Title: Physics-Informed Conditional Diffusion for Multi-Modal PDEs
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Diffusion Models, Physics-Informed Models, Quantum Physics, Eigenfunctions, Partial Differential Equations, ML for Science

Abstract: Many physical systems that are represented by partial differential equations (PDEs) admit multiple valid solutions, such as eigenstates of differential operators, or wave modes, yet most neural PDE surrogates are deterministic and collapse to averages. This multiplicity of solutions is especially predominant in various engineering and scientific domains ranging from acoustics and seismology to quantum systems. With the ability to generate or complete sparse measurements, diffusion-based approaches to solve PDEs by sampling physically valid solutions are gaining traction as an alternative to traditional numerical solvers. In this paper, we present a novel physics-informed conditional diffusion framework for multi-modal PDEs, called PDEDIFF, that learns distributions over solution fields from sparse, irregular samples while enforcing governing equations and boundary conditions through mesh-free residual penalties computed by automatic differentiation. PDEDIFF is capable of effectively solving PDEs with multiple valid solutions by learning $\mathbb{P}[Y|X]$, i.e., it learns a solution field $Y$ for a corresponding input spatial information $X$. Unlike Physics‑Informed Neural Networks (PINNs), which minimize residuals around expected values $\mathbb{E}[Y|X]$ and hence tend to regress toward a conditional mean, PDEDIFF samples diverse physically consistent solutions by integrating PDE residuals directly into the diffusion objective. Our results indicate that generative, physics-informed diffusion is a practical tool for uncertainty-aware and multi-modal PDE modeling in low-to-moderate dimensions.

TL;DR: PDEDIFF is a mesh-free, physics-informed conditional diffusion model that samples the distribution of multi-modal PDEs' solution; autograd replaces finite differences,& beats PINNs & diffusion model baselines while recalling valid multiple solutions.