# FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling arXiv: https://arxiv.org/abs/2506.07902 ยท HTML: https://mlsanity.com/paper/2506.07902/ ## Key facts - Cited by 20 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 66 of its 94 references that are in the map. - First posted on arXiv 9 Jun 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 66) - [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017) - [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020) - [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) - [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) - [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022) - [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022) - [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016) - [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) - [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019) - [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020) - [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020) - [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019) - [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022) - [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020) - [Perceiver: General Perception with Iterative Attention](https://mlsanity.com/paper/2103.03206/) (2021) - [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020) - [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020) - [ConvNeXt V2: Co-designing and Scaling ConvNets with Masked Autoencoders](https://mlsanity.com/paper/2301.00808/) (2023) - [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021) - [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022) - [3DShape2VecSet: A 3D Shape Representation for Neural Fields and Generative Diffusion Models](https://mlsanity.com/paper/2301.11445/) (2023) - [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015) - [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022) - [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021) - [From data to functa: Your data point is a function and you can treat it like one](https://mlsanity.com/paper/2201.12204/) (2022) ## Built on by (most cited first, 20 of 20) - [PI-MFM: Physics-informed multimodal foundation model for solving partial differential equations](https://mlsanity.com/paper/2512.23056/) (2025) - [Scale-Adaptive Generative Flows for Multiscale Scientific Data](https://mlsanity.com/paper/2509.02971/) (2025) - [RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion](https://mlsanity.com/paper/2509.21659/) (2025) - [Physics-informed diffusion models in spectral space](https://mlsanity.com/paper/2602.09708/) (2026) - [RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations](https://mlsanity.com/paper/2509.01234/) (2025) - [Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing](https://mlsanity.com/paper/2604.07366/) (2026) - [Divergence-Free Diffusion Models for Incompressible Fluid Flows](https://mlsanity.com/paper/2601.19368/) (2026) - [PILD: Physics-Informed Learning via Diffusion](https://mlsanity.com/paper/2601.21284/) (2026) - [Decoupled Latent Optimization of Diffusion Models for Full Waveform Inversion](https://mlsanity.com/paper/2606.14139/) (2026) - [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026) - [GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries](https://mlsanity.com/paper/2509.24117/) (2025) - [Reconstructing Multi-Scale Physical Fields from Extremely Sparse Measurements with an Autoencoder-Diffusion Cascade](https://mlsanity.com/paper/2512.01572/) (2025) - [FunPhase: A Periodic Functional Autoencoder for Motion Generation via Phase Manifolds](https://mlsanity.com/paper/2512.09423/) (2025) - [pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learning](https://mlsanity.com/paper/2603.16757/) (2026) - [PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics](https://mlsanity.com/paper/2605.03548/) (2026) - [StreamPhy: Streaming Inference of High-Dimensional Physical Dynamics via State Space Models](https://mlsanity.com/paper/2605.07384/) (2026) - [(HB-ARFM) History-Bootstrapped Flow Matching for Inverse Boiling Reconstruction](https://mlsanity.com/paper/2606.00349/) (2026) - [Multiphase-Diff: Diffusion-Based Generative Modeling for High-Contrast Multiphase Physical Systems with Sharp Interfaces](https://mlsanity.com/paper/2608.13669/) (2026) - [PhysDEM: Physics-Defined Energy-Matching Diffusion for Spatiotemporal Field Generation under Scarce Measurements](https://mlsanity.com/paper/2610.01759/) (2026) - [Measuring Effective Data Resolution in Guided Diffusion Posteriors](https://mlsanity.com/paper/2610.04422/) (2026) ## Abstract (arXiv) Recent advances in generative modeling -- particularly diffusion models and flow matching -- have achieved remarkable success in synthesizing discrete data such as images and videos. However, adapting these models to physical applications remains challenging, as the quantities of interest are continuous functions governed by complex physical laws. Here, we introduce FunDiff, a novel framework for generative modeling in function spaces. FunDiff combines a latent diffusion process with a function autoencoder architecture to handle input functions with varying discretizations, generate continuous functions evaluable at arbitrary locations, and seamlessly incorporate physical priors. These priors are enforced through architectural constraints or physics-informed loss functions, ensuring that generated samples satisfy fundamental physical laws. We theoretically establish minimax optimality guarantees for density estimation in function spaces, showing that diffusion-based estimators achieve optimal convergence rates under suitable regularity conditions. We demonstrate the practical effectiveness of FunDiff across diverse applications in fluid dynamics and solid mechanics. Empirical results show that our method generates physically consistent samples with high fidelity to the target distribution and exhibits robustness to noisy and low-resolution data. Code and datasets are publicly available at https://github.com/sifanexisted/fundiff.