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

arXiv: https://arxiv.org/abs/2606.08343 · HTML: https://mlsanity.com/paper/2606.08343/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 32 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 21 of 21)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems](https://mlsanity.com/paper/2109.00092/) (2021)
- [LNO: Laplace Neural Operator for Solving Differential Equations](https://mlsanity.com/paper/2303.10528/) (2023)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [Efficiently Parameterized Neural Metriplectic Systems](https://mlsanity.com/paper/2405.16305/) (2024)
- [Metriplectic Conditional Flow Matching for Dissipative Dynamics](https://mlsanity.com/paper/2509.19526/) (2025)
- [Metriplector: From Field Theory to Neural Architecture](https://mlsanity.com/paper/2603.29496/) (2026)

## Built on by (most cited first, 1 of 1)

- [Poisson-GENERIC Neural Operators: Exact Metriplectic Structure in Function Space via Casimir Entropies](https://mlsanity.com/paper/2610.05570/) (2026)

## Abstract (arXiv)

We propose GENERIC-FNO, a neural operator that embeds the metriplectic (GENERIC) degeneracy structure of nonequilibrium thermodynamics in function space, coupling reversible, energy-conserving dynamics to irreversible, entropy-producing dynamics via the degeneracy conditions. Prior structure-preserving neural operators enforce at most one conservation law or a Hamiltonian form, and thermodynamically consistent learning has been confined to finite-dimensional, graph, or particle systems. GENERIC-FNO learns the energy and entropy functionals as neural operators and builds the reversible and irreversible operators as diagonal Fourier multipliers flanked by rank-one projections that enforce both degeneracy conditions exactly, by construction, with no penalty, update projection, or residual; the Jacobi identity is not enforced. The identities hold to machine precision (~10^-13) for any initialization, dimension, or resolution, so the continuous-time dynamics conserve the learned energy and produce the learned entropy exactly, with explicit time stepping adding only an O(dt^2) drift. These are guarantees about the learned functionals within GENERIC's scope of closed conservative-dissipative dynamics, not a certificate of physical accuracy, and the (E,S,L,M) decomposition is not unique; we make this gauge freedom explicit and propose a gauge-invariant dissipation diagnostic independent of the learned functionals. Across three backbones (1D/2D FNO, DeepONet) and four canonical scalar PDEs, the guarantees transfer zero-shot over a 4x super-resolution range and hold in 3D; the diagnostic identifies the reversible and the most dissipative system in every backbone; and over 200-step rollouts, where every unconstrained model we test diverges or collapses, GENERIC-FNO stays bounded, at half the parameters but 4-10x the compute, while losing accuracy on pure transport and on the smallest 1D backbone.