FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation

TL;DR

FC-PINO enhances physics-informed neural operators by integrating Fourier continuation, enabling high-precision solutions for non-periodic and non-smooth PDEs with improved accuracy and efficiency.

Contribution

The paper introduces Fourier continuation into PINO, allowing it to effectively handle non-periodic and non-smooth PDEs, overcoming limitations of spectral differentiation.

Findings

FC-PINO outperforms standard PINO on non-periodic PDEs

Fourier continuation improves derivative accuracy for non-smooth functions

FC-PINO achieves high-precision solutions across challenging benchmarks

Abstract

The physics-informed neural operator (PINO) is a machine learning paradigm that has demonstrated promising results for learning solutions to partial differential equations (PDEs). It leverages the Fourier Neural Operator to learn solution operators in function spaces and leverages physics losses during training to penalize deviations from known physics laws. Spectral differentiation provides an efficient way to compute derivatives for the physics losses, but it inherently assumes periodicity. When applied to non-periodic functions, this assumption can lead to significant errors, including Gibbs phenomena near domain boundaries which degrade the accuracy of both function representations and derivative computations. To overcome this limitation, we introduce the FC-PINO (Fourier-Continuation-based Physics-Informed Neural Operator) architecture which extends the accuracy and efficiency of PINO and spectral differentiation to non-periodic and non-smooth PDEs. In FC-PINO, we propose integrating Fourier continuation into the PINO framework, and test two different continuation approaches: FC-Legendre and FC-Gram. By transforming non-periodic signals into periodic functions on extended domains in a well-conditioned manner, Fourier continuation enables fast and accurate derivative computations. This approach avoids the discretization sensitivity of finite differences and the memory overhead of automatic differentiation. We demonstrate that standard PINO fails (without padding) or struggles (even with padding) to solve non-periodic and non-smooth PDEs with high precision, across challenging benchmarks. In contrast, the proposed FC-PINO provides accurate, robust, and scalable solutions, substantially outperforming PINO alternatives, and demonstrating that Fourier continuation is critical for extending PINO to a wider range of PDE problems when high-precision solutions are needed.