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

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 of the flow while maintaining an efficient $\mathcal{O}(N)$ algorithmic complexity. We extensively evaluate our framework across various 1D and 2D physical systems, including the chaotic Navier-Stokes equations. Our method achieves stable generation without structural divergence, significantly improves the stability-accuracy trade-off compared to state-of-the-art methods, and substantially accelerates inference with respect to exact projection strategies. Code available at: https://github.com/ispamm/Projection-free-energy-guided-flow-matching.