Dynamic operator-guided flow matching: A generative physical inverse solver for arbitrary sparse observations

Abstract

Solving ill-posed physical inverse problems under extremely sparse observations remains a fundamental engineering challenge. Conventional inversion frameworks require repeated, computationally prohibitive forward simulations. While deep learning-accelerated approaches (e.g., Diffusion Posterior Sampling) mitigate these costs, they still incur significant inference latency and frequently suffer from catastrophic posterior collapse. To address these challenges, we propose the Dynamic Operator-Guided Inversion Solver (DOGIS). DOGIS couples deep generative priors with governing PDEs by embedding a pretrained, fully differentiable Fourier Neural Operator (FNO) as a forward surrogate into the continuous normalizing flow trajectory during amortized training. Inspired by the Ensemble Smoother with Multiple Data Assimilation (ES-MDA), DOGIS employs a time-adaptive loss-weighting strategy that prevents premature generative distortion while encouraging the model to internalize complex physical manifolds. At inference time, a lightweight, gradient-based physical projection executes a continuous Kalman update, strictly anchoring the generated field to available sparse measurements. Comprehensive evaluations across Darcy flow and Structural Health Monitoring (SHM) scenarios demonstrate DOGIS's robustness. Unlike baseline models, DOGIS successfully circumvents mode collapse and exhibits graceful degradation when reconstructing 64 by 64 resolution fields from merely 16 sensors (an extreme 0.39% sparsity). Furthermore, it provides precise, physics-aware uncertainty quantification (UQ) localized around genuinely ambiguous topological boundaries. By shifting the computational burden to amortized training, DOGIS achieves an average 8x inference speedup over standard DPS, offering a mathematically rigorous and highly agile backend for reliable optimal sensor deployment in engineering scenarios.