PODNO: Proper Orthogonal Decomposition Neural Operators

Abstract. 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 the potential to outperform FNO in both accuracy and computational efficiency for high-frequency problems. From an analysis point of view, we established the universality of a generalization of PODNO, termed generalized spectral operators. In addition, we evaluate PODNO’s performance numerically on dispersive equations such as the nonlinear Schrödinger equation and the Kadomtsev–Petviashvili equation. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/Zilan-Cheng/PODNO and in the supplementary material (PODNO-main.zip PODNO-main.zip [45 KB]). [Formula: see text]