Neuro-Symbolic AI for Analytical Solutions of Differential Equations

Analytical solutions to differential equations offer exact, interpretable insight but are rarely available because discovering them requires expert intuition or exhaustive search in combinatorial spaces. We introduce SIGS, a neuro-symbolic framework that automates this process. SIGS uses a formal grammar to generate only syntactically valid building blocks, embeds these expressions into a continuous 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. SIGS is the first neuro-symbolic method to (i) analytically solve coupled systems of nonlinear PDEs, (ii) discover solutions under grammar misspecification, and (iii) produce accurate symbolic approximations for PDEs lacking known closed-form solutions. Overall, SIGS achieves orders-of-magnitude improvements in accuracy and efficiency over existing symbolic methods on standard benchmarks.