SirenFNO: Efficient and Full Frequency Learning of Fourier Neural Operators

2026-06-09

AI Summary

This work addresses the limitation of conventional Fourier Neural Operators (FNOs), which exhibit a low-frequency bias due to spectral truncation and struggle to accurately approximate solutions of partial differential equations (PDEs) featuring strong high-frequency oscillations. To overcome this, the authors propose SirenFNO, a novel framework that integrates Sinusoidal Representation Networks (SIRENs) into the FNO architecture for the first time. By enabling mode-wise kernel parameterization, SirenFNO achieves full-spectrum learning without frequency truncation while preserving discretization invariance. Coupled with functional tensor decomposition, the method substantially enhances parameter efficiency: across multiple PDE benchmarks, SirenFNO maintains or improves accuracy with 4–15 times fewer parameters, and its decomposed variant reduces parameter count by up to 73-fold while achieving superior accuracy.

Abstract

Fourier neural operators (FNOs) are effective and efficient surrogates for approximating solutions of PDEs and generalize across discretizations. However, owing to the reliance on frequency truncation to maintain learning efficiency of FNOs, empirical studies suggest that FNOs exhibit spectral bias toward low-frequency information, which may hinder the learning capability especially for certain PDEs with strong high-frequency oscillations. To address this limitation, we propose SirenFNO, a novel framework that leverages sinusoidal representation networks (SIRENs) to learn implicit neural representations and performs mode-wise kernel parameterization. Our SIREN parameterization learns a full-grid spectrum with a constant and discretization-independent parameter count, thereby eliminating the need for frequency truncation. We further extend SirenFNO with functional tensor decompositions to enhance parameter and learning efficiency. Empirical results show that our SirenFNO consistently outperforms FNO with approximately $4$ to $15$ times parameter reductions with preserved discretization invariance, and our functional decomposition variants obtain performance improvements with a maximum of $73$ times fewer parameters across multiple PDE benchmarks.

Problem

• Fourier neural operators
• spectral bias
• high-frequency oscillations
• frequency truncation
• PDEs

Innovation

• Fourier Neural Operators
• SIREN
• frequency truncation
• functional tensor decomposition
• discretization invariance