发表机构
University of Michigan; Los Alamos National Laboratory（密歇根大学; 洛斯阿拉莫斯国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对稀疏观测的PDE逆问题，提出输入空间优化方法FunBCS，通过后向一致采样确保物理一致性，在四个逆问题中误差降低27%-64%，速度提升1.4-2.1倍。
AI 中文摘要
从极其稀疏的观测中恢复偏微分方程（PDE）系数场是一个严重不适定的逆问题，而生成式机器学习方法（如扩散模型）已成为编码先验信息的主要方式。近期最先进的扩散求解器将这些先验提升到函数空间，在扩散去噪器的输出空间中寻找物理一致的解。我们证明，在不连续的PDE设定下，输出空间方法可能无法适当地以正确的系数场最小化未观测误差。因此，我们提出函数空间后向一致采样（FunBCS），一种用于求解PDE问题的输入空间优化方法，旨在找到最佳输入，使得去噪器重建结果具有物理一致性。我们进一步证明，与输出空间优化方法不同，FunBCS能适当地最小化未观测误差。根据我们的理论分析，我们还提供了在采样过程中动态分配输入空间优化步骤数量的见解。我们在四个PDE逆问题（包括不连续的达西流）上的评估表明，与当前最先进方法相比，FunBCS将重建误差降低了27%-64%，同时运行速度提高了1.4-2.1倍。
英文摘要
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.
