Agentic Program Evolution of Executable Numerical Operators for PDE Dynamics

Time-dependent partial differential equation (PDE) simulations support forecasting and design in fluid dynamics, pattern formation, wave propagation, and many other physical systems. Although the governing equations may be known, computational budgets require them to be realized on finite grids and time steps. Accuracy and stability therefore depend on expert choices of derivative approximation, nonlinear discretization, filtering, operator splitting, time integration, and numerical constants, especially when relevant scales are unresolved. We introduce APEX (Agentic Program Evolution of eXecutable numerical operators), a framework that treats these coupled choices as a typed executable program and improves them through evaluator-guided evolution. A deterministic compiler rejects candidates that violate type or resource constraints, while a frozen evaluator supplies structured diagnostics of accuracy, stability, spectral behavior, and cost. A Program Evolution Graph retains successful and failed trials, and recurring subprograms are promoted to a reusable operator library. Our analysis characterizes finite-horizon error propagation, structured search state, and semantics-preserving operator abstraction. Experiments across canonical PDE families show that APEX improves long-horizon accuracy over learned neural operators while remaining competitive with task-specific coarse solvers. Combined library-and-history reuse improves a related PDE under the same target-side call budget. In one seven-call synthesis run on under-resolved 2D Navier–Stokes, APEX synthesizes a near-cutoff shadow-state program whose final error is $19.3\times$ lower than the standard coarse solver. APEX provides an explicit, auditable, and reusable approach to numerical-method design.