Newton Informed Neural Operator for Solving Nonlinear Partial Differential Equations

Solving nonlinear partial differential equations (PDEs) with multiple solutions is essential in various fields, including physics, biology, and engineering. These methods can become computationally expensive, especially when relying on solvers like Newton's method, which may struggle with ill-posedness near bifurcation points.In this paper, we propose a novel approach, the Newton Informed Neural Operator, which learns the Newton solver for nonlinear PDEs. This approach allows us to compute multiple solutions in a single learning process while requiring fewer supervised data points than existing neural network methods.