Artificial intelligence (AI) shows great potential to reduce the huge cost of solving partial differential equations (PDEs). However, it is not fully realized in practice as neural networks are defined and trained on fixed domains and boundaries. Herein, we propose local neural operator (LNO) for solving transient PDEs on varied domains. It comes together with a handy strategy including boundary treatments, enabling one pre-trained LNO to predict solutions on different domains. For demonstration, LNO learns Navier-Stokes equations from randomly generated data samples, and then the pre-trained LNO is used as an explicit numerical time-marching scheme to solve the flow of fluid on unseen domains, e.g., the flow in a lid-driven cavity and the flow across the cascade of airfoils. It is about 1000$\times$ faster than the conventional finite element method to calculate the flow across the cascade of airfoils. The solving process with pre-trained LNO achieves great efficiency, with significant potential to accelerate numerical calculations in practice.

翻译：人工智能（AI）在降低求解偏微分方程（PDEs）的巨大成本方面展现出巨大潜力。然而，由于神经网络在固定域和边界上定义和训练，这一潜力在实践中尚未完全实现。为此，我们提出局部神经算子（LNO）以解决变域上的瞬态PDEs。该方法结合了包含边界处理在内的便捷策略，使得单一预训练LNO能够预测不同域上的解。作为演示，LNO从随机生成的数据样本中学习Navier-Stokes方程，随后将预训练LNO用作显式数值时间推进方案，求解未见域上的流体流动，例如顶盖驱动空腔流和穿过叶栅的流动。与传统的有限元方法相比，该算子计算穿过叶栅的流动速度快约1000倍。基于预训练LNO的求解过程实现了极高的效率，在实际应用中具有加速数值计算的巨大潜力。