Neuro-Symbolic AI for Analytical Solutions of Differential Equations

The understanding of physical processes has been a long-standing effort for scientists and engineers. A key step in this endeavor is to translate physical insights (laws) into precise mathematical relationships that capture the underlying phenomena. These relationships are then tested through experiments, which either validate the proposed hypothesis or suggest refinements. Among such mathematical formulations, differential equations (DEs) are especially ubiquitous across disciplines, as they describe how physical quantities evolve over time and space. Finding analytical (also referred to as explicit or closed-form) solutions to these equations, that is, a mathematical expression that satisfies the differential equation along with the given initial and boundary conditions, provides a structured way to compare theoretical predictions with experimental measurements. Moreover, analytical solutions often reveal intrinsic properties of physical systems, such as stability, periodicity, underlying symmetries and asymptotic behavior. Thus, analytical solutions provide deep insight into how these systems behave in time and space. Despite intense efforts over centuries, there are very few methods to construct analytical solutions of differential equations. All of them can be viewed as fundamentally compositional: They break complex equations into simpler, more manageable pieces and then systematically recombine those pieces into a final solution.