Guided Flow Matching for Forward and Inverse PDE Problems with Sparse Observations: Algorithm and Theory
2026-05-25 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce FM4PDE, a new method to predict solutions of partial differential equations (PDEs) even when only limited data is available. Their approach learns how PDE inputs and outputs relate together, allowing it to simulate forward results or recover inputs from sparse observations. They provide mathematical guarantees for their sampling methods, including deterministic and stochastic versions, ensuring accuracy and efficiency. Tests on various PDE problems show their method is both accurate and faster than other state-of-the-art generative models.
partial differential equationsPDE coefficientsinverse problemsflow matchinggenerative modelsdeterministic samplingstochastic samplingerror boundsadaptive guidancescientific computing
Authors
Xifeng Zhang, Jin Zhao
Abstract
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 support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.