Physics-Informed Diffusion Models in Spectral Space

Davide Gallon, Philippe Von Wurstemberger, Patrick Cheridito, Arnulf Jentzen
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:32814-32831, 2026.

Abstract

We propose physics-informed spectral diffusion (PISD), a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems. We learn the joint distribution of PDE parameters and solutions via a diffusion process in a latent space of scaled spectral representations, where Gaussian noise corresponds to functions with controlled regularity. This spectral formulation enables significant dimensionality reduction compared to grid-based diffusion models and ensures that the induced process in function space remains within a class of functions for which the PDE operators are well defined. Building on diffusion posterior sampling, we enforce physics-informed constraints and measurement conditions during inference, applying Adam-based updates at each diffusion step. We evaluate the proposed approach on Poisson, Helmholtz, and incompressible Navier-Stokes equations, demonstrating improved accuracy and computational efficiency compared with existing diffusion-based PDE solvers, which are state of the art for sparse observations. Code is available at https://github.com/deeplearningmethods/PISD.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-gallon26a, title = {Physics-Informed Diffusion Models in Spectral Space}, author = {Gallon, Davide and Wurstemberger, Philippe Von and Cheridito, Patrick and Jentzen, Arnulf}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {32814--32831}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/gallon26a/gallon26a.pdf}, url = {https://proceedings.mlr.press/v306/gallon26a.html}, abstract = {We propose physics-informed spectral diffusion (PISD), a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems. We learn the joint distribution of PDE parameters and solutions via a diffusion process in a latent space of scaled spectral representations, where Gaussian noise corresponds to functions with controlled regularity. This spectral formulation enables significant dimensionality reduction compared to grid-based diffusion models and ensures that the induced process in function space remains within a class of functions for which the PDE operators are well defined. Building on diffusion posterior sampling, we enforce physics-informed constraints and measurement conditions during inference, applying Adam-based updates at each diffusion step. We evaluate the proposed approach on Poisson, Helmholtz, and incompressible Navier-Stokes equations, demonstrating improved accuracy and computational efficiency compared with existing diffusion-based PDE solvers, which are state of the art for sparse observations. Code is available at https://github.com/deeplearningmethods/PISD.} }
Endnote
%0 Conference Paper %T Physics-Informed Diffusion Models in Spectral Space %A Davide Gallon %A Philippe Von Wurstemberger %A Patrick Cheridito %A Arnulf Jentzen %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-gallon26a %I PMLR %P 32814--32831 %U https://proceedings.mlr.press/v306/gallon26a.html %V 306 %X We propose physics-informed spectral diffusion (PISD), a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems. We learn the joint distribution of PDE parameters and solutions via a diffusion process in a latent space of scaled spectral representations, where Gaussian noise corresponds to functions with controlled regularity. This spectral formulation enables significant dimensionality reduction compared to grid-based diffusion models and ensures that the induced process in function space remains within a class of functions for which the PDE operators are well defined. Building on diffusion posterior sampling, we enforce physics-informed constraints and measurement conditions during inference, applying Adam-based updates at each diffusion step. We evaluate the proposed approach on Poisson, Helmholtz, and incompressible Navier-Stokes equations, demonstrating improved accuracy and computational efficiency compared with existing diffusion-based PDE solvers, which are state of the art for sparse observations. Code is available at https://github.com/deeplearningmethods/PISD.
APA
Gallon, D., Wurstemberger, P.V., Cheridito, P. & Jentzen, A.. (2026). Physics-Informed Diffusion Models in Spectral Space. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:32814-32831 Available from https://proceedings.mlr.press/v306/gallon26a.html.

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