Physics constraints without hard projection: energy-guided transport keeps 100% success on full-PDE 2D Navier-Stokes
Synopsis
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.
Interpretation
Physical constraints are imposed as a continuous dissipative force injected along the generative trajectory instead of discrete algebraic projections, so generation is no longer interrupted mid-flow. Earlier zero-shot methods relied either on gradient approximations or on explicit geometric projections at each step; the latter work for sparse boundary conditions but are costly on dense, full-domain PDE residuals and can push intermediate states off the learned data manifold. In the full-PDE stress test on 2D Navier-Stokes, the explicit projection method PCFM reaches a 0% success rate, while Energy-Guided Transport holds 100%, completes a sample in about 0.94 s and reaches a physical residual of 1.63e-3.
The method writes the PDE residual as a smooth potential-energy function, so the generative trajectory slides downhill on an energy landscape. It does not materialize the constraint Jacobian; reverse-mode automatic differentiation computes the energy gradient with respect to the state as a vector-Jacobian product, keeping memory on the order of the state itself. Evaluation covers four systems, the 1D heat equation, 1D reaction-diffusion, 1D Burgers' equation and 2D Navier-Stokes, all sharing the same pretrained functional flow-matching backbone and initial noise with the baselines, under both boundary/initial-condition-only and full-PDE constraint regimes.
Continuous guidance also changes speed, with inference one to two orders of magnitude faster than exact projection. Explicit projection needs about 94 s per sample on the partial-constraint 2D Navier-Stokes task and runs out of memory or diverges numerically under dense constraints. Energy-Guided Transport completes a 2D Navier-Stokes sample in about 0.94 s under full-PDE constraints and keeps a 100% success rate across all four systems.
Perspective
The demonstrated scope is four synthetic physical systems, the 1D heat equation, 1D reaction-diffusion, 1D Burgers' equation and 2D Navier-Stokes, under boundary/initial-condition-only and full-domain PDE constraints, with white-noise and smooth-kernel priors; 2D Navier-Stokes uses only the white-noise prior because the dense covariance matrix exceeds GPU memory.
Energy-Guided Transport is a continuous dissipative mechanism rather than a hard constraint, so it does not guarantee exact convergence, needs optional terminal refinement, and relies on the pretrained flow-matching model to supply a structurally reasonable transport prior; excessively large guidance scales increase ODE stiffness; the current energy functional minimizes only pointwise residuals and does not explicitly constrain macroscopic statistics such as shock gradient distributions or energy spectra.