---
title: "Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates"
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: ["Optimization Algorithm","Potential-Energy-Surface Fitting","Geodesic Acceleration","Physics-Informed Neural Networks","Levenberg-Marquardt Method","Riemann Normal Coordinates"]
---

# Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates

## AI commentary

> This paper tackles a long-overlooked pain point in nonlinear optimization—parameter-effects curvature—from a differential geometry perspective. For practitioners struggling with PINN convergence issues, RNC-LM offers an elegant and practical solution. Particularly striking is the order-of-magnitude speedup in large-scale potential-energy-surface fitting, which is highly useful for materials simulation and molecular dynamics. Open-sourcing the code would be a great boon.

## At a glance

Nonlinear least-squares optimization is central to regression, physics-informed neural networks (PINNs), and other ML tasks. The standard Levenberg-Marquardt (LM) method suffers from parameter-effects curvature, making its tangent-space step inconsistent for finite updates. This paper proposes RNC-LM, which extends geodesic acceleration to arbitrary-order corrections via Riemann normal coordinates, constructing finite-step updates that eliminate residual acceleration order by order. The method improves convergence and robustness on benchmarks, reducing relative L2 error to 1e-3 in a PINN failure-mode problem and achieving a 34-fold speedup on large-scale potential-energy-surface fitting.

## Key points

- Identifies the fundamental inconsistency in standard LM updates due to parameter-effects curvature
- Proposes RNC-LM, extending geodesic acceleration to arbitrary-order corrections via Riemann normal coordinates
- Validates improved convergence and robustness on classic least-squares benchmarks and PINN failure-mode tests
- Achieves a 34-fold speedup on large-scale machine-learning potential-energy-surface fitting

## Full text

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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