发表机构
Tianjin University; Jilin University（天津大学; 吉林大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究针对神经偏微分方程求解器，提出能量流形自然梯度下降框架\EMNGD，将参数更新与函数空间能量曲率对齐，在黎曼流形上优化，证明相关性质，实验显示比现有基线精度更高、收敛更快，还能量化精度与成本权衡。
AI 中文摘要
能量自然梯度下降（ENGD）使参数更新与基础函数空间能量的曲率对齐，但现有公式假设参数域为无约束欧几里得空间。我们引入了\EMNGDfull{}，这是一个用于物理信息和变分神经偏微分方程求解器的流形优化框架，其参数位于黎曼流形上。EMNGD将能量诱导的二次模型限制在可行的切向方向，并使用回缩在整个优化过程中保持参数约束。在强制性条件下，我们证明了无阻尼EMNGD方向的前推是能量度量中函数空间牛顿向量的最佳可行近似。我们建立了坐标不变性，在欧几里得空间中精确简化为ENGD，通过阿米尔乔回溯实现全局一阶收敛，并对不精确的切向求解具有鲁棒性。对于二次残差能量和广义高斯 - 牛顿拉回，伍德伯里恒等式将切向系统转移到样本空间而不改变方向。奈斯特罗姆近似提供了具有可控方向误差的可扩展样本空间求解，并在迭代收敛后恢复精确方向。在评估的神经偏微分方程基准测试中，EMNGD比比较的现有最先进基线实现了更高的精度和更快的收敛。伍德伯里保持EMNGD方向，而可扩展求解器诊断量化了预处理和残差子采样的精度 - 成本权衡。
英文摘要
Energy natural gradient descent (ENGD) aligns parameter updates with the curvature of an underlying function-space energy, but existing formulations assume an unconstrained Euclidean parameter domain. We introduce \EMNGDfull{}, a manifold optimization framework for physics-informed and variational neural PDE solvers whose parameters lie on a Riemannian manifold. EMNGD restricts the energy-induced quadratic model to feasible tangent directions and uses retractions to preserve parameter constraints throughout optimization. Under coercivity, we prove that the push-forward of the undamped EMNGD direction is the best feasible approximation to the function-space Newton vector in the energy metric. We establish coordinate invariance, exact reduction to ENGD in Euclidean space, global first-order convergence with Armijo backtracking, and robustness to inexact tangent solves. For quadratic residual energies and generalized Gauss--Newton pullbacks, the Woodbury identity transfers the tangent system to sample space without changing the direction. Nyström approximation provides scalable sample-space solves with controlled direction error and recovers the exact direction after iterative convergence. On the evaluated neural PDE benchmarks, EMNGD achieves higher accuracy and faster convergence than the compared state-of-the-art baselines. Woodbury preserves the EMNGD direction, while scalable-solver diagnostics quantify the accuracy--cost trade-off of preconditioning and residual subsampling.
