New Symbolic Equation Solver (SES) needs zero training data

Solving complex equations without known analytical solutions has long forced scientists to rely on numerical methods, which yield approximations rather than explicit symbolic formulas. Now, researchers from an unnamed institution (authors Sergei Garmaev, Vinay Sharma, Olga Fink) present the Symbolic Equation Solver (SES), a novel framework that reformulates equation solving as an optimization problem over differentiable symbolic models. The key innovation: SES does not need any paired input-output training data. Instead, it constructs its objective directly from the governing equation and initial or boundary conditions, learning a symbolic expression that satisfies the constraints.

The method was tested on a diverse set of problems including a system of algebraic equations, an equation with transcendental terms, an ordinary differential equation, and partial differential equations under various initial/boundary conditions. In all cases, SES recovered compact symbolic expressions that matched the known analytical solutions. The work eliminates a major bottleneck in symbolic regression—the reliance on expensive or unavailable training data—and opens the door to automatically discovering closed-form solutions for equations where only the equation itself is known.