Learning When to Refine: Long-Horizon Reinforcement Learning for Budgeted Neural-Operator PDE Solvers

A budgeted adaptive neural-operator solver uses a global Fourier neural operator, a local patch-wise correction operator, and a set-aware selector guided by a macro policy to allocate a finite refinement budget over time. Rollout-verified policy improvement (RV-PI) evaluates feasible refinement counts through continuation rollouts and accepts updates only when held-out trajectory error improves, achieving a three-seed mean trajectory relative L2 error of 0.6910 on the shallow-water benchmark with a 32-intervention budget.