Poisson-GENERIC Neural Operators: Exact Metriplectic Structure in Function Space via Casimir Entropies

Abstract

Existing thermodynamically consistent neural operators impose the GENERIC degeneracy conditions by projecting the reversible operator onto the complement of the entropy gradient. This makes the operator state-dependent and forfeits the Jacobi identity, so the result is metriplectic-degenerate rather than metriplectic. We instead obtain degeneracy the way GENERIC does. For nonlinear transport, the reversible operator $L$ is the compatible Lie-Poisson pencil $αD+λ(uD+Du)$; otherwise it is a constant, trivially Poisson Fourier multiplier. On an augmented state $(u,s)$ with a latent entropy density, $S=\int s$ is a Casimir of $L$, so $L\,δS/δz=0$ holds identically without projection. The energy combines a fixed mechanical quadratic, a learned gauge-free potential, and a convex internal energy. The friction operator $M=AA^\top$ satisfies $M\,δE/δz=0$ pointwise, and its Onsager parity structure permits diffusion and damping while provably excluding transport. For any parameters, skewness, positivity, both degeneracies, and the Jacobi identity (on the resolved band for the Lie-Poisson term) hold to machine precision. Heat conduction and damped waves admit exact closed-form friction operators, the second law bounds physical energy under a checkable curvature condition, and a discrete-gradient integrator yields exact discrete first and second laws. On four PDEs in 1D and 2D with three backbones (FNO, Transolver, CNO), the model wins 61 of 72 seed-level comparisons against same-backbone unconstrained baselines, learns the exact transport and wave symbols, matches the true dissipation rate within 13% on heat and Burgers, and dissipates nothing on advection. A constant-$L$ ablation isolates the cost of exact Jacobi as the loss of Burgers, while a learned-entropy ablation injects energy on every reversible-irreversible problem.

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