从非凸自协调正则化到物理信息神经网络的可扩展拟牛顿训练

中文摘要 AI 辅助

物理信息神经网络（PINNs）通常需要高精度的拟牛顿优化来获得可靠的偏微分方程解，但其残差目标函数可能呈现不定、近奇异且尺度不佳的局部曲率。正则化拟牛顿方法提供了稳定割线模型的成熟机制，而自协调方法提供了依赖曲率的步长选择的局部度量规则。基于这两条研究路线，我们提出了SCORE——一种受自协调启发的拟牛顿方法，具有用于PINN训练的减量耦合移位割线几何。其独特机制是，从学习到的逆度量计算出的单个拟牛顿减量共同确定了经过强沃尔夫条件检验的候选步长和用于定义下一个割线几何的自适应移位。移位位移表示沿接受步长的平均移位度量的作用，且无需构造海森矩阵或计算海森-向量乘积。在局部谱等价条件下，我们证明拟牛顿减量和候选步长仍可与正移位度量中的对应量相媲美，并在匹配度量情况下恢复归一化自协调规则。强沃尔夫接受、 fallback线搜索和标准曲率保障措施在不修改底层PINN目标函数的情况下提供了全局收敛性。对粘性伯格斯方程、仓本-西瓦什金斯基方程、科特韦格-德弗里斯方程和复金兹堡-朗道方程的实验表明，SCORE达到的最终误差低于所测试的BFGS和自缩放布罗伊登基线。伯格斯方程的消融实验进一步表明，移位曲率稳定和基于减量的步长选择对高精度优化做出了互补贡献。

英文摘要

Physics-informed neural networks (PINNs) often require high-accuracy quasi-Newton refinement to obtain reliable partial differential equation solutions, but their residual objectives can exhibit indefinite, nearly singular, and poorly scaled local curvature. Regularized quasi-Newton methods provide established mechanisms for stabilizing secant models, while self-concordant methods provide local-metric rules for curvature-dependent step selection. Building on these two lines of work, we propose SCORE, a self-concordance-inspired quasi-Newton method with decrement-coupled shifted secant geometry for PINN training. Its distinguishing mechanism is that a single quasi-Newton decrement computed from the learned inverse metric jointly determines a strong-Wolfe-tested candidate step and an adaptive shift used to define the next secant geometry. The shifted displacement represents the action of an averaged shifted metric along the accepted step, while requiring neither Hessian construction nor Hessian-vector products. Under a local spectral-equivalence condition, we show that the quasi-Newton decrement and candidate step remain comparable to their counterparts in a positive shifted metric, and recover the normalized self-concordant rule in the matched-metric case. Strong Wolfe acceptance, fallback line search, and standard curvature safeguards provide globalization without modifying the underlying PINN objective. Experiments on the viscous Burgers, Kuramoto--Sivashinsky, Korteweg--de Vries, and complex Ginzburg--Landau equations show that SCORE attains lower final errors than the tested BFGS and self-scaled Broyden baselines. The Burgers ablation further indicates that shifted curvature stabilization and decrement-based step selection make complementary contributions to high-accuracy refinement.

发表机构

• School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
• Rice University（莱斯大学）

机构由 AI 辅助整理，请以论文原文为准。