Backward-Consistent Diffusion Sampling for Sparsely Observed PDE Inverse Problems

2026-10-03

AI Summary

This study addresses the ill-posed nature of inverse problems for partial differential equations (PDEs) under sparse observations and the uncontrollable unobserved errors inherent in existing diffusion models that optimize within the output space. To overcome these limitations, this work proposes Functional Backward Consistent Sampling (FunBCS), which shifts optimization from the output space to the coefficient field in the input space. This paradigm is theoretically proven, for the first time, to minimize unobserved errors, complemented by a dynamic step allocation strategy ensuring physically consistent reconstructions. Evaluated across four PDE inverse problem benchmarks, FunBCS reduces reconstruction errors by 27%–64% and accelerates computation by 1.4–2.1× compared to state-of-the-art methods.

Abstract

Recovering Partial Differential Equation (PDE) coefficient fields from extremely sparse observations is a severely ill-posed inverse problem for which generative machine learning methods (e.g., diffusion models) have become a leading way to encode the prior. Recent state-of-the-art diffusion solvers lift these priors to function spaces, finding a physics-consistent reconstruction in the output space of the diffusion denoiser. We prove that, in a discontinuous PDE setting, output space methods can result in failure to appropriately minimize the unobserved error with the correct coefficient field. Consequently, we propose Function space Backward-Consistent Sampling (FunBCS), an input space optimization approach for solving PDE problems which aims to find the best input such that the denoiser reconstruction is physics-consistent. We then prove that FunBCS appropriately minimizes the unobserved error, unlike output space optimization methods. Per our theoretical analysis, we also provide insights on how to dynamically allocate the number of input space optimization steps used throughout the sampling process. Our evaluations, across four PDE inverse problems (including the discontinuous Darcy flow), demonstrate that FunBCS reduces the reconstruction error by $27$-$64\%$ while running $1.4$-$2.1\times$ faster when compared to the current state-of-the-art.

Problem

• PDE inverse problems
• sparse observations
• diffusion models
• coefficient field recovery
• ill-posed problem

Innovation

• Diffusion Models
• PDE Inverse Problems
• Function Space Optimization
• Backward-Consistent Sampling
• Sparse Observations