Adaptive Coordinate Transforms for Neural Operators

TL;DR

The paper introduces the Adaptive Coordinate Transform (ACT) block, a neural module that learns data-driven coordinate systems to improve neural operator performance on PDEs by better capturing evolving structures.

Contribution

It proposes a novel differentiable module that learns adaptive coordinate transformations within neural operators, addressing spatial misalignment issues in PDE modeling.

Findings

ACT improves predictive accuracy across diverse PDE benchmarks.

Learning coordinate systems reduces operator complexity.

Experimental results show significant performance gains.

Abstract

Neural operators have achieved promising performance on partial differential equations (PDEs), but most existing models are built on fixed Eulerian coordinates. This mismatch between evolving physical structures and static coordinates creates spatial misalignment, leading to unnecessarily non-local operator mappings and reinforcing a smoothness preference near sharp transitions. Inspired by adaptive coordinate transformations in classical PDE analysis, we propose the Adaptive Coordinate Transform (ACT) block, a plug-and-play module for data-driven geometric adaptation in neural operators. ACT blocks resolve this structural limitation by learning adaptive coordinate systems within the operator learning pipeline. Specifically, given an input feature, the ACT block learns a coordinate transformation and represents the same feature under the transformed coordinates via differentiable sampling. This operation preserves the underlying signal while changing its spatial representation, equivalent to expressing the same physical quantity in different coordinate systems. By adapting the coordinate system to the data, ACT allows the network to better track evolving structures, reduce operator complexity, and dynamically focus on critical features to improve learning. We evaluate the proposed approach across diverse PDE benchmarks and multiple neural operator architectures. Experimental results demonstrate consistent and significant improvements in predictive accuracy, indicating that learning coordinate systems provides a powerful mechanism for enhancing operator learning.