发表机构
University of Illinois at Chicago; Georgia Tech Research Institute（伊利诺伊大学芝加哥分校; 佐治亚理工学院研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出 Poisson-GENERIC 神经算子，通过 Casimir 熵在函数空间中精确满足度量-辛结构的退化条件与 Jacobi 恒等式，在多个 PDE 上优于同骨干基线。
AI 中文摘要
现有的热力学一致神经算子通过将可逆算子投影到熵梯度的补空间上来施加 GENERIC 退化条件。这使得算子依赖于状态，并丧失 Jacobi 恒等式，因此结果是度量-辛退化的而非度量-辛的。我们转而采用 GENERIC 本身的方式获得退化性。对于非线性输运，可逆算子 $L$ 是相容的 Lie-Poisson 铅笔 $\alpha D+\lambda(uD+Du)$；否则它是常数，即平凡的 Poisson 傅里叶乘子。在带有潜熵密度的增广状态 $(u,s)$ 上，$S=\int s$ 是 $L$ 的 Casimir，因此 $L\\\\,\delta S/\delta z=0$ 无需投影即恒成立。能量由固定的力学二次项、学习到的无规范势和凸内能组成。摩擦算子 $M=AA^\top$ 逐点满足 $M\\\\,\delta E/\delta z=0$，其 Onsager 奇偶结构允许扩散和阻尼，同时可证明地排除输运。对于任意参数，偏斜性、正定性、两个退化条件以及 Jacobi 恒等式（对于 Lie-Poisson 项在解析频带上）均达到机器精度。热传导和阻尼波具有精确的闭式摩擦算子，第二定律在可检验的曲率条件下约束物理能量，离散梯度积分器产生精确的离散第一和第二定律。在 1D 和 2D 的四个 PDE 上，使用三种骨干网络（FNO、Transolver、CNO），该模型在与同骨干无约束基线的 72 次种子级比较中赢得 61 次，学习到精确的输运和波符号，在热和 Burgers 问题上匹配真实耗散率（误差在 13% 以内），并在平流问题上不产生耗散。常数 $L$ 的消融实验将精确 Jacobi 的成本隔离为 Burgers 问题的损失，而学习熵的消融实验在每个可逆-不可逆问题上注入能量。
英文摘要
Existing thermodynamically consistent neural operators impose the GENERIC degeneracy conditions by projecting the reversible operator onto the complement of the entropy gradient. This makes the operator state-dependent and forfeits the Jacobi identity, so the result is metriplectic-degenerate rather than metriplectic. We instead obtain degeneracy the way GENERIC does. For nonlinear transport, the reversible operator $L$ is the compatible Lie-Poisson pencil $αD+λ(uD+Du)$; otherwise it is a constant, trivially Poisson Fourier multiplier. On an augmented state $(u,s)$ with a latent entropy density, $S=\int s$ is a Casimir of $L$, so $L\,δS/δz=0$ holds identically without projection. The energy combines a fixed mechanical quadratic, a learned gauge-free potential, and a convex internal energy. The friction operator $M=AA^\top$ satisfies $M\,δE/δz=0$ pointwise, and its Onsager parity structure permits diffusion and damping while provably excluding transport. For any parameters, skewness, positivity, both degeneracies, and the Jacobi identity (on the resolved band for the Lie-Poisson term) hold to machine precision. Heat conduction and damped waves admit exact closed-form friction operators, the second law bounds physical energy under a checkable curvature condition, and a discrete-gradient integrator yields exact discrete first and second laws. On four PDEs in 1D and 2D with three backbones (FNO, Transolver, CNO), the model wins 61 of 72 seed-level comparisons against same-backbone unconstrained baselines, learns the exact transport and wave symbols, matches the true dissipation rate within 13% on heat and Burgers, and dissipates nothing on advection. A constant-$L$ ablation isolates the cost of exact Jacobi as the loss of Burgers, while a learned-entropy ablation injects energy on every reversible-irreversible problem.
CommentsPreprint. Under Review
