Metriplectic Conditional Flow Matching for Dissipative Dynamics

Metriplectic conditional flow matching (MCFM) learns dissipative dynamics without violating first principles. Neural surrogates often inject energy and destabilize long-horizon rollouts; MCFM instead builds the conservative–dissipative split into both the vector field and a structure-preserving sampler. MCFM trains via conditional flow matching on short transitions, avoiding long-rollout adjoints. In inference, a Strang-prox scheme alternates a symplectic update with a proximal metric step, ensuring discrete energy decay; an optional projection enforces strict decay when a trusted energy is available. We provide continuous- and discrete-time guarantees linking this parameterization and sampler to conservation, monotonic dissipation, and stable rollouts. On a controlled mechanical benchmark, MCFM yields phase portraits closer to ground truth and markedly fewer energy-increase and positive energy-rate events than an equally expressive unconstrained neural flow, while matching terminal distributional fit.