🤖 AI Summary
This work addresses the challenge that existing neural operators struggle to simultaneously satisfy energy conservation and entropy production structures dictated by nonequilibrium thermodynamics in function space. The authors embed the full GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework into a Fourier neural operator, learning energy and entropy functionals while parameterizing Poisson and friction operators via diagonal Fourier multipliers combined with rank-one projections to rigorously enforce thermodynamic degeneracy conditions. This approach is the first to guarantee thermodynamic consistency at machine precision in function space without requiring penalty terms or post-processing. Additionally, it introduces a gauge-invariant dissipation diagnostic to disentangle reversible and dissipative dynamics. On 1D and 2D problems, the model achieves zero-shot structural fidelity at 4× super-resolution, accurately recovers the physical dissipation hierarchy, and matches or outperforms existing unconstrained and energy-penalized baselines.
📝 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 reversible (Hamiltonian) structure, while thermodynamically consistent learning has 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 exactly, by construction, with no penalty term, update projection, or residual. The degeneracy identities hold to machine precision (residuals ~10^-13) for any initialization, dimension, or resolution, so the continuous-time dynamics conserve the learned energy and produce entropy exactly; the explicit time stepping adds only a small O(dt^2) drift (per-step residual ~10^-6). We further note that the (E,S,L,M) decomposition of a given flow is not unique, and introduce a gauge-invariant dissipation diagnostic separating reversible from dissipative dynamics independently of the learned functionals. Across three operator backbones (1D/2D FNOs and DeepONet) and four PDEs spanning reversible, dissipative, and mixed regimes, GENERIC-FNO preserves its exact structural guarantees zero-shot across a 4x super-resolution range (64 to 256), recovers the ground-truth ordering of physical dissipation, and is competitive with strong unconstrained and energy-penalized baselines, outperforming them on several dissipative and mixed problems at comparable or fewer parameters.