A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers
PositiveArtificial Intelligence
The introduction of the Neural Preconditioned Newton (NP-Newton) method marks a significant advancement in solving parametric nonlinear systems of equations. By incorporating a Fixed-Point Neural Operator (FPNO), this method adeptly navigates the challenges posed by unbalanced nonlinearities, which often lead to stagnation or instability in traditional Newton iterations. The FPNO innovatively employs negative step sizes to enhance convergence, setting it apart from conventional line-search or trust-region algorithms. Numerical experiments have demonstrated the NP-Newton method's computational efficiency and robustness, particularly in real-world applications characterized by strong nonlinearities. This development not only showcases the potential of neural networks in mathematical problem-solving but also paves the way for more effective solutions in complex nonlinear systems.
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