---
title: "基于黎曼法坐标的莱文贝格-马夸特方法高阶几何更新"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2607.07623v1"
permalink: "https://feed.astralor.com/items/itm_f487c73e5800b081d050587d"
published: "2026-07-08T16:39:03.000Z"
score: 81
channel: "ai"
tags: ["优化算法","势能面拟合","测地线加速度","物理信息神经网络","莱文贝格-马夸特方法","黎曼法坐标"]
---

# 基于黎曼法坐标的莱文贝格-马夸特方法高阶几何更新

## AI 点评

> 这篇论文从微分几何出发，切中了非线性优化中一个长期被忽视的痛点——参数效应曲率。对于经常卡在PINN收敛困境的实践者，RNC-LM提供了一个优雅且可落地的解法。尤其值得关注的是它在大型势能面拟合中实现的数量级加速，这对材料模拟和分子动力学领域非常实用。论文代码若开源，将是极大的贡献。

## 速览

非线性最小二乘优化是回归、物理信息神经网络等机器学习任务的核心。标准LM方法受参数效应曲率影响，其切线空间步长在参数坐标中更新可能导致不一致。本文提出RNC-LM方法，通过黎曼法坐标将测地线加速度扩展至任意阶修正，构建有限步更新，消除逐阶的曲线残余加速度，使目标函数下降更贴合线性模型预测。在经典基准和PINN故障模式测试中，RNC-LM显著提升收敛性与鲁棒性，将相对L2误差降至1e-3量级；在机器学习势能面拟合中实现34倍加速。

## 关键点

- 揭示标准LM方法中参数效应曲率导致有限步长更新不一致的根本问题
- 提出RNC-LM，通过黎曼法坐标将测地线加速度扩展到任意阶修正
- 在经典非线性最小二乘基准和PINN故障模式测试中验证了收敛性与鲁棒性的提升
- 在大规模机器学习势能面拟合任务中实现34倍加速

## 全文

Nonlinear least-squares optimization is central to regression, physics-informed neural networks, and other machine-learning tasks. Such problems have a natural geometric interpretation, model predictions form a manifold in data space, while the chosen parameterization can introduce parameter-effects curvature that becomes a dominant source of nonlinearity. This exposes a limitation of the Levenberg-Marquardt (LM) method, its tangent-space step is applied as a straight update in parameter coordinates. Geodesic acceleration gives a second-order correction, but its removal of parameter-effect curvature is exact only in the infinitesimal-step limit. We propose a Riemann-normal-coordinate Levenberg-Marquardt method (RNC-LM) to improve this consistency for finite optimization steps. By reformulating the geodesic equation, RNC-LM extends geodesic acceleration to arbitrary-order corrections and constructs finite-step updates with progressively higher reparameterization consistency. A line search along the resulting RNC curve controls the traveled distance while keeping the cost close to standard LM. The method eliminates the tangential component of residual acceleration order by order in a moving tangent frame, making the actual objective reduction more consistent with the linear model prediction of LM. On classical nonlinear least-squares benchmarks, RNC-LM improves convergence and robustness in curved valleys and rank-deficient problems. On a reaction-diffusion PINN failure-mode benchmark, it reduces the relative L2 error to the order of 1e-3 and recovers a physically meaningful solution. On a large-scale machine-learning potential-energy-surface fitting task, it achieves a 34-fold speedup over standard LM.

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引用前请核对原文;本文由星际流动整理:https://feed.astralor.com/items/itm_f487c73e5800b081d050587d