Zongmin Yu, Liu Yang
ASYS is a prior-guided framework where an agent translates PDE theory and problem constraints into differentiable symbolic programs via evolutionary search and gradient-based optimization, discovering interpretable analytical forms.
Mathematicians understand PDE solutions through mathematical structures, not numerical tables. Existing methods (numerical simulation, neural networks) do not directly produce such structures, requiring manual analysis per problem.
ASYS uses an agent to convert PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. Evolutionary search refines mathematical forms, and gradient-based optimization fits continuous parameters. This automates inductive-bias injection rather than blind symbolic regression.
Across five problems (bounded dynamics, finite-time blow-up, free-boundary focusing), ASYS produced interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, where no closed-form description existed. This demonstrates a new paradigm beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.