# Energy-Guided Transport

Pretrained flow-matching models learn only the data distribution and often violate conservation laws and boundary conditions at inference; earlier zero-shot fixes used algebraic projections that hit memory bottlenecks and disturb the probability path on dense, full-domain PDEs. Energy-Guided Transport writes the physical residual as a smooth potential energy and applies a continuous dissipative force to intermediate states during integration, obtaining the gradient implicitly through reverse-mode automatic differentiation. Across the heat equation, reaction-diffusion, Burgers' equation and 2D Navier-Stokes, it holds a 100% success rate under full-PDE constraints, generating a 2D Navier-Stokes sample in about 0.94 s where explicit projection fails.

Pretrained flow-matching models learn only the data distribution and often violate conservation laws and boundary conditions at inference; earlier zero-shot fixes used algebraic projections that hit memory bottlenecks and disturb the probability path on dense, full-domain PDEs. Energy-Guided Transport writes the physical residual as a smooth potential energy and applies a continuous dissipative force to intermediate states during integration, obtaining the gradient implicitly through reverse-mode automatic differentiation. Across the heat equation, reaction-diffusion, Burgers' equation and 2D Navier-Stokes, it holds a 100% success rate under full-PDE constraints, generating a 2D Navier-Stokes sample in about 0.94 s where explicit projection fails.