发表机构
CSIRO Environment, Hobart, Australia; CSIRO Technology, Sydney, Australia（澳大利亚联邦科学与工业研究组织环境部; 澳大利亚联邦科学与工业研究组织技术部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对粗粒化随机强迫湍流，证明物理信息损失中仅线性约束有效，提出对FNO输出施加闭式投影以精确满足连续性及动量平衡，成本低且显著提升稳定性与精度。
AI 中文摘要
物理信息损失函数将离散化的偏微分方程残差添加到数据项中，但仅当所强制的方程在待拟合的场上闭合时，该损失函数才严格有效。对于粗粒化且随机强迫的数据，这一条件不成立：偏微分方程中的非线性项通常不与滤波器交换，因此粗粒化后的场不满足原始方程，将其残差驱动至零相当于在学习的算子中隐式编码了闭合模型。在周期域上经过归一化、平移不变的滤波后，常系数线性约束在滤波网格上仍然有效。对于二维不可压缩纳维-斯托克斯方程，两个这样的约束是连续性方程和全局动量平衡。我们通过在傅里叶神经算子（FNO）的输出上应用闭式投影，以$O(N \log N)$的计算成本精确强制这些约束。在二维各向同性湍流中，其强迫被截断至完全位于截止波数之外，该投影实现了$5.0\times10^{-7}$的连续性误差，而普通FNO为0.51，物理信息FNO为0.17；全局动量平衡误差为$8.6\times10^{-11}$，相比之下分别为$7.5\times10^{-4}$和$8.3\times10^{-4}$。该投影仅额外增加了2%的训练时间，并将448步内能量保持有界的滚动轨迹比例从0.18提高到0.87。在解析的科尔莫戈罗夫流上，测试时优化每轨迹的成本约为投影的3,300倍，同时最大散度高出三个数量级。非线性约束也可以在对其未闭合项建模后被强制：通过从粗训练数据单独拟合的恒定亚网格通量来闭合全局能量平衡，消除了强制未闭合平衡所产生的20.9%能量赤字，并使每个自由运行的滚动轨迹在2,000步内保持有界。
英文摘要
A physics-informed loss adds a discretised PDE residual to the data term, but is only strictly valid if the equations being enforced are closed on the fields being fitted. This fails on coarse-grained, stochastically forced data: nonlinear terms in the PDE generally do not commute with the filter, so the coarse-grained fields do not satisfy the original equations, and driving their residual to zero encodes an implicit closure into the learned operator. Under normalised, translation-invariant filtering on a periodic domain, constant-coefficient linear constraints remain valid on the filtered grid. For the 2D incompressible Navier-Stokes equations, two such constraints are continuity and global momentum balance. We enforce these exactly by applying a closed-form projection on the output of a Fourier neural operator (FNO) at $O(N \log N)$ cost. On two-dimensional isotropic turbulence, truncated so that its forcing lies entirely beyond the cutoff wavenumber, the projection achieves a continuity error of $5.0\times10^{-7}$ against 0.51 for a plain FNO and 0.17 for a physics-informed FNO, and a global momentum balance error of $8.6\times10^{-11}$ against $7.5\times10^{-4}$ and $8.3\times10^{-4}$. The projection only costs an extra 2% of training time, and takes the fraction of rollouts whose energy remains bounded at 448 steps from 0.18 to 0.87. On resolved Kolmogorov flow, test-time optimisation costs approximately 3,300$\times$ as much per trajectory as the projection, while leaving maximum divergence three orders of magnitude higher. Nonlinear constraints can also be enforced once their unclosed terms are modelled: closing the global energy balance with a constant subgrid flux, fitted from the coarse training data alone, removes the 20.9% energy deficit that enforcing the unclosed balance produces and keeps every free-running rollout bounded over 2,000 steps.
CommentsAccepted at the NeurIPS 2026 Workshop on AI for Stochastic Dynamics (STODY). Code: https://github.com/m-groom/physics-constrained-neural-operators
