Diffusion sampling method improves sparse data recovery in pdes
Backward-Consistent Diffusion Sampling for Sparsely Observed PDE Inverse Problems
Machine Learning
Summary
Recovering tricky physical properties from very limited data is hard. The authors show that existing methods can fail when dealing with sudden changes in these properties. They propose a new technique, FunBCS, that works backward on the original inputs to better match the physical rules. Their method reduces errors significantly and runs faster in tests on several problems.
What this means in practice
- •For environmental engineers: Improve estimation of underground water flow properties from limited sensors using FunBCS for more accurate and efficient subsurface mapping.
- •For simulation software developers: Incorporate FunBCS to enhance inverse PDE solvers that reconstruct physical parameters from sparse data, reducing error and run time.
Authors
Yida Pan, Muhammad H. Ashiq, Chanyong Jung, Yixuan Jia, Jonah M. Miller, Qing Qu, Ismail Alkhouri
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.