PDE Solutions Get Analytical

For decades, understanding Partial Differential Equation (PDE) solutions has been the exclusive domain of rigorous mathematical analysis, a painstaking, problem-by-problem endeavor. Traditional numerical simulations and even modern neural networks fall short, failing to directly produce the underlying mathematical structures that provide true insight. A new framework, Agentic Symbolic Search (ASYS), proposes a paradigm shift.

Automated Inductive Bias Injection for Symbolic Discovery

ASYS operates as a prior-guided framework where an agent synthesizes PDE theory, problem constraints, and past search experience into differentiable symbolic programs. This approach refines mathematical forms through evolutionary search while simultaneously fitting continuous parameters via gradient-based optimization. Crucially, this transforms the search into an automated form of inductive-bias injection, moving away from brute-force symbolic regression. The framework naturally recovers known analytical forms and constructs novel analytical approximations for problems where none previously existed, offering valuable guidance for mathematicians.

Bridging the Gap: From Computation to Mathematical Insight

The impact of ASYS is demonstrated across diverse PDE problems, including bounded dynamics, finite-time blow-up, and free-boundary phenomena. The framework generated interpretable representations such as a geometric interface formula for 2D Allen-Cahn dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up. These results signify a potential new era in characterizing PDE solutions, offering a powerful complement to handcrafted analytical solutions, mesh-based numerical methods, and black-box neural network approximations.