# Generative Latent Neural PDE Solver using Flow Matching

arXiv: https://arxiv.org/abs/2503.22600 · HTML: https://mlsanity.com/paper/2503.22600/

## Key facts

- Cited by 28 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 68 of its 105 references that are in the map.
- First posted on arXiv 28 Mar 2025.

## 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 68)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (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)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [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)
- [Common Diffusion Noise Schedules and Sample Steps are Flawed](https://mlsanity.com/paper/2305.08891/) (2023)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [Efficient Diffusion Training via Min-SNR Weighting Strategy](https://mlsanity.com/paper/2303.09556/) (2023)

## Built on by (most cited first, 28 of 28)

- [Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulation](https://mlsanity.com/paper/2507.02608/) (2025)
- [D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matching](https://mlsanity.com/paper/2602.21469/) (2026)
- [Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria](https://mlsanity.com/paper/2508.07102/) (2025)
- [Particle-Guided Diffusion Models for Partial Differential Equations](https://mlsanity.com/paper/2601.23262/) (2026)
- [FlowRefiner: Flow Matching-Based Iterative Refinement for 3D Turbulent Flow Simulation](https://mlsanity.com/paper/2604.17149/) (2026)
- [Flow marching for a generative PDE foundation model](https://mlsanity.com/paper/2509.18611/) (2025)
- [Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing](https://mlsanity.com/paper/2604.07366/) (2026)
- [Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders](https://mlsanity.com/paper/2505.13569/) (2025)
- [EquiReg: Equivariance Regularized Diffusion for Inverse Problems](https://mlsanity.com/paper/2505.22973/) (2025)
- [Latent Generative Solvers for Generalizable Long-Term Physics Simulation](https://mlsanity.com/paper/2602.11229/) (2026)
- [Reframing Generative Models for Physical Systems using Stochastic Interpolants](https://mlsanity.com/paper/2509.26282/) (2025)
- [Efficient Generative Transformer Operators For Million-Point PDEs](https://mlsanity.com/paper/2512.04974/) (2025)
- [Residual-augmented flow matching operators for probabilistic partial differential equations](https://mlsanity.com/paper/2512.12749/) (2025)
- [Flow Matching and Diffusion Models via PointNet for Generating Fluid Fields on Irregular Geometries](https://mlsanity.com/paper/2601.03030/) (2026)
- [Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations](https://mlsanity.com/paper/2605.13790/) (2026)
- [Efficient probabilistic surrogate modeling techniques for partially-observed large-scale dynamical systems](https://mlsanity.com/paper/2511.04641/) (2025)
- [Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver](https://mlsanity.com/paper/2602.18248/) (2026)
- [Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency](https://mlsanity.com/paper/2605.08454/) (2026)
- [Hamiltonian Neural PDE Solvers through Functional Approximation](https://mlsanity.com/paper/2505.13275/) (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)
- [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)
- [Learning to Discretize: Diffusion-Based Adaptive Mesh with Spectral Guidance](https://mlsanity.com/paper/2607.11974/) (2026)
- [Round-Trip Consistency: Bidirectional Diffusion Models Can Predict Their Own Rollout Errors](https://mlsanity.com/paper/2608.00675/) (2026)
- [Composing Flow-Matching Energies with Known Physics: Generation, OOD Detection, and Inversion on PDE Fields](https://mlsanity.com/paper/2608.18004/) (2026)
- [FLARE: Flow Matching with Local Axis-Angle Representations for Stochastic Micromagnetic Evolution](https://mlsanity.com/paper/2609.34070/) (2026)
- [Cross-Domain Pretraining for Steady-State Neural CFD Surrogates](https://mlsanity.com/paper/2610.10398/) (2026)

## Abstract (arXiv)

Autoregressive next-step prediction models have become the de-facto standard for building data-driven neural solvers to forecast time-dependent partial differential equations (PDEs). Denoise training that is closely related to diffusion probabilistic model has been shown to enhance the temporal stability of neural solvers, while its stochastic inference mechanism enables ensemble predictions and uncertainty quantification. In principle, such training involves sampling a series of discretized diffusion timesteps during both training and inference, inevitably increasing computational overhead. In addition, most diffusion models apply isotropic Gaussian noise on structured, uniform grids, limiting their adaptability to irregular domains. We propose a latent diffusion model for PDE simulation that embeds the PDE state in a lower-dimensional latent space, which significantly reduces computational costs. Our framework uses an autoencoder to map different types of meshes onto a unified structured latent grid, capturing complex geometries. By analyzing common diffusion paths, we propose to use a coarsely sampled noise schedule from flow matching for both training and testing. Numerical experiments show that the proposed model outperforms several deterministic baselines in both accuracy and long-term stability, highlighting the potential of diffusion-based approaches for robust data-driven PDE learning.