Neuro-Symbolic AI for Analytical Solutions of Differential Equations

TL;DR

This work introduces SIGS, a grammar-guided neuro-symbolic framework for discovering analytical solutions to differential equations. It combines a context-free grammar-based library of solution atoms with a Grammar Variational Autoencoder to embed candidate expressions into a smooth latent space, enabling a two-stage structure search and coefficient refinement guided by the PDE residual. SIGS achieves state-of-the-art performance by recovering exact closed-form solutions when available and delivering accurate symbolic approximations for problems lacking closed forms, outperforming leading symbolic and neural baselines in both accuracy and speed. The approach emphasizes interpretability and principled search over brute-force symbolic regression, with potential for Automating Ansatz design via language models and integration with hybrid numerical-symbolic strategies in future work.

Abstract

Analytical solutions of differential equations offer exact insights into fundamental behaviors of physical processes. Their application, however, is limited as finding these solutions is difficult. To overcome this limitation, we combine two key insights. First, constructing an analytical solution requires a composition of foundational solution components. Second, iterative solvers define parameterized function spaces with constraint-based updates. Our approach merges compositional differential equation solution techniques with iterative refinement by using formal grammars, building a rich space of candidate solutions that are embedded into a low-dimensional (continuous) latent manifold for probabilistic exploration. This integration unifies numerical and symbolic differential equation solvers via a neuro-symbolic AI framework to find analytical solutions of a wide variety of differential equations. By systematically constructing candidate expressions and applying constraint-based refinement, we overcome longstanding barriers to extract such closed-form solutions. We illustrate advantages over commercial solvers, symbolic methods, and approximate neural networks on a diverse set of problems, demonstrating both generality and accuracy.

Figures (9)

• Figure 1: Overview over the proposed Symbolic Iterative Grammar Solver (SIGS). A. Terminal symbols $\Phi$ and rules $R$, together with non-terminals $N$ and starting symbol $S$, form the grammar $\mathcal{G}$ which generates the mathematical expressions in the library $\mathcal{L(G)}$. B. Each expression $w\in\mathcal{L(G)}$ is identified with a function $u$ in the finite set of candidate functions $\mathcal{U}$. C. The encoder $\mathcal{E}$ and decoder $\mathcal{D}$ of the Grammar Variational Autoencoder (GVAE, kusner2017grammar) embed the finite $\mathcal{L(G)}$ into the continuous latent space $Z$. D. Given a differential equation and system conditions, a structure search is performed over $z\in Z$ using iterative clustering, followed by a separate optimizations of the constants in the final structure, optimizing for lowest residual $\mathcal{R}$ of the corresponding candidate function $u=\mathcal{I\circ D}(z)\in\mathcal{U}$.
• Figure 2: From left to right: source term $F(x,y)$ for the Poisson–Gauss problem; finite-element solution $u_h$ (FEniCS); symbolic approximation $u_{\mathrm{sigs}}$ (SIGS); absolute error $\lvert u_h - u_{\mathrm{sigs}}\rvert$
• Figure 3: Comparison of different methods for solving the damped wave equation at $t=2.5$. All methods show the same physical domain $x,y \in [-8,8]$ with wave center at $(-5,5)$. Parameters: $k=0.5$, $\omega=0.4$, $\alpha=0.45$.
• Figure 4: Contour plot of the learned solution $u(x,t)$ for the Burgers equation. The horizontal axis represents the spatial domain $x \in [-5, 5]$, the vertical axis represents the temporal domain $t \in [0, 2]$, and the colormap indicates the solution magnitude ranging from 0.26 to 1.46. The solution is computed on a $128 \times 128$ discretization grid
• Figure 5: Contour plot of the learned solution $u(x,t)$ for the Diffusion equation. The horizontal axis represents the spatial domain $x \in [0, 1.4]$, the vertical axis represents the temporal domain $t \in [0, 1]$, and the colormap indicates the solution magnitude ranging from $-1.5$ to $11.9$. The solution is computed on a $128 \times 128$ discretization grid.