论文标题
局部神经操作员，用于解决各个域上的瞬态偏微分方程
Local neural operator for solving transient partial differential equations on varied domains
论文作者
论文摘要
人工智能（AI）具有降低解决部分微分方程（PDE）的巨大成本的巨大潜力。但是，由于神经网络是在固定域和边界上定义和训练的，因此在实践中并未完全实现。本文中，我们提出了局部神经操作员（LNO），以求解各种域上的瞬态PDE。它带有一个方便的策略，包括边界处理，使一个预先训练的LNO可以预测不同领域的解决方案。为了进行演示，LNO从随机生成的数据样本中学习Navier-Stokes方程，然后将预训练的LNO用作明确的数值时间构造方案，以解决在看不见的域上流体流动的流动，例如，在lid驱动器中的流动流动和跨层层流的流动。它比传统的有限元方法快1000美元$ \ times $ $ $ $ $ $，可以计算跨翼型级联的流量。具有预训练的LNO的解决过程可实现巨大的效率，在实践中有很大的潜力加速数值计算。
Artificial intelligence (AI) shows great potential to reduce the huge cost of solving partial differential equations (PDEs). However, it is not fully realized in practice as neural networks are defined and trained on fixed domains and boundaries. Herein, we propose local neural operator (LNO) for solving transient PDEs on varied domains. It comes together with a handy strategy including boundary treatments, enabling one pre-trained LNO to predict solutions on different domains. For demonstration, LNO learns Navier-Stokes equations from randomly generated data samples, and then the pre-trained LNO is used as an explicit numerical time-marching scheme to solve the flow of fluid on unseen domains, e.g., the flow in a lid-driven cavity and the flow across the cascade of airfoils. It is about 1000$\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.
