[Submitted on 19 Sep 2026]
Title:Learning When to Refine: Long-Horizon Reinforcement Learning for Budgeted Neural-Operator PDE Solvers
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Abstract:Neural operators provide fast surrogates for time-dependent PDEs, but autoregressive deployment creates a refinement-allocation problem: prediction errors vary over space and time, while only a finite number of local corrections can be committed along a trajectory. We formulate this as budgeted adaptive neural-operator solving. A global Fourier neural operator advances the full field, a local operator proposes patch-wise residual corrections, and a set-aware selector chooses where to refine. A macro policy decides when and how much of the remaining refinement budget to spend. We introduce rollout-verified policy improvement (RV-PI), which evaluates feasible refinement counts through actual continuation rollouts of the learned PDE solver, converts long-horizon advantages into conservative policy targets, and accepts an update only when held-out trajectory error improves. On the shallow-water benchmark with a 32-intervention budget, RV-PI achieves a three-seed mean trajectory relative L2 error of 0.6910, improving over immediate-only policy improvement by 5.37% and RandomMacro by 2.41%. On the forcing-driven Brusselator benchmark with a 76-intervention budget, RV-PI attains 0.09954, improving over immediate-only policy improvement by 2.31% and RandomMacro by 5.32%. These results show that, under a fixed refinement budget, the value of a local correction depends on its downstream effect on the autoregressive trajectory, not only on its immediate error reduction.
Comments: 21 pages, 4 figures, 2 tables
Subjects:
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
Cite as: arXiv:2610.06883 [cs.LG]
(or arXiv:2610.06883v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.06883
arXiv-issued DOI via DataCite
Submission history
From: Ange Tong [view email] [v1] Sat, 19 Sep 2026 12:17:32 UTC (2,400 KB)
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