| Abstract Scope |
I will introduce a novel approach, the Newton Informed Neural Operator (NINO), which learns the Newton solver for nonlinear PDEs. Our method combines traditional numerical techniques with the Newton iteration scheme, efficiently approximating the nonlinear mapping at each step. This framework enables the computation of multiple solutions within a single learning process while requiring fewer supervised data points than existing neural network-based methods. In addition, I will present the Laplacian Eigenfunction-Based Neural Operator (LE-NO), a framework designed for efficiently learning nonlinear terms in PDEs, with a particular emphasis on nonlinear parabolic equations. By adopting a data-driven approach to model the nonlinear right-hand side, LE-NO employs Laplacian eigenfunctions as basis functions, providing an efficient and accurate approximation of the nonlinear operator. |