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

arXiv: https://arxiv.org/abs/2211.15960 · HTML: https://mlsanity.com/paper/2211.15960/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Nov 2022.

## Built on by (most cited first, 13 of 13)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [High precision PINNs in unbounded domains: application to singularity formulation in PDEs](https://mlsanity.com/paper/2506.19243/) (2025)
- [Enabling Automatic Differentiation with Mollified Graph Neural Operators](https://mlsanity.com/paper/2504.08277/) (2025)
- [Projected Neural Differential Equations for Learning Constrained Dynamics](https://mlsanity.com/paper/2410.23667/) (2024)
- [Learning Generic Solutions for Multiphase Transport in Porous Media via the Flux Functions Operator](https://mlsanity.com/paper/2307.01354/) (2023)
- [NOBLE -- Neural Operator with Biologically-informed Latent Embeddings to Capture Experimental Variability in Biological Neuron Models](https://mlsanity.com/paper/2506.04536/) (2025)
- [FNOangle θ: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems](https://mlsanity.com/paper/2604.05187/) (2026)
- [Fourier Feature Pyramids for Physics-Informed Neural Networks](https://mlsanity.com/paper/2605.24278/) (2026)
- [Equation Recast for Canonical Operator Learning Across Parametric PDEs](https://mlsanity.com/paper/2609.02982/) (2026)
- [Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches](https://mlsanity.com/paper/2604.16842/) (2026)

## Abstract (arXiv)

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.