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

arXiv: https://arxiv.org/abs/2610.06883 · HTML: https://mlsanity.com/paper/2610.06883/

## 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.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 18 of 18)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Advantage-Weighted Regression: Simple and Scalable Off-Policy Reinforcement Learning](https://mlsanity.com/paper/1910.00177/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [Reinforcement Learning for Adaptive Mesh Refinement](https://mlsanity.com/paper/2103.01342/) (2021)
- [Differentiability in Unrolled Training of Neural Physics Simulators on Transient Dynamics](https://mlsanity.com/paper/2402.12971/) (2024)
- [Deep Reinforcement Learning for Adaptive Mesh Refinement](https://mlsanity.com/paper/2209.12351/) (2022)
- [Local neural operator for solving transient partial differential equations on varied domains](https://mlsanity.com/paper/2203.08145/) (2022)
- [TANTE: Time-Adaptive Operator Learning via Neural Taylor Expansion](https://mlsanity.com/paper/2502.08574/) (2025)
- [Accelerating PDE Surrogates via RL-Guided Mesh Optimization](https://mlsanity.com/paper/2603.02066/) (2026)
- [Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation](https://mlsanity.com/paper/2605.24041/) (2026)
- [Enforcing the Principle of Locality for Physical Simulations with Neural Operators](https://mlsanity.com/paper/2405.01319/) (2024)

## 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.