Don't Fix the Basis -- Learn It: Spectral Representation with Adaptive Basis Learning for PDEs

2026-05-11

AI Summary

This work addresses the limitations of conventional spectral neural operators, which rely on fixed global bases and struggle to capture spatial heterogeneity and multiscale dynamics. The authors propose the Adaptive Basis Learning (ABLE) framework—the first approach to enable end-to-end learning of spectral bases. ABLE constructs data-driven, spatially adaptive Parseval frames that preserve invertibility and maintain O(N log N) computational complexity, effectively shifting representational capacity from spectral coefficients to the basis functions themselves. The framework leverages an FFT-based efficient implementation, incorporates learnable auxiliary density functions, and can seamlessly replace spectral layers in existing neural operators. Experiments demonstrate that ABLE significantly outperforms strong baselines across multiple PDE benchmarks, particularly excelling in scenarios with sharp gradients and multiscale features. Moreover, when integrated as a plug-in module into models such as U-FNO and HPM, it consistently enhances performance.

Abstract

Spectral neural operators achieve strong performance for PDE learning, but rely on fixed global bases that limit their ability to represent spatially heterogeneous and multiscale dynamics. We propose Adaptive Basis Learning (ABLE), a framework that learns data-dependent spectral representations instead of relying on predefined bases. ABLE constructs a spatially adaptive Parseval frame via a learned ancillary density, enabling the operator to act in a lifted spectral space while preserving invertibility and maintaining $O(N\log N)$ complexity through FFT-based implementation. This shifts the source of expressivity from spectral coefficients to the representation itself, allowing the model to capture localized structures and non-translation-invariant interactions more efficiently. ABLE integrates seamlessly into existing neural operator architectures as a drop-in replacement for spectral layers. Across a range of benchmarks ABLE improves accuracy over strong baselines, with the largest gains in regimes characterized by sharp gradients and multiscale behavior. Moreover, augmenting existing models (e.g., U-FNO, HPM) with ABLE further enhances their performance, demonstrating its role as a general and complementary spectral refinement. Our results highlight that the data-driven choice of representation, rather than operator complexity alone, is a key bottleneck in neural operator design. By learning the basis itself, ABLE provides a principled and efficient framework for improving spectral methods in PDE learning.

Problem

• spectral representation
• neural operators
• adaptive basis
• PDE learning
• multiscale dynamics

Innovation

• Adaptive Basis Learning
• Spectral Neural Operators
• Parseval Frame
• Data-Driven Representation
• PDE Learning