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

arXiv 2610.06883 ↗First posted 19 Sep 2026cs.LG cs.AIcited by 0

On the map: Deep reinforcement learning methods → RL theory and sample complexity

Abstract (arXiv)

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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods. Builds on 18 of its 25 references that are in the map. First posted on arXiv 19 Sep 2026.

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References · 18

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2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
2022Message Passing Neural PDE Solverscited by 386
2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 216
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2022Blending Neural Operators and Relaxation Methods in PDE Numerical Solverscited by 54
2021Reinforcement Learning for Adaptive Mesh Refinementcited by 39
2024Differentiability in Unrolled Training of Neural Physics Simulators on Transient Dynamicscited by 27

6 more

2022Deep Reinforcement Learning for Adaptive Mesh Refinementcited by 20
2022Local neural operator for solving transient partial differential equations on varied domainscited by 9
2025TANTE: Time-Adaptive Operator Learning via Neural Taylor Expansioncited by 8
2026Accelerating PDE Surrogates via RL-Guided Mesh Optimizationcited by 2
2026Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigationcited by 2
2024Enforcing the Principle of Locality for Physical Simulations with Neural Operatorscited by 1

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