将能量梯度下降拓展到流形优化，提升神经PDE求解器的精度与收敛速度。
Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers
- 在流形上定义能量自然梯度，约束参数满足物理规律
- 理论证明其方向逼近函数空间牛顿方向，且收敛速度快于现有方法
- 适用于物理信息神经网络，尤其适合追求高精度的科学计算场景
能量自然梯度下降（ENGD）使参数更新顺应函数空间能量的曲率，但现有方法假设参数空间为无约束欧氏空间。本文提出 EMNGDfull{}，一种用于物理信息和变分神经偏微分方程（PDE）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方向，并使用重映射保持参数约束。在强制性条件下，我们证明了未阻尼 EMNGD 方向的前推是函数空间牛顿向量在能量度量下的最优可行逼近。建立了坐标不变性、退化为欧氏空间中的 ENGD、使用阿米乔回溯法的全局一阶收敛性，以及对切方向求解不精确的鲁棒性。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系统转移到样本空间而不改变方向。 Nystrom 近似提供可扩展的样本空间求解，控制方向误差，并在迭代收敛后恢复精确方向。在评估的神经PDE基准测试中， EMNGD 在精度和收敛速度上均优于当前最先进的基线方法。伍德伯里恒等式保持了 EMNGD 方向，而可扩展求解器诊断量化了预处理和残差采样的精度-成本权衡。
原文摘要 · Abstract (English)
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.
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