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

arXiv: https://arxiv.org/abs/2610.05570 · HTML: https://mlsanity.com/paper/2610.05570/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 4 Oct 2026.

## Builds on (most cited first, 19 of 19)

- [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)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [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)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks](https://mlsanity.com/paper/2012.03133/) (2020)
- [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)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (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)
- [GENERIC-FNO: Embedding Energy Conservation and Entropy Production into Fourier Neural Operators](https://mlsanity.com/paper/2606.08343/) (2026)

## Abstract (arXiv)

Existing thermodynamically consistent neural operators enforce GENERIC degeneracy by projecting the reversible operator, introducing state dependence without guaranteeing Jacobi. We instead use a compatible Lie--Poisson pencil L=α D+λ(uD+Du) for nonlinear transport and constant Poisson multipliers otherwise. Augmenting the observed field u with latent entropy density s makes S=int s a Casimir, so L,δ S/δ z=0 without projection. Energy combines a fixed mechanical quadratic, learned gauge-free potential, and convex internal energy. The friction operator M=AA^top enforces M,δ E/δ z=0 pointwise and supports diffusion and damping without an independent reversible transport generator. Skewness, positivity, and both degeneracies hold exactly; the continuum pencil satisfies Jacobi, verified numerically on the alias-free band. The parameterization represents the closed-form heat and damped-wave friction operators within its admissible range. Physical energy is bounded under coercivity assumptions, while a discrete-gradient integrator preserves both laws when solved to convergence. Across four PDEs in 1D and 2D and three backbones (FNO, Transolver, CNO), the model wins 61/72 seed-level comparisons against matched unconstrained baselines, recovers exact transport and wave symbols, matches heat and Burgers dissipation within 13%, and exhibits negligible advection dissipation. A constant-L ablation exposes the cost of nonlinear transport, while a learned-energy ablation shows that conserving learned energy need not constrain physical energy.