The authors propose a Riemann-normal-coordinate Levenberg-Marquardt method (RNC-LM) to address the limitation of standard LM's tangent-space step, which applies straight updates in parameter coordinates. By reformulating the geodesic equation, RNC-LM extends geodesic acceleration to arbitrary-order corrections and constructs finite-step updates with progressively higher reparameterization consistency.
- The method eliminates the tangential component of residual acceleration order by order in a moving tangent frame.
- A line search along the resulting RNC curve controls traveled distance while keeping the cost close to standard 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.
RNC-LM makes the actual objective reduction more consistent with the linear model prediction of LM, enhancing performance in complex optimization landscapes.