A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers
Youngkyu Lee, Shanqing Liu, Jerome Darbon, George Em Karniadakis
我们提出了一种新的神经预后牛顿(NP-牛顿)方法,用于求解参数非线性方程系统。 为了克服由不平衡非线性引起的牛顿迭代的停滞或不稳定,我们引入了一个定点神经运算符(FPNO),它通过模拟定点迭代来学习从当前迭代到解决方案的直接映射。 与传统线搜索或信任区域算法不同,建议的FPNO自适应使用负步尺寸来有效减轻不平衡非线性的影响。 通过数值实验,我们展示了拟议的NP-Newton方法在多个实际应用中的计算效率和稳健性,特别是对于非常强的非线性。
We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a fixed-point neural operator (FPNO) that learns the direct mapping from the current iterate to the solution by emulating fixed-point iterations. Unlike traditional line-search or trust-region algorithms, the proposed FPNO adaptively employs negative step sizes to ...
