用神经算子加速非线性方程求解，解决传统方法不稳定问题。
A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers
- 用固定点神经算子学习迭代到解的映射，自适应负步长应对强非线性。
- 在强非线性场景下，计算效率和鲁棒性显著优于传统方法。
- 适合需要快速稳定求解复杂非线性系统的工程与科学计算场景。
我们提出一种新的神经预处理牛顿（NP-Newton）方法，用于求解参数化非线性方程组。为克服因非线性不平衡导致的牛顿迭代停滞或不稳定的难题，引入固定点神经算子（FPNO），通过模拟固定点迭代学习从当前迭代值到解的直接映射。与传统线搜索或信赖域算法不同，所提出的FPNO能自适应采用负步长，有效缓解非线性不平衡的影响。通过数值实验，我们在多个真实应用场景中验证了所提方法在计算效率和鲁棒性上的优势，尤其在极强非线性情况下表现突出。
原文摘要 · Abstract (English)
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 effectively mitigate the effects of unbalanced nonlinearities. Through numerical experiments we demonstrate the computational efficiency and robustness of the proposed NP-Newton method across multiple real-world applications, especially for very strong nonlinearities.
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