PODNO: Proper Orthogonal Decomposition Neural Operators
This work addresses computational challenges in solving high-frequency PDEs for scientific computing applications, representing an incremental improvement over existing Fourier Neural Operators.
The paper tackles solving partial differential equations with high-frequency components by introducing Proper Orthogonal Decomposition Neural Operators (PODNO), which replaces Fourier transforms with POD-derived transforms in neural operators, showing potential for improved accuracy and efficiency in numerical tests on dispersive equations like the Nonlinear Schrodinger and Kadomtsev-Petviashvili equations.
In this paper, we introduce Proper Orthogonal Decomposition Neural Operators (PODNO) for solving partial differential equations (PDEs) dominated by high-frequency components. Building on the structure of Fourier Neural Operators (FNO), PODNO replaces the Fourier transform with (inverse) orthonormal transforms derived from the Proper Orthogonal Decomposition (POD) method to construct the integral kernel. Due to the optimality of POD basis, the PODNO has potential to outperform FNO in both accuracy and computational efficiency for high-frequency problems. From analysis point of view, we established the universality of a generalization of PODNO, termed as Generalized Spectral Operator (GSO). In addition, we evaluate PODNO's performance numerically on dispersive equations such as the Nonlinear Schrodinger (NLS) equation and the Kadomtsev-Petviashvili (KP) equation.