FunBCS：输入空间优化求解稀疏观测 PDE 反问题，误差降低 27-64%
Backward-Consistent Diffusion Sampling for Sparsely Observed PDE Inverse Problems
扩散模型解 PDE 反问题有坑，这篇论文证明输出空间优化会失准，换成 FunBCS 在输入空间优化，误差降 27-64% 还更快。
论文提出 Function space Backward-Consistent Sampling（FunBCS），用于从极稀疏观测中重建 PDE 系数场。作者证明在非连续 PDE 设定下，现有扩散求解器的输出空间优化方法无法正确最小化未观测误差。FunBCS 改为在输入空间优化，寻找使去噪器重建结果物理一致的输入，并从理论上证明其能正确最小化未观测误差。在包括 Darcy Flow 在内的四个 PDE 反问题上，FunBCS 将重建误差降低 27-64%，同时速度比当前 SOTA 快 1.4-2.1 倍。
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) 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.
