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

Jason Sulskis; Sathya Ravi

cs.LGOct 4, 2026

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

Authors: Jason Sulskis, Sathya Ravi

Organizations: Georgia Tech Research Institute, Atlanta, GA, USA

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.

Figures & tables

Panel of Figure 1 — Figure 1: One step of the Poisson-GENERIC neural operator. The state $z=(u,s)$ combines the observed field with a latent entropy density, with $S=\int s$ a Casimir of $L$ . The energy combines a fixed mechanical quadratic with learned pointwise terms. The Poisson operator governs the reversible dynamics, while the projected $AA^{\!\top}$ friction operator enforces $M\,\delta E/\delta z=0$ and excludes transport. The channels are advanced by a split step, or by a discrete-gradient step that conserves $E_{\theta}$ and produces $S$ exactly.

	1D heat		2D Burgers		1D wave		2D wave
	Lrn	Phys	Lrn	Phys	Lrn	Phys	Lrn	Phys
$\langle v,Lv\rangle$ (skewness)	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$
$\langle w,Mv\rangle-\langle v,Mw\rangle$ (symmetry)	$2\!\cdot\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$1\!\cdot\!10^{-16}$	$1\!\cdot\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$
$\\|L\,\delta S/\delta z\\|/\\|\delta S/\delta z\\|$	$4\!\cdot\!10^{-18}$	$0$	$8\!\cdot\!10^{-18}$	$0$	$2\!\cdot\!10^{-18}$	$0$	$8\!\cdot\!10^{-19}$	$0$
$\\|M\,\delta E/\delta z\\|/\\|\delta E/\delta z\\|$	$0$	$3\!\cdot\!10^{-18}$	$0$	$2\!\cdot\!10^{-18}$	$0$	$1\!\cdot\!10^{-19}$	$0$	$4\!\cdot\!10^{-20}$
$\langle\delta E/\delta z,\partial_{t}z\rangle$ ( $\dot{E}$ )	$1\!\cdot\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-16}$	$\leq\!10^{-17}$	$\leq\!10^{-16}$	$\leq\!10^{-17}$

Table 1: Structural identities at random initialization, double precision,

16^{d}

grid, maximum over a batch; Lrn is the learned-

E,S

ablation, Phys the physical model of record.

\mathcal{J}

is the normalized Jacobi residual of § A.6 ; “generic” and “band-limited” refer to the test states and functionals. The Lie–Poisson rows are empty for the wave, whose operator has no pencil; the projected-

L

row uses a free learned entropy in both columns, since with a Casimir entropy the projection is trivial. Discrete-gradient entries are at the tolerance of the fixed-point solve, not the floating-point floor. Entries at or below the double-precision floor are written

\leq\!10^{-16}

		Persist.	Baseline	Learned $E,S$	PG-phys	PG-phys ( $\lambda\equiv 0$ )	Params
1D heat	FNO	0.184	$\mathbf{0.0014\pm 0.0003}$	$0.0096\pm 0.0066$	$0.017\pm 0.005$	$0.0056\pm 0.0011$	44 / 18
	Transolver	0.184	$0.088\pm 0.050$	$0.018\pm 0.006$	$0.015\pm 0.007$	$\mathbf{0.0087\pm 0.0049}$	34 / 41
	CNO	0.184	$0.015\pm 0.011$	$0.0097\pm 0.0031$	$0.013\pm 0.006$	$\mathbf{0.0046\pm 0.0008}$	21 / 12
1D advection	FNO	0.863	$0.0040\pm 0.0004$	$0.019\pm 0.006$	$0.0015\pm 0.0007$	$\mathbf{0.0001\pm 0.0000}$	44 / 18
	Transolver	0.863	$0.029\pm 0.004$	$0.094\pm 0.096$	$0.0016\pm 0.0006$	$\mathbf{0.0001\pm 0.0001}$	34 / 41
	CNO	0.863	$0.012\pm 0.002$	$0.035\pm 0.008$	$0.0017\pm 0.0010$	$\mathbf{0.0001\pm 0.0001}$	21 / 12

Table 2: Ten-frame rollout relative

L^{2}

error, mean

\pm

std over three seeds, with the persistence baseline and parameter counts (baseline / PG-phys, thousands). Bold: best of the four models (for the wave, whose operator has no pencil, the two PG columns are one model trained twice and are both marked; their differences are run-to-run noise).

\dagger

: one of three seeds diverged; mean

\pm

std of the other two.

\ddagger

: one seed finite but unstable under the split step (

\max_{t}Q/\text{bound}=1.67

); three-seed statistics shown. Neither failure reproduced on retraining.

Figure 2: Physical energy

Q_{t}/Q^{\star}_{0}

along a 40-frame rollout from held-out initial conditions (1D, FNO backbone, seed 0; grey lines mark the 4-frame training horizon and the 10-frame evaluation horizon). PG-phys stays on the truth on all three equations. The learned-energy ablation conserves its own

E_{\theta}

while losing a quarter of the physical energy on advection and multiplying it six- and thirty-fold on Burgers and the wave. The unconstrained baseline matches the truth on advection and Burgers and drifts on the wave.

		$\Pi_{\mathrm{model}}/\Pi^{\star}$	$Q_{10}/Q_{10}^{\star}$	$\max_{t}Q/\text{bound}$
1D	heat	0.94 / 0.99 / 0.96	1.02 / 1.00 / 1.01	0.16 / 0.17 / 0.14
	Burgers	1.00 / 1.00 / 1.00	1.00 / 1.00 / 1.00	0.13 / 0.14 / 0.10
	wave †	0.75 / 0.86 / 0.76	1.08 / 1.04 / 1.08	0.89 / 0.83 / 0.87
	advection	$\|\Pi_{\mathrm{model}}\|\leq 10^{-4}$	1.000 / 1.000 / 1.000	0.44 / 0.28 / 0.42
2D	heat ‡	0.87 / 1.12 / 0.88	1.07 / 0.94 / 1.06	0.35 / 0.13 / 0.35
	Burgers	0.93 / 0.93 / 0.95	1.05 / 1.04 / 1.04	0.40 / 0.25 / 0.36

Table 3: Dissipation calibration of PG-phys (three-seed means; FNO / Transolver / CNO): fractional one-step decay of the physical energy against the truth, physical energy after ten frames relative to the truth, and

\max_{t}Q

relative to the bound equation 3 , evaluated with

a=0

and

C_{\theta}=-\inf\varphi_{\theta}

over the data range (Remark 2 ).

\dagger

: two seeds (the third diverged in training).

\ddagger

: two seeds (the third was unstable under the split step,

\max_{t}Q/\text{bound}=1.67

); neither failure reproduced on retraining (see Accuracy).

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