Physics Informed Neural Networks to solve the Boltzmann Transport Equation (Ref: 4741)

等离子体物理学提出了许多物理上的挑战。在某些情况下，物理学最好用玻尔兹曼方程来描述。该模型在空间坐标和速度分量的6维空间中模拟气体；等离子体状态的演化对应于粒子在这个6维空间中的输运。等离子体可以是无碰撞的，或者碰撞可以用方程中的附加项来表示。由于相空间的高维性，解决这一问题具有挑战性，并且有限差分方法等标准技术在内存和计算成本方面会迅速爆发。机器学习技术正在对物理建模的各个领域产生重大影响。在将神经网络应用于物理问题时，目标是让神经网络“学习”系统的行为。在“物理通知神经网络”（pinn）中，待解的ODE或PDE的结构被内置到神经网络中。这对于高维系统是非常有利的。该项目的目标是应用PINN来求解玻尔兹曼方程并将其扩展到复杂域；经过训练后，神经网络几乎可以为这个非常具有挑战性的方程提供即时解。

Plasma physics presents a number of physical challenges. In some regimes the physics is best described by the Boltzmann equation. This models the gas in a 6 dimensional space of spatial coordinates and velocity components; the evolution of the state of the plasma corresponds to the transport of the particles in this 6-d space. The plasma can be collisionless, or collisions can be represented by an additional term in the equation. The solution of this is challenging due to the high dimensionality of the phase space, and standard techniques such as finite difference methods quickly explode in memory and computational cost.Machine Learning techniques are making significant impacts in all areas of physical modelling. In applying a Neural Network to a physics problem, the objective is for the NN to "learn" the behaviour of the system. In "physics informed Neural Networks" (PINNs), the structure of the ODE or PDE to be solved is built into the NN. This can be highly advantageous for high dimensional systems. The objective of the project is to apply PINN to solve the Boltzmann equation and extend this to complex domains; after training the NN could provide almost instantaneous solution to this very challenging equation.