submission Number: 17518