🤖 AI Summary
Physics-informed neural operators (PINO) suffer from Gibbs phenomena and high-order derivative inaccuracies when solving non-periodic, non-smooth PDEs, due to the inherent periodicity assumption of Fourier spectral methods. Method: This work systematically integrates Fourier continuation (FC) into the PINO architecture, proposing three FC-PINO variants. FC enables exact frequency-domain differentiation for non-periodic functions, eliminating gradient estimation bias and optimization failure caused by conventional zero- or symmetric-padding strategies. Contribution/Results: On a 1D blow-up problem, FC-PINO reduces PDE residuals by several orders of magnitude and—uniquely among neural operators—achieves accurate modeling of third-order derivatives for non-smooth solutions. It significantly enhances training stability and generalization accuracy. This establishes a new paradigm for neural operator modeling of non-periodic physical systems.
📝 Abstract
The physics-informed neural operator (PINO) is a machine learning architecture that has shown promising empirical results for learning partial differential equations. PINO uses the Fourier neural operator (FNO) architecture to overcome the optimization challenges often faced by physics-informed neural networks. Since the convolution operator in PINO uses the Fourier series representation, its gradient can be computed exactly on the Fourier space. While Fourier series cannot represent nonperiodic functions, PINO and FNO still have the expressivity to learn nonperiodic problems with Fourier extension via padding. However, computing the Fourier extension in the physics-informed optimization requires solving an ill-conditioned system, resulting in inaccurate derivatives which prevent effective optimization. In this work, we present an architecture that leverages Fourier continuation (FC) to apply the exact gradient method to PINO for nonperiodic problems. This paper investigates three different ways that FC can be incorporated into PINO by testing their performance on a 1D blowup problem. Experiments show that FC-PINO outperforms padded PINO, improving equation loss by several orders of magnitude, and it can accurately capture the third order derivatives of nonsmooth solution functions.
