Energy-Guided Transport for Projection-Free Physics-Informed Flow Matching

Giuliano Galadini, Aurelio Uncini, Danilo Comminiello

Deep generative models, particularly Flow Matching frameworks, have emerged as powerful tools for synthesizing complex physical systems. However, standard architectures remain purely statistically driven and agnostic to the underlying governing equations, often violating fundamental physical constraints during inference. Existing zero-shot approaches designed to enforce these constraints typically rely on rigid algebraic projections. While effective for sparse boundary conditions, applying such discrete geometric corrections to dense, full-domain PDEs may disrupt the probability path, \rev{potentially deviating from the learned manifold and introducing substantial computational overhead}. To address these challenges, we introduce Energy-Guided Transport, a novel training-free framework that formulates non-linear physical constraints as a smooth potential energy landscape. Rather than interrupting the ODE integration, our method applies a continuous dissipative force directly to the intermediate generative states, steering them toward the physical manifold. By evaluating this force implicitly via reverse-mode automatic differentiation, our framework preserves the natural dynamics…