PODNO: Proper Orthogonal Decomposition Neural Operators

2025-04-25

AI Summary

This work addresses the low accuracy and poor efficiency in solving high-frequency-dominated partial differential equations (PDEs), such as the nonlinear Schrödinger (NLS) and Kadomtsev–Petviashvili (KP) equations. We propose a neural operator framework grounded in Proper Orthogonal Decomposition (POD). Our key innovation is the first integration of POD-derived optimal low-dimensional orthogonal bases—instead of conventional Fourier bases—into the kernel design of neural operators, enabling more compact and physically interpretable spectral representations of high-frequency modes. We theoretically establish the universality of the proposed Generalized Spectral Operator (GSO). Experiments on NLS and KP equations demonstrate that our method significantly outperforms the Fourier Neural Operator (FNO) in both prediction accuracy and computational efficiency, validating the superiority and generalizability of POD bases for modeling high-frequency PDEs.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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 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.

Problem

• Solving high-frequency dominated PDEs efficiently
• Improving accuracy over Fourier Neural Operators
• Evaluating performance on dispersive equations

Innovation

• Uses Proper Orthogonal Decomposition basis
• Replaces Fourier transform with orthonormal transforms
• Optimizes accuracy for high-frequency PDEs