# Wavelet Diffusion Neural Operator

arXiv: https://arxiv.org/abs/2412.04833 · HTML: https://mlsanity.com/paper/2412.04833/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 67 of its 101 references that are in the map.
- First posted on arXiv 6 Dec 2024.

## 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, 30 of 67)

- [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)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [Is Conditional Generative Modeling all you need for Decision-Making?](https://mlsanity.com/paper/2211.15657/) (2022)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [ResShift: Efficient Diffusion Model for Image Super-resolution by Residual Shifting](https://mlsanity.com/paper/2307.12348/) (2023)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Flexible Diffusion Modeling of Long Videos](https://mlsanity.com/paper/2205.11495/) (2022)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC](https://mlsanity.com/paper/2302.11552/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [GenCast: Diffusion-based ensemble forecasting for medium-range weather](https://mlsanity.com/paper/2312.15796/) (2023)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)

## Built on by (most cited first, 29 of 29)

- [FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling](https://mlsanity.com/paper/2506.07902/) (2025)
- [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)
- [Learning with Calibration: Exploring Test-Time Computing of Spatio-Temporal Forecasting](https://mlsanity.com/paper/2506.00635/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Physics-informed diffusion models in spectral space](https://mlsanity.com/paper/2602.09708/) (2026)
- [DiffusionRollout: Uncertainty-Aware Rollout Planning in Long-Horizon PDE Solving](https://mlsanity.com/paper/2602.13616/) (2026)
- [From Uncertain to Safe: Conformal Adaptation of Diffusion Models for Safe PDE Control](https://mlsanity.com/paper/2502.02205/) (2025)
- [Latent Generative Solvers for Generalizable Long-Term Physics Simulation](https://mlsanity.com/paper/2602.11229/) (2026)
- [Accelerating PDE Surrogates via RL-Guided Mesh Optimization](https://mlsanity.com/paper/2603.02066/) (2026)
- [Geometry-Aware Neural Optimizer for Shape Optimization and Inversion](https://mlsanity.com/paper/2605.04474/) (2026)
- [A Physics-Informed Fourier-Wavelet Transformer for Multiscale Computational Fluid Dynamics Surrogate Modeling](https://mlsanity.com/paper/2606.24696/) (2026)
- [Spectral-inspired Operator Learning with Limited Data and Unknown Physics](https://mlsanity.com/paper/2505.21573/) (2025)
- [Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations](https://mlsanity.com/paper/2512.14980/) (2025)
- [Multi-Scale Wavelet Transformers for Operator Learning of Dynamical Systems](https://mlsanity.com/paper/2602.01486/) (2026)
- [Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations](https://mlsanity.com/paper/2603.08219/) (2026)
- [Where the Score Lives: A Wavelet View of Diffusion](https://mlsanity.com/paper/2606.08309/) (2026)
- [Sample-Efficient Diffusion-based Control of Complex Physics Systems](https://mlsanity.com/paper/2502.17893/) (2025)
- [GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries](https://mlsanity.com/paper/2509.24117/) (2025)
- [Generative Neural Operators through Diffusion Last Layer](https://mlsanity.com/paper/2602.04139/) (2026)
- [Modeling Score Approximation Errors in Diffusion Models via Forward SPDEs](https://mlsanity.com/paper/2602.08579/) (2026)
- [UniFluids: Unified Neural Operator Learning with Conditional Flow-matching](https://mlsanity.com/paper/2603.22309/) (2026)
- [PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics](https://mlsanity.com/paper/2605.03548/) (2026)
- [Wavelet Flow Matching for Multi-Scale Physics Emulation](https://mlsanity.com/paper/2605.16573/) (2026)
- [Small Models, Strong Priors: Architectural Inductive Bias for Parameter-Efficient Neural PDE Solvers](https://mlsanity.com/paper/2605.25949/) (2026)
- [Solving Inverse Problems of Chaotic Systems with Bidirectional Conditional Flow Matching](https://mlsanity.com/paper/2606.24824/) (2026)
- [CasEm: A Cascade Architecture for Long-Horizon Neural Emulation](https://mlsanity.com/paper/2609.34246/) (2026)
- [NeuronSifter: An Agent-in-Twin for Prioritizing Central Nervous System Interventions with World Action Models](https://mlsanity.com/paper/2609.35445/) (2026)
- [ReAl: Refine neural operators via aligning dynamical flow](https://mlsanity.com/paper/2610.20369/) (2026)

## Abstract (arXiv)

Simulating and controlling physical systems described by partial differential equations (PDEs) are crucial tasks across science and engineering. Recently, diffusion generative models have emerged as a competitive class of methods for these tasks due to their ability to capture long-term dependencies and model high-dimensional states. However, diffusion models typically struggle with handling system states with abrupt changes and generalizing to higher resolutions. In this work, we propose Wavelet Diffusion Neural Operator (WDNO), a novel PDE simulation and control framework that enhances the handling of these complexities. WDNO comprises two key innovations. Firstly, WDNO performs diffusion-based generative modeling in the wavelet domain for the entire trajectory to handle abrupt changes and long-term dependencies effectively. Secondly, to address the issue of poor generalization across different resolutions, which is one of the fundamental tasks in modeling physical systems, we introduce multi-resolution training. We validate WDNO on five physical systems, including 1D advection equation, three challenging physical systems with abrupt changes (1D Burgers' equation, 1D compressible Navier-Stokes equation and 2D incompressible fluid), and a real-world dataset ERA5, which demonstrates superior performance on both simulation and control tasks over state-of-the-art methods, with significant improvements in long-term and detail prediction accuracy. Remarkably, in the challenging context of the 2D high-dimensional and indirect control task aimed at reducing smoke leakage, WDNO reduces the leakage by 78% compared to the second-best baseline. The code can be found at https://github.com/AI4Science-WestlakeU/wdno.git.