发表机构
Max Planck Institute for Plasma Physics; Université de Strasbourg, CNRS, Inria, IRMA; CEA, IRFM（马普等离子体物理研究所; 斯特拉斯堡大学、法国国家科学研究中心、Inria、IRMA; 法国原子能和替代能源委员会、核聚变研究设施）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对输运主导问题，提出基于流映射的拉格朗日PINNs方法，通过多PINN近似流映射并利用半群性质组合，在四类方程数值实验中展现出优于经典PINNs及数值格式的性能。
AI 中文摘要
我们提出了一种针对输运主导问题的新型数值方法，该方法采用物理信息神经网络（PINNs）。在这类问题中，解会随时间发展出大梯度和精细结构，这对表示解本身的经典PINNs构成了特殊挑战。在我们的方法中，我们选择近似方程的流映射而非解本身，通过拉回精确初始条件来恢复解，从而使解的精细结构由映射生成，而非由网络表示。我们的方法通过多个PINNs近似方程的流映射来构建，每个时间子区间对应一个PINN，并利用流的半群性质将它们组合，使得每个网络仅需表示接近恒等映射的映射。为突出我们方法相比经典PINNs或经典数值格式的优势，我们给出了线性平流方程、涡度形式的不可压缩欧拉方程、弗拉索夫-泊松方程以及漂移动理学方程的数值结果。
英文摘要
We propose a novel numerical method for transport-dominated problems using Physics-Informed Neural Networks (PINNs). In such problems, the solution can develop large gradients and fine structures over time. This is particularly challenging for classical PINNs, which represent the solution itself. In our approach, rather than the solution, we choose to approximate the flow map of the equation. The solution is then recovered by pulling back the exact initial condition, so that its fine structures are produced by the map rather than represented by the network. Our method is constructed by approximating the flow map of the equation with multiple PINNs, one per time subinterval, and composing them using the semigroup property of the flow, so that each network only has to represent a map close to the identity. To highlight the advantages of our method compared to classical PINNs or classical numerical schemes, we present numerical results on the linear advection equation, the incompressible Euler equation in vorticity formulation, the Vlasov-Poisson equation, and finally a drift-kinetic equation.
