Fourier neural operators (FNOs) can learn highly nonlinear mappings between function spaces, and have recently become a popular tool for learning responses of complex physical systems. However, to achieve good accuracy and efficiency, FNOs rely on the Fast Fourier transform (FFT), which is restricted to modeling problems on rectangular domains. To lift such a restriction and permit FFT on irregular geometries as well as topology changes, we introduce domain agnostic Fourier neural operator (DAFNO), a novel neural operator architecture for learning surrogates with irregular geometries and evolving domains. The key idea is to incorporate a smoothed characteristic function in the integral layer architecture of FNOs, and leverage FFT to achieve rapid computations, in such a way that the geometric information is explicitly encoded in the architecture. In our empirical evaluation, DAFNO has achieved state-of-the-art accuracy as compared to baseline neural operator models on two benchmark datasets of material modeling and airfoil simulation. To further demonstrate the capability and generalizability of DAFNO in handling complex domains with topology changes, we consider a brittle material fracture evolution problem. With only one training crack simulation sample, DAFNO has achieved generalizability to unseen loading scenarios and substantially different crack patterns from the trained scenario.

翻译：傅里叶神经算子（FNO）能够学习函数空间之间的高度非线性映射，近年来已成为研究复杂物理系统响应的热门工具。然而，为达到良好的精度与效率，FNO依赖快速傅里叶变换（FFT），这使其仅适用于矩形域上的建模问题。为突破这一限制，并在不规则几何及拓扑变化场景中实现FFT，我们提出域无关傅里叶神经算子（DAFNO）——一种新型神经算子架构，用于学习非规则几何与演化域上的替代模型。其核心思想是将平滑特征函数融入FNO的积分层架构，并利用FFT实现快速计算，从而将几何信息显式编码于网络结构中。在实证评估中，相较于基线神经算子模型，DAFNO在材料建模与翼型仿真两个基准数据集上取得了最先进精度。为进一步展示DAFNO处理含拓扑变化复杂域的能力与泛化性，我们将其应用于脆性材料断裂演化问题。仅需一个训练裂纹仿真样本，DAFNO即可实现对未见加载场景及与训练场景显著不同的裂纹模式的泛化能力。