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.

翻译：神经算子在偏微分方程求解中取得了显著成效，但现有模型大多基于固定的欧拉坐标构建。这种演化物理结构与静态坐标之间的失配会导致空间错位，引发不必要的非局部算子映射，并在突变界面附近强化平滑性偏好。受经典偏微分方程分析中自适应坐标变换思想的启发，我们提出自适应坐标变换模块——一种即插即用的数据驱动几何自适应组件。ACT模块通过在算子学习框架中构建自适应坐标系统来突破结构限制：给定输入特征，该模块学习坐标变换并利用可微采样在变换后的坐标下重新表征相同特征。该操作在保持原始信号的同时改变其空间表征，等价于在不同坐标系下表达相同的物理量。通过使坐标系统自适应于数据，ACT能够帮助网络更好地追踪演化结构、降低算子复杂度，并动态聚焦关键特征以提升学习效果。我们在多个偏微分方程基准测试和多种神经算子架构上评估了该方案，实验结果表明预测精度获得持续且显著的提升，说明学习坐标系统为增强算子学习提供了有效机制。