# Backward-Consistent Diffusion Sampling for Sparsely Observed PDE Inverse Problems

arXiv: https://arxiv.org/abs/2610.04624 · HTML: https://mlsanity.com/paper/2610.04624/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- First posted on arXiv 3 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Abstract (arXiv)

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× faster when compared to the current state-of-the-art.