GENERIC-FNO: Embedding Energy Conservation and Entropy Production into Fourier Neural Operators

TMLR Paper9539 Authors

Abstract:

We introduce GENERIC-FNO, the first neural operator to embed the full GENERIC
(metriplectic) structure of nonequilibrium thermodynamics---reversible,
energy-conserving dynamics and irreversible, entropy-producing dynamics coupled
through the degeneracy conditions---directly in function space. Existing
structure-preserving neural operators enforce at most a single conservation law
or a purely reversible (Hamiltonian) structure, while thermodynamically
consistent learning has so far been confined to finite-dimensional, graph, or
particle systems. GENERIC-FNO closes this gap: it learns the energy and entropy
functionals as neural operators and parameterizes the Poisson and friction
operators as diagonal Fourier multipliers sandwiched between rank-one
projections that enforce the degeneracy conditions \emph{exactly, by
construction}---with no penalty term, no projection of the predicted update, and
no free residual. The degeneracy identities therefore hold to machine precision (residuals
$\sim\!10^{-13}$) for any initialization, spatial dimension, or grid resolution,
so the continuous-time dynamics conserve the learned energy and produce entropy
exactly; the explicit time stepping we deploy adds only a small
$\mathcal{O}(\Delta t^2)$ drift (per-step energy residual $\sim\!10^{-6}$). We further observe that the $(E,S,L,M)$
decomposition realizing a given flow is not unique, and introduce a
gauge-invariant dissipation diagnostic that separates genuinely reversible from
dissipative dynamics independently of the learned functionals. Across three
operator backbones (1D and 2D Fourier neural operators and DeepONet) and four
PDEs spanning reversible, dissipative, and mixed regimes, GENERIC-FNO preserves
its exact structural guarantees zero-shot across a $4\times$ super-resolution
range ($64\!\to\!256$), recovers the correct ground-truth ordering of physical
dissipation, and remains competitive with strong unconstrained and
energy-penalized baselines---outperforming them on several dissipative and mixed
problems at comparable or fewer parameters.

Submission Type: Long submission (more than 12 pages of main content)

Changes Since Last Submission: Added some missed recent work citations.

Assigned Action Editor: ~Sayan_Ranu2

Submission Number: 9539