Newton Informed Neural Operator for Computing Multiple Solutions of Nonlinear Partials Differential Equations
This addresses a domain-specific problem for researchers and practitioners in physics, biology, and engineering who need to compute multiple solutions of nonlinear PDEs, offering an incremental improvement over prior neural network techniques.
The paper tackles the problem of solving nonlinear partial differential equations with multiple solutions, which is challenging for existing neural network methods like PINN and DeepONet due to ill-posedness issues. The proposed Newton Informed Neural Operator efficiently learns multiple solutions in a single process and requires fewer supervised data points compared to existing methods.
Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods for solving nonlinear PDEs, such as Physics-Informed Neural Networks (PINN), Deep Ritz methods, and DeepONet, often encounter challenges when confronted with the presence of multiple solutions inherent in the nonlinear problem. These methods may encounter ill-posedness issues. In this paper, we propose a novel approach called the Newton Informed Neural Operator, which builds upon existing neural network techniques to tackle nonlinearities. Our method combines classical Newton methods, addressing well-posed problems, and efficiently learns multiple solutions in a single learning process while requiring fewer supervised data points compared to existing neural network methods.