Backward-Consistent Diffusion Sampling for Sparsely Observed PDE Inverse Problems

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 (DMs)) have become a leading way to supply the prior. Recent state-of-the-art DM-based solvers lift these priors to function spaces and keep the learned prior separate from the physics. At inference, a differentiable surrogate (e.g., a neural operator) is used to guide the sampling. Existing solvers enforce measurement consistency after computing the denoiser prediction. Under sparse observations, these updates can leave unobserved components of the denoiser prediction unchanged, so denoiser errors in the measurement operator's nullspace may persist. Consequently, we propose Function space Backward-Consistent Sampling (FunBCS), where we optimize the denoiser input, altering the denoiser prediction and thereby allowing unobserved components to change. On four PDE inverse problems with only $3$% of the solution field observed, our method improves reconstruction accuracy on every task (notably by $2.5\times$ on discontinuous coefficients), while using one hyperparameter configuration for all tasks and running about $15$% faster per sample.