将热力学结构嵌入神经算子，实现能量守恒与熵产生的精确模拟。
GENERIC-FNO: Embedding Energy Conservation and Entropy Production into Fourier Neural Operators
- 通过傅里叶乘子构造满足退化条件的泊松与摩擦算子，直接在函数空间中建模
- 能量守恒与熵产生误差低于10^-13，时间步长误差为10^-6量级，精度极高
- 适用于多类偏微分方程，超分辨率4倍下仍保持结构精确，适合物理约束学习
我们提出GENERIC-FNO，首个将非平衡热力学完整GENERIC（度量-斜对称）结构——可逆的能量守恒动力学与不可逆的熵产生动力学——直接嵌入函数空间的神经算子。现有保结构神经算子最多仅能维持单一守恒律或哈密顿结构，而热力学一致性学习此前局限于有限维、图或粒子系统。GENERIC-FNO通过神经算子学习能量与熵泛函，并将泊松与摩擦算子参数化为夹在秩一投影间的对角傅里叶乘子，严格构造满足退化条件，无需惩罚项、更新投影或残差项。退化恒等式在任意初始化、维度与分辨率下均达到机器精度（残差~10^-13），连续时间动力学精确守恒能量并产生熵；显式时间积分仅引入小量O(dt^2)漂移（单步残差~10^-6）。我们进一步指出，给定流的（E,S,L,M）分解不唯一，提出规范不变耗散诊断，独立分离可逆与耗散动力学。在三种算子骨干（1D/2D FNO与DeepONet）及四类涵盖可逆、耗散与混合区间的偏微分方程上，GENERIC-FNO零样本实现4倍超分辨率（64至256），准确恢复物理耗散排序，性能优于强约束与能量惩罚基线，在相同或更少参数下表现更优。
原文摘要 · Abstract (English)
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.
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