Physics-Informed Generative Solver: Bridging Data-Driven Priors and Conservation Laws for Stable Spatiotemporal Field Reconstruction

2026-05-21

AI Summary

This work addresses the challenge that existing generative models often violate physical conservation laws when reconstructing continuous physical fields from sparse measurements. To enforce strict adherence to physically admissible manifolds without retraining, the authors propose Martingale-regularized score matching combined with physics-informed implicit score sampling. The method leverages Fokker–Planck equation regularization and gradient guidance from physical residual terms to steer the denoising trajectory, ensuring generated solutions satisfy underlying dynamical laws. It achieves joint reconstruction of acoustic pressure and particle velocity, enabling aliasing-resistant virtual dense array synthesis, and further demonstrates robust generalization by successfully reconstructing ERA5 meteorological fields under extremely sparse observation conditions, thereby validating its broad applicability and effectiveness.

Abstract

Reconstructing continuous physical fields from sparse measurements is a central inverse problem, but data-driven generative models can produce states that violate governing dynamics. We introduce a physics-informed generative solver that separates stable prior learning from inference-time enforcement of conservation laws. Martingale-Regularized Score Matching regularizes score pretraining with a Score Fokker-Planck constraint, yielding a dynamically stable prior. Physics-Informed Implicit Score Sampling then guides denoising trajectories by gradients of physical residuals, projecting samples toward admissible manifolds without retraining. In acoustics, the method co-generates pressure and particle velocity from sparse sensors, enabling dense virtual arrays that suppress spatial aliasing. The same framework generalizes to real-world ERA5 meteorological fields under extreme sparsity. Together, this work establishes a rigorous and generalizable paradigm for solving high-dimensional inverse problems, bridging the gap between generative artificial intelligence and first-principles science.

Problem

• spatiotemporal field reconstruction
• inverse problem
• physics-informed
• conservation laws
• sparse measurements

Innovation

• Physics-Informed Generative Modeling
• Score-Based Generative Models
• Conservation Laws
• Spatiotemporal Field Reconstruction
• Inverse Problems