  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.
