Laplacian Eigenfunction-Based Neural Operator for Learning Nonlinear Partial Differential Equations

Abstract: Learning nonlinear partial differential equations (PDEs) is emerging in many scientific and engineering disciplines, driving advancements in areas such as fluid dynamics, materials science, and biological systems. In this work, we introduce the Laplacian Eigenfunction-Based Neural Operator (LE-NO), a framework for efficiently learning nonlinear terms in PDEs, with a particular focus on nonlinear parabolic equations. By leveraging data-driven approaches to model the right-hand-side nonlinear operator, the LE-NO framework uses Laplacian eigenfunctions as basis functions, enabling efficient approximation of the nonlinear term. This approach reduces the computational complexity of the problem by enabling direct computation of the inverse Laplacian matrix and helps overcome challenges related to limited data and large neural network architectures--common hurdles in operator learning. We demonstrate the capability of our approach to generalize across various boundary conditions and provide insights into its potential applications in mathematical physics. Our results highlight the promise of LE-NO in capturing complex nonlinear behaviors, offering a robust tool for the discovery and prediction of underlying dynamics in PDEs.