Physics-Guided Conditional Flow Matching with Energy Regularization for Robust PDE Inverse Problems
Abstract
We consider partial differential equation (PDE) inverse problems from sparse, noisy, and corrupted observations, with the aim of recovering unknown coefficient fields and associated state variables in a mesh-free setting. Under such sparse and corrupted observations, standard physics-informed and generative approaches typically treat all samples indiscriminately and therefore lack a principled mechanism for reconciling physical laws with contaminated data. We address this difficulty with a two-stage flow-matching framework. In the first stage, we develop physics-guided conditional flow matching (PG-CFM), which incorporates strong-form PDE information through residual regularization along the generative trajectories together with intermittent global collocation constraints. In the second stage, we introduce energy-regularized flow matching (ERFM), which fine-tunes the Stage-1 model by assigning each observation a physics--data energy score from a frozen teacher and reweighting the flow-matching objective to reduce the influence of high-energy, PDE-inconsistent samples. We show that the resulting Stage-2 objective is equivalent to flow matching under a teacher-induced reweighted data distribution, which gives a population-level interpretation of the robustness mechanism. Numerical experiments on several inverse benchmarks, including Poisson and Navier--Stokes problems, show that the proposed framework yields more accurate coefficient recovery than robust PINN variants and competing generative baselines.
Related
- Data-driven Smooth-to-hard Spectral Regularization for General Linear Inverse Problems
- Data-driven stochastic asymptotic regularization for ill-posed inverse problems with a posteriori and heuristic stopping
- Data-driven Regularization for Inverse Problems
- Direct Conditional Transition Sampling for Diffusion Inverse Problems
- Reconstruction-Aware Distribution Matching for Inverse Problems with Unpaired Data
BibTeX
@misc{chen2026physicsguided,
title = {{Physics-Guided Conditional Flow Matching with Energy Regularization for Robust PDE Inverse Problems}},
author = {Yongsheng Chen and Shuo Lu and Wei Guo and Xinghui Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15536},
eprint = {2609.15536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15536},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Physics-Guided Conditional Flow Matching with Energy Regularization for Robust PDE Inverse Problems
AU - Yongsheng Chen
AU - Shuo Lu
AU - Wei Guo
AU - Xinghui Zhong
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15536
UR - https://arxiv.org/abs/2609.15536
AB - We consider partial differential equation (PDE) inverse problems from sparse, noisy, and corrupted observations, with the aim of recovering unknown coefficient fields and associated state variables in a mesh-free setting. Under such sparse and corrupted observations, standard physics-informed and generative approaches typically treat all samples indiscriminately and therefore lack a principled mechanism for reconciling physical laws with contaminated data. We address this difficulty with a two-stage flow-matching framework. In the first stage, we develop physics-guided conditional flow matching (PG-CFM), which incorporates strong-form PDE information through residual regularization along the generative trajectories together with intermittent global collocation constraints. In the second stage, we introduce energy-regularized flow matching (ERFM), which fine-tunes the Stage-1 model by assigning each observation a physics--data energy score from a frozen teacher and reweighting the flow-matching objective to reduce the influence of high-energy, PDE-inconsistent samples. We show that the resulting Stage-2 objective is equivalent to flow matching under a teacher-induced reweighted data distribution, which gives a population-level interpretation of the robustness mechanism. Numerical experiments on several inverse benchmarks, including Poisson and Navier--Stokes problems, show that the proposed framework yields more accurate coefficient recovery than robust PINN variants and competing generative baselines.
ER -