Neural operators, which can act as implicit solution operators of hidden governing equations, have recently become popular tools for learning the responses of complex real-world physical systems. Nevertheless, most neural operator applications have thus far been data-driven and neglect the intrinsic preservation of fundamental physical laws in data. In this work, we introduce a novel integral neural operator architecture called the Peridynamic Neural Operator (PNO) that learns a nonlocal constitutive law from data. This neural operator provides a forward model in the form of state-based peridynamics, with objectivity and momentum balance laws automatically guaranteed. As applications, we demonstrate the expressivity and efficacy of our model in learning complex material behaviors from both synthetic and experimental data sets. We show that, owing to its ability to capture complex responses, our learned neural operator achieves improved accuracy and efficiency compared to baseline models that use predefined constitutive laws. Moreover, by preserving the essential physical laws within the neural network architecture, the PNO is robust in treating noisy data. The method shows generalizability to different domain configurations, external loadings, and discretizations.

翻译：神经算子能够作为隐式控制方程的隐式求解算子，近年来已成为学习复杂现实物理系统响应的主流工具。然而，大多数神经算子应用至今仍属数据驱动，且忽略了在数据中内在保留基本物理规律。本研究提出一种新型积分神经算子架构——近场动力学神经算子(PNO)，该算子从数据中学习非局部本构定律。该神经算子以基于状态的近场动力学形式提供正向模型，自动保证了客观性与动量守恒定律。在应用层面，我们通过合成数据集与实验数据集展示了模型在学习复杂材料行为方面的表现力与有效性。研究表明，由于能够捕捉复杂响应，相较于使用预定义本构定律的基准模型，我们训练的神经算子实现了更优的准确性与效率。此外，通过在神经网络架构中保留基本物理规律，PNO在噪声数据处理中展现出鲁棒性。该方法对不同域配置、外部载荷与离散化方案均具有可泛化性。