A pullback-corrected scalar auxiliary variable optimizer with momentum and adaptive mobility

2026-09-11

AI Summary

本文提出了一种具有动量和自适应流动性的回拉校正标量辅助变量优化方法，以解决科学机器学习中多目标函数优化问题。

Abstract

Objectives in scientific machine learning are often prescribed as a sum of several terms, such as the residual, boundary, initial, and data losses of a physics-informed neural network. In the pullback-corrected scalar auxiliary variable (PB--SAV) method, one scalar tracks the shifted objective while the component gradients build a positive semidefinite curvature correction of rank at most the number of components. We carry that correction into an optimizer with momentum and an adaptive mobility, applying it to the gradient and the stored momentum in a single implicit solve. A mobility that is nonincreasing in the Loewner order yields an exact modified energy law, covering Euclidean and AMSGrad-type choices; the corresponding identity for momentum appended after the solve carries a cross term of indefinite sign. For a fixed mobility we give a necessary and sufficient condition for local stability at a stationary point, depending on the Hessian minus twice the correction, and show that it also gives local geometric convergence for every scalar relaxation sequence. The implicit solve reduces to a dense system whose order is the number of components. In the forward Burgers comparison, four components reduce the mean tail objective by 64.7% and the final solution error by 50.2% relative to one component at the same learning rate and momentum settings.

Problem

• scientific machine learning
• multi-component objective function
• optimization
• physics-informed neural network

Innovation

• pullback-corrected scalar auxiliary variable
• momentum
• adaptive mobility
• implicit solve
• geometric convergence