Neural operators, which emerge as implicit solution operators of hidden governing equations, have recently become popular tools for learning responses of complex real-world physical systems. Nevertheless, the majority of neural operator applications has thus far been data-driven, which neglects the intrinsic preservation of fundamental physical laws in data. In this paper, we introduce a novel integral neural operator architecture, to learn physical models with fundamental conservation laws automatically guaranteed. In particular, by replacing the frame-dependent position information with its invariant counterpart in the kernel space, the proposed neural operator is by design translation- and rotation-invariant, and consequently abides by the conservation laws of linear and angular momentums. As applications, we demonstrate the expressivity and efficacy of our model in learning complex material behaviors from both synthetic and experimental datasets, and show that, by automatically satisfying these essential physical laws, our learned neural operator is not only generalizable in handling translated and rotated datasets, but also achieves state-of-the-art accuracy and efficiency as compared to baseline neural operator models.

翻译：神经算子作为隐式求解底层控制方程的新兴方法，近年来已成为学习复杂现实物理系统响应的流行工具。然而，目前大多数神经算子应用仍基于数据驱动，忽视了数据中物理基本定律的内在保持性。本文提出一种新型积分神经算子架构，用于自动保证具有基本守恒定律的物理模型学习。具体而言，通过将依赖框架的位置信息替换为核空间中的不变对应量，所提出的神经算子天然具有平移和旋转不变性，从而遵循线动量与角动量的守恒定律。作为应用实例，我们在合成数据集和实验数据集上展示了该模型在学习复杂材料行为方面的表现力与有效性，表明通过自动满足这些基本物理定律，所学习的神经算子不仅能够泛化处理平移和旋转后的数据集，而且与基准神经算子模型相比，实现了最先进的精度与效率。