Sábháilte in:
| Príomhchruthaitheoirí: | , , , |
|---|---|
| Formáid: | Preprint |
| Foilsithe / Cruthaithe: |
2026
|
| Ábhair: | |
| Rochtain ar líne: | https://arxiv.org/abs/2602.09708 |
| Clibeanna: |
Cuir clib leis
Níl clibeanna ann, Bí ar an gcéad duine le clib a chur leis an taifead seo!
|
| _version_ | 1866908826073038848 |
|---|---|
| author | Gallon, Davide von Wurstemberger, Philippe Cheridito, Patrick Jentzen, Arnulf |
| author_facet | Gallon, Davide von Wurstemberger, Philippe Cheridito, Patrick Jentzen, Arnulf |
| contents | We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09708 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Physics-informed diffusion models in spectral space Gallon, Davide von Wurstemberger, Philippe Cheridito, Patrick Jentzen, Arnulf Machine Learning Artificial Intelligence Computer Vision and Pattern Recognition Numerical Analysis We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric 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. |
| title | Physics-informed diffusion models in spectral space |
| topic | Machine Learning Artificial Intelligence Computer Vision and Pattern Recognition Numerical Analysis |
| url | https://arxiv.org/abs/2602.09708 |