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

等离子体物理学带来了许多物理挑战。在某些区域，用玻尔兹曼方程来描述物理学是最好的。这是在空间坐标和速度分量的6维空间中对气体进行建模；等离子体状态的演变对应于粒子在这6维空间中的传输。等离子体可以是无碰撞的，或者碰撞可以用方程中的一个附加项来表示。由于相空间的高维性，解决这一问题具有挑战性，而有限差分方法等标准技术在内存和计算成本上迅速爆炸。机器学习技术在物理建模的所有领域都产生了重大影响。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“物理信息神经网络”(PINN)中，要求解的常微分方程组或偏微分方程组的结构被嵌入到神经网络中。这对于高维系统非常有利。该项目的目标是应用Pinn来求解Boltzmann方程，并将其扩展到复杂的区域；经过训练，神经网络可以提供这个非常具有挑战性的方程的几乎瞬时解。

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.