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

2026-07-08 · Source: Takara TLDR - Daily AI Papers · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, quick

Summary

The Riemann-Normal-Coordinate Levenberg-Marquardt method (RNC-LM) is a novel optimization technique designed to enhance the standard Levenberg-Marquardt (LM) method for nonlinear least-squares problems, which are crucial in regression and physics-informed neural networks (PINNs). RNC-LM addresses LM's limitation of applying tangent-space steps as straight updates in parameter coordinates, which struggles with parameter-effects curvature. By extending geodesic acceleration through Riemann Normal Coordinates, RNC-LM constructs finite-step updates with progressively higher reparameterization consistency, eliminating the tangential component of residual acceleration order by order. Benchmarks show RNC-LM improves convergence and robustness in challenging scenarios, reduces relative L2 error to the order of 1e-3 on a reaction-diffusion PINN benchmark, and achieves a 34-fold speedup over standard LM on a large-scale machine-learning potential-energy-surface fitting task.

Key takeaway

For Machine Learning Engineers optimizing complex models or Research Scientists working with physics-informed neural networks, adopting the Riemann-Normal-Coordinate Levenberg-Marquardt (RNC-LM) method can significantly enhance optimization performance. You should consider RNC-LM to overcome limitations of standard LM in highly nonlinear or rank-deficient problems, potentially achieving substantial speedups and more accurate, physically meaningful solutions, such as reducing L2 error to 1e-3. Evaluate RNC-LM for tasks requiring robust convergence and efficiency.

Key insights

RNC-LM improves nonlinear least-squares optimization by using higher-order geometric updates for enhanced reparameterization consistency and faster convergence.

Principles

• Parameter-effects curvature limits LM.
• Higher-order geometric updates improve consistency.

Method

RNC-LM reformulates the geodesic equation to extend geodesic acceleration, constructing finite-step updates with arbitrary-order corrections and a line search to control distance.

In practice

• Improve PINN solution accuracy.
• Speed up large-scale ML potential fitting.

Topics

• Nonlinear Least-Squares
• Levenberg-Marquardt Method
• Riemann Normal Coordinates
• Optimization Algorithms
• Physics-Informed Neural Networks
• Machine Learning Potentials

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Takara TLDR - Daily AI Papers.