# ReAl: Refine neural operators via aligning dynamical flow

arXiv: https://arxiv.org/abs/2610.20369 · HTML: https://mlsanity.com/paper/2610.20369/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 29 of its 36 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 29 of 29)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Training Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2305.13301/) (2023)
- [Flow-GRPO: Training Flow Matching Models via Online RL](https://mlsanity.com/paper/2505.05470/) (2025)
- [DPOK: Reinforcement Learning for Fine-tuning Text-to-Image Diffusion Models](https://mlsanity.com/paper/2305.16381/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [DiffusionNFT: Online Diffusion Reinforcement with Forward Process](https://mlsanity.com/paper/2509.16117/) (2025)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecasting](https://mlsanity.com/paper/2306.01984/) (2023)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation](https://mlsanity.com/paper/2309.01745/) (2023)
- [On conditional diffusion models for PDE simulations](https://mlsanity.com/paper/2410.16415/) (2024)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [RealPDEBench: A Benchmark for Complex Physical Systems with Real-World Data](https://mlsanity.com/paper/2601.01829/) (2026)
- [Hierarchical Implicit Neural Emulators](https://mlsanity.com/paper/2506.04528/) (2025)
- [DiffusionRollout: Uncertainty-Aware Rollout Planning in Long-Horizon PDE Solving](https://mlsanity.com/paper/2602.13616/) (2026)
- [FlowRefiner: Flow Matching-Based Iterative Refinement for 3D Turbulent Flow Simulation](https://mlsanity.com/paper/2604.17149/) (2026)
- [PDESpectralRefiner: Achieving More Accurate Long Rollouts with Spectral Adjustment](https://mlsanity.com/paper/2506.10711/) (2025)
- [Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations](https://mlsanity.com/paper/2605.13790/) (2026)

## Abstract (arXiv)

Neural partial differential equation (PDE) surrogates trained with mean squared error (MSE) can capture dominant dynamics but often lose high-frequency information and fine-scale structures. Generative models (e.g., Flow Matching) can capture the distribution of fine-scale details, yet stochastic errors may accumulate during autoregressive rollouts and compromise long-term stability. To address these limitations, we introduce Refine and Align(ReAl), a model-agnostic plug-in that augments pretrained PDE surrogates to improve the reconstruction of high-frequency structures associated with complex physical phenomena. In the refinement stage, a conditional flow-matching model learns the residual distribution around the deterministic base prediction, recovering missing fine-scale structures while retaining the dominant dynamics. In the alignment stage, to mitigate the generative discrepancies amplified in the autoregressive rollout, particularly in chaotic systems, we generate multiple trajectories from the same input and reinforce the refiner with trajectory-level rewards, thereby achieving long-horizon dynamical alignment. Experiments across diverse surrogate architectures and PDE tasks demonstrate that ReAl consistently improves high-frequency fidelity, forecasting accuracy, and long-horizon stability.