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

Every existing construction of thermodynamically consistent neural operators imposes the two GENERIC degeneracy conditions on a learned energy and entropy, but degeneracy is always obtained by projecting the reversible operator onto the complement of the entropy gradient. The consequence is an operator that depends on state and a Jacobi identity that is forfeited, so the resulting structure is metriplectic-degenerate rather than metriplectic. Our approach closes this gap by taking degeneracy from the same place GENERIC itself takes it. For nonlinear transport, the reversible operator $L$ is the compatible Lie–Poisson pencil $\alpha D+\lambda (uD+Du)$; otherwise it is a constant Fourier multiplier, which is trivially Poisson. On an augmented state $(u,s)$ carrying a latent entropy density, the entropy $S=\int s$ is a Casimir of $L$, so $L\delta S/\delta z=0$ holds identically, with no projection required. A fixed mechanical quadratic, a learned gauge-free potential, and a convex internal energy together make up the energy. The friction operator $M=A{A}^{\top }$ satisfies $M\delta E/\delta z=0$ pointwise, and its Onsager parity structure permits diffusion and damping while provably ruling out transport. For any parameters, skewness, positivity, both degeneracies, and the Jacobi identity (on the resolved band for the Lie–Poisson term) all hold to machine precision. Heat conduction and damped waves have closed-form friction operators that are representable exactly, the second law bounds physical energy under a checkable curvature condition, and exact discrete first and second laws follow from a discrete-gradient integrator. The model was tested on four PDEs in one and two dimensions with three backbones (FNO, Transolver, CNO): with the same backbone it wins 61 of 72 seed-level comparisons against unconstrained baselines, learns the exact transport and wave symbols, reproduces the true dissipation rate to within 13% on heat and Burgers, and dissipates nothing on advection. Whereas a constant-$L$ ablation isolates the price of exact Jacobi as the loss of Burgers, a learned-entropy ablation injects energy on every reversible–irreversible problem.