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

TL;DR

ABLE introduces a data-driven spectral representation for PDE neural operators, enabling better modeling of localized and multiscale dynamics by learning adaptive bases.

Contribution

It proposes a novel framework that learns data-dependent spectral bases, improving expressivity and accuracy of neural operators for PDEs, especially in complex regimes.

Findings

ABLE improves accuracy over strong baselines in PDE benchmarks.

It enhances existing models like U-FNO and HPM when integrated.

ABLE captures localized structures and multiscale behaviors more efficiently.

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.