AI RESEARCH
Guided Flow Matching for Forward and Inverse PDE Problems with Sparse Observations: Algorithm and Theory
arXiv CS.LG
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ArXi:2605.25509v1 Announce Type: cross Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we deterministic, stochastic, and hybrid samplers.