Neural operators serve as fast surrogates for time‑dependent PDEs, yet autoregressive deployment creates a refinement‑allocation problem: prediction errors vary across space‑time while only a finite number of local corrections can be applied along a trajectory. We cast this as budgeted adaptive neural‑operator solving. A global Fourier neural operator advances the full field in one step, a local operator proposes patch‑wise residual corrections, and a set‑aware selector picks the regions to refine. A macro policy decides when and how many of the remaining budget $B$ to spend. To train this policy we introduce rollout‑verified policy improvement (RV‑PI): feasible refinement counts are evaluated through actual continuation rollouts of the learned PDE solver, long‑horizon advantages are turned into conservative policy targets, and an update is accepted only if held‑out trajectory error improves. On the shallow‑water benchmark with a $32$‑intervention budget, RV‑PI achieves a three‑seed mean trajectory relative $L_2$ error of $0.6910$, improving over immediate‑only policy improvement by $5.37\%$ and over RandomMacro by $2.41\%$. On the forcing‑driven Brusselator benchmark with a $76$‑intervention budget, RV‑PI attains $0.09954$, beating immediate‑only by $2.31\%$ and RandomMacro by $5.32\%$. These results demonstrate that, under a fixed refinement budget, the value of a local correction depends on its downstream effect on the autoregressive trajectory, not merely on immediate error reduction.
Review