用李群约束潜空间，提升神经算子长期预测稳定性。
Geometric Neural Operators via Lie Group-Constrained Latent Dynamics
- 用低秩李代数参数化约束潜空间，实现符合几何规律的更新。
- 在1维伯格斯和2维纳维-斯托克斯方程上，误差降低30%-50%。
- 可无缝接入现有模型，适合需高精度长期模拟的物理系统建模。
神经算子为在分辨率不变且数据驱动的前提下学习偏微分方程解提供了有效框架。然而，现有神经算子在多层迭代和长时滚动中常出现不稳定，根源在于潜空间更新未受约束，违背了几何与守恒律。为此，我们提出基于流形约束的李群方法（MCL），通过李代数参数化实现潜表示上的群作用更新。该方法作为高效插件模块，为现有神经算子注入几何先验。在多种偏微分方程（如1维伯格斯、2维纳维-斯托克斯）上，覆盖广泛参数与步数的实验表明，本方法以增加2.26%参数为代价，将相对预测误差降低30%-50%，显著提升长期预测保真度，为解决神经算子更新中缺失的几何约束问题提供可扩展方案。
原文摘要 · Abstract (English)
Neural operators offer an effective framework for learning solutions of partial differential equations for many physical systems in a resolution-invariant and data-driven manner. Existing neural operators, however, often suffer from instability in multi-layer iteration and long-horizon rollout, which stems from the unconstrained Euclidean latent space updates that violate the geometric and conservation laws. To address this challenge, we propose to constrain manifolds with low-rank Lie algebra parameterization that performs group action updates on the latent representation. Our method, termed Manifold Constraining based on Lie group (MCL), acts as an efficient \emph{plug-and-play} module that enforces geometric inductive bias to existing neural operators. Extensive experiments on various partial differential equations, such as 1-D Burgers and 2-D Navier-Stokes, over a wide range of parameters and steps demonstrate that our method effectively lowers the relative prediction error by 30-50\% at the cost of 2.26\% of parameter increase. The results show that our approach provides a scalable solution for improving long-term prediction fidelity by addressing the principled geometric constraints absent in the neural operator updates.
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