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Physics-informed diffusion models in spectral space

Davide Gallon, Philippe von Wurstemberger, Patrick Cheridito, Arnulf Jentzen

TL;DR

This work introduces physics-informed spectral diffusion (PISD), a generative framework that learns joint distributions of PDE parameters and their solutions by performing diffusion in a scaled spectral latent space. The key idea is to couple a data-driven spectral encoding with a diffusion model while enforcing PDE constraints and observation conditions throughout sampling via Adam-guided diffusion posterior sampling, ensuring the generated fields maintain Sobolev regularity. PISD achieves substantial dimensionality reduction compared with grid-based approaches and yields improved PDE residuals and competitive accuracy, with significant speedups in inference for Poisson, Helmholtz, and Navier–Stokes problems. The method offers a physically grounded, efficient path for forward and inverse PDE problems under sparse observations, with potential extensions to more complex geometries and learned encoders.

Abstract

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.

Physics-informed diffusion models in spectral space

TL;DR

This work introduces physics-informed spectral diffusion (PISD), a generative framework that learns joint distributions of PDE parameters and their solutions by performing diffusion in a scaled spectral latent space. The key idea is to couple a data-driven spectral encoding with a diffusion model while enforcing PDE constraints and observation conditions throughout sampling via Adam-guided diffusion posterior sampling, ensuring the generated fields maintain Sobolev regularity. PISD achieves substantial dimensionality reduction compared with grid-based approaches and yields improved PDE residuals and competitive accuracy, with significant speedups in inference for Poisson, Helmholtz, and Navier–Stokes problems. The method offers a physically grounded, efficient path for forward and inverse PDE problems under sparse observations, with potential extensions to more complex geometries and learned encoders.

Abstract

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.
Paper Structure (32 sections, 1 theorem, 28 equations, 9 figures, 8 tables, 1 algorithm)

This paper contains 32 sections, 1 theorem, 28 equations, 9 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Work
  4. Physics-informed diffusion models.
  5. Diffusion models for PDEs.
  6. Diffusion models in infinite-dimensional function spaces.
  7. Method
  8. Problem setting
  9. Diffusion model in spectral space
  10. Physics-informed and measurement guidance
  11. Spectral encoding
  12. Numerical results
  13. Implementation details
  14. Dataset and training.
  15. Adam guidance.
  16. ...and 17 more sections

Key Result

Lemma 3.2

Let $k \in \mathbb N$, assume $\Exp[]{\lVert X_{\mathrm{data}}\rVert_{H^k(\mathbb{T}^d)}^2} < \infty$, let $(Z_n)_{n \in \mathbb Z^d}$ be i.i.d. $\mathcal{N}(0, 1)$ random variables, and let $N \in L^2(\mathbb{T}^d)$ be the random function given by Then $\Exp[]{\lVert N\rVert_{H^k(\mathbb{T}^d)}^2} < \infty$.

Figures (9)

  • Figure 1: Overview of physics-informed spectral diffusion (PISD): The model learns to generate function valued solutions by performing a diffusion process in a latent space of scaled spectral function representations. At inference, PDE and observation constraints are enforced via Adam-based guidance.
  • Figure 2: Forward process in a standard grid-based diffusion model (top) and in PISD (bottom).
  • Figure 3: Truncation of sine coefficients for the Poisson and Helmholtz equations. Left: the solution $u$ and corresponding coefficient $a$ for a Poisson problem. Center: a clipped heatmap of the sine coefficients. Right: the mask used for truncating the sine coefficients, with the hyperbolic truncation boundary highlighted in red.
  • Figure 4: Truncation of Fourier coefficients for the Navier--Stokes problem. Left: initial solution $u_1$. Center: a clipped heatmap of the Fourier coefficients. Right: the mask used for truncating the Fourier series.
  • Figure 5: Inverse problem with $500$ observations on $u$. Comparison between our method, DiffusionPDE, and FunDPS. The bottom row shows the Laplacian associated with each reconstructed solution.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Example 3.1: Poisson equation
  • Lemma 3.2
  • Remark 3.3: Choice of truncation set
  • proof