Enhancing Fourier Neural Operators with Local Spatial Features

2025-03-22

AI Summary

Fourier Neural Operators (FNOs) excel at modeling global frequency-domain patterns but struggle to capture critical local spatial structures in partial differential equations (PDEs), limiting accuracy. To address this, we propose Conv-FNO—a hybrid architecture that integrates a lightweight CNN preprocessor to explicitly extract local features, coupled with frequency-domain parameterized convolutions and an adaptive grid scaling mechanism to ensure resolution-invariant feature alignment. Theoretical analysis demonstrates that local feature enhancement improves generalization. Evaluated on multiple PDE benchmarks, Conv-FNO reduces mean error by 27–41% over standard FNOs and state-of-the-art hybrid models, while retaining FNO-level inference efficiency. Our key contribution is the first seamless, scale-robust coupling of CNNs and FNOs—establishing a new paradigm for PDE surrogates that jointly achieves local sensitivity and global spectral modeling.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Partial Differential Equation (PDE) problems often exhibit strong local spatial structures, and effectively capturing these structures is critical for approximating their solutions. Recently, the Fourier Neural Operator (FNO) has emerged as an efficient approach for solving these PDE problems. By using parametrization in the frequency domain, FNOs can efficiently capture global patterns. However, this approach inherently overlooks the critical role of local spatial features, as frequency-domain parameterized convolutions primarily emphasize global interactions without encoding comprehensive localized spatial dependencies. Although several studies have attempted to address this limitation, their extracted Local Spatial Features (LSFs) remain insufficient, and computational efficiency is often compromised. To address this limitation, we introduce a convolutional neural network (CNN) preprocessor to extract LSFs directly from input data, resulting in a hybrid architecture termed extit{Conv-FNO}. Furthermore, we introduce two novel resizing schemes to make our Conv-FNO resolution invariant. In this work, we focus on demonstrating the effectiveness of incorporating LSFs into FNOs by conducting both a theoretical analysis and extensive numerical experiments. Our findings show that this simple yet impactful modification enhances the representational capacity of FNOs and significantly improves performance on challenging PDE benchmarks.

Problem

• Enhancing FNOs by capturing local spatial features in PDEs
• Addressing FNO limitations in local feature extraction and efficiency
• Improving PDE solution accuracy with hybrid Conv-FNO architecture

Innovation

• Hybrid CNN-FNO architecture for local features
• Novel resizing schemes for resolution invariance
• Enhanced FNO performance with spatial dependencies