论文标题
Oppinn：具有操作员的物理信息的神经网络学习将解决方案近似于Fokker-Planck-Landau方程
opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation
论文作者
论文摘要
我们提出了一个混合框架Oppinn：物理信息的神经网络（PINN），其中运算符学习近似于Fokker-Planck-Landau（FPL）方程的解决方案。 Oppinn框架分为两个步骤：步骤1和步骤2。在步骤1期间对操作员替代模型进行训练后，Pinn可以使用预先训练的替代模型在步骤2期间有效地近似于FPL方程的解决方案。操作员替代模型通过近似FPL方程中的复杂Landau碰撞积分来大大降低计算成本并提高PINN。操作员替代模型也可以与传统的数值方案结合使用。当速度模式变大时，它在计算时间内提供了高效率。使用Oppinn框架，我们在各种类型的初始条件下为FPL方程提供了神经网络解决方案，以及两个和三个维度的相互作用模型。此外，基于FPL方程的理论属性，我们表明，随着预定义损耗函数的降低，近似的神经网络解决方案会收敛到FPL方程的先验经典解决方案。
We propose a hybrid framework opPINN: physics-informed neural network (PINN) with operator learning for approximating the solution to the Fokker-Planck-Landau (FPL) equation. The opPINN framework is divided into two steps: Step 1 and Step 2. After the operator surrogate models are trained during Step 1, PINN can effectively approximate the solution to the FPL equation during Step 2 by using the pre-trained surrogate models. The operator surrogate models greatly reduce the computational cost and boost PINN by approximating the complex Landau collision integral in the FPL equation. The operator surrogate models can also be combined with the traditional numerical schemes. It provides a high efficiency in computational time when the number of velocity modes becomes larger. Using the opPINN framework, we provide the neural network solutions for the FPL equation under the various types of initial conditions, and interaction models in two and three dimensions. Furthermore, based on the theoretical properties of the FPL equation, we show that the approximated neural network solution converges to the a priori classical solution of the FPL equation as the pre-defined loss function is reduced.
