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

TL;DR

FC-PINO addresses the high-precision PDE solution challenge by integrating Fourier continuation (FC) into the Physics-Informed Neural Operator (PINO) framework, enabling accurate spectral differentiation on non-periodic domains. It introduces two FC schemes, FC--Legendre and FC--Gram, to extend signals to a periodic domain on an enlarged axis while maintaining well-conditioned extensions, and analyzes derivative convergence as $ subscript{k}$-th derivatives scale like $ subscript{O}(N^{-(d-k)})$. Across 1D Burgers, 2D Burgers, and 3D Navier–Stokes benchmarks, FC-PINO substantially outperforms standard PINO and padding baselines, achieving PDE residuals down to $10^{-12}$ and delivering robust, high-precision solutions. The results demonstrate that FC-based extensions are crucial for transferring PINO to broader classes of non-periodic and non-smooth PDE problems while preserving the efficiency benefits of spectral methods. The implemented FC-PINO framework and FC methods are public, offering a principled, scalable path for high-precision operator learning in physics-informed ML.

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.

Figures (21)

• Figure 1: Examples of periodic extensions obtained using the Fourier continuation methods. (top) Extensions of $f(x) = \sin(16x) - \cos(8x)$ from $[0,1]$ to $[-0.25,1.25]$. (bottom) Extensions of $f(x,y) = \sin(12x) - \cos(14y) + 3xy$ from $[0,1]^2$ to $[-0.25,1.25]^2$.
• Figure 2: The Fourier Neural Operator (FNO) architecture (extracted from FNO_OG).
• Figure 3: Visualization of the Gibbs phenomenon when applying spectral differentiation to a non-periodic solution $U(y)$ to the 1D Burgers' equation \ref{['eq: 1D Burger']}. (top) Applying spectral differentiation on the extended domain after using FC--Gram. (bottom two) Applying spectral differentiation on the original domain without Fourier continuation. We observe large spurious oscillations, whose magnitude grows rapidly with the derivative order.
• Figure 4: Illustration of the various variables used in the Fourier continuation. The given signal $f$ of $n$ points given on $[a,b]$ is extended to a periodic signal with $(n+c)$ points, and the extension is split and appended on both sides (each with $c/2$ points). FC--Legendre and FC--Gram construct periodic extension using only the information contained in the boundary vectors $f_{\ell}$ and $f_r$ of width $d$, and not the remaining part $f_{\text{int}}$ of the signal.
• Figure 5: PDE residual loss across different $\lambda$'s when fine-tuning the pretrained FC--PINO. The black curve represents the pretrain phase where we use the Adam optimizer for 3000 epochs, learning a solution operator over multiple $\lambda$ values. We subsequently switch to the L--BFGS optimizer, as signified by the dashed line, and fine-tune the operator individually on 5 different PDE instances corresponding to different values of $\lambda$.