Neuro-Symbolic AI for Analytical Solutions of Differential Equations
This addresses a fundamental problem in physics and engineering by enabling exact insights into physical processes, though it appears incremental as it builds on existing compositional and iterative methods.
The paper tackles the difficulty of finding analytical solutions to differential equations by developing a neuro-symbolic AI framework that merges compositional techniques with iterative refinement, demonstrating advantages in generality and accuracy over existing methods.
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.