Adaptive Coordinate Transforms for Neural Operators

2026-05-07

AI Summary

Existing neural operators rely on fixed Eulerian coordinates, which struggle to capture evolving physical structures and often lead to spatial misalignment, amplified non-local mappings, and excessive smoothing in regions with sharp transitions. To address this limitation, this work proposes the Adaptive Coordinate Transformation (ACT) module—a plug-and-play component that learns data-driven coordinate transformations and integrates differentiable sampling to reconstruct feature representations, enabling neural operators to model dynamic processes in more suitable coordinate systems. Inspired by adaptive meshing techniques in classical PDE solvers, ACT introduces, for the first time, end-to-end learnable coordinate systems into neural operators, thereby overcoming the constraints of fixed grids. Extensive experiments across multiple PDE benchmarks and mainstream architectures—including FNO and DeepONet—demonstrate that ACT consistently enhances predictive accuracy, underscoring the critical role of coordinate learning in improving operator generalization.

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.

Problem

• neural operators
• coordinate transform
• partial differential equations
• spatial misalignment
• adaptive representation

Innovation

• Adaptive Coordinate Transform
• Neural Operators
• Coordinate Learning
• Differentiable Sampling
• PDE Solvers