Particle-Guided Diffusion Models for Partial Differential Equations

TL;DR

The work tackles probabilistic PDE solving by marrying physics-informed residuals with pretrained diffusion priors through a Sequential Monte Carlo (SMC) framework. It introduces Guided Euler–Maruyama (GEM) and Second-Order Stochastic Guided (SOSaG) proposals, along with a pseudo-bootstrap (pBS) variant that bridges diffusion priors with PINNs. A tempering-based Bridging to PINNs and an evolutionary-algorithm interpretation of SMC are proposed to enhance diversity and robustness. Across benchmark PDEs and multiphysics systems with observation noise, the proposed approach achieves lower relative errors than state-of-the-art generative PDE methods, demonstrating scalable uncertainty-aware PDE solving with physics-informed regularization.

Abstract

We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples remain physically admissible. We embed this sampling procedure within a new Sequential Monte Carlo (SMC) framework, yielding a scalable generative PDE solver. Across multiple benchmark PDE systems as well as multiphysics and interacting PDE systems, our method produces solution fields with lower numerical error than existing state-of-the-art generative methods.

Figures (10)

• Figure 1: Contour plot of the ground truth parameters overlayed with the observed values (highlighted pixels) for each PDE. The top row shows the coefficient field and the bottom row shows the solution field.
• Figure 2: The top two rows show contour plot of the generated PDE solutions and coefficients using different methods. The bottom two rows show the corresponding (relative) error when compared to the ground truth (GT). We especially observe that the most erroneous part are around the edges. This is consistent with the common problem of diffusion models that they tend to smooth out the high-frequency information of the generated samples.
• Figure 3: Averaged ESS for the EM TDS algorithm ($\beta = 0$) across 20 runs. We can see that the sampler degenerates towards the end of the sampling process often leading to only one particle contributing to the target estimate.
• Figure 4: Example likelihood values for the same particles generated by SMC. The figure on the left shows the likelihood values with a larger variance, while the left figure shows the likelihood values with a smaller variance.
• Figure 5: Reconstruction comparison for the benchmark PDE experiments. The top row shows the contour plots of the reconstruction of $a$ while the bottom row shows the reconstruction of $u$. The final two plots on the right show the ground truth of the respective fields. The figures show a single example run of each method.