The Boltzmann equation is a partial differential equation that describes the variation of particles in a non-equilibrium thermodynamic system. In the field of gas insulation and plasma discharge, the Boltzmann equation can be used to accurately describe the electron transport in gas discharge plasmas, and its solution is the basis of many plasma simulation models. However, the traditional numerical methods for solving the Boltzmann equation all require meshing on the computational domain, and the solution accuracy significantly depends on the quality of the meshing. The physics-informed neural networks (PINNs), as a new method for solving Boltzmann equation, overcomes the shortcomings of traditional numerical methods in mesh generation and equation discretization, but its training is inefficient when dealing with multi-tasks because of huge parameter space of PINNs. To address this issue, this paper proposes a Meta-PINN network with two loops of PINNs based on meta learning. Through the training in inner and outer loops, Meta-PINN solve the Boltzmann equation in multi-tasks accurately and efficiently.
In the Meta-PINN, there are two types of networks, which are the PINN network and the meta network. In the inner loop, the PINN network solve the Boltzmann equation in multi-tasks by minimizing the loss function of PINN. After all multi-tasks are optimized, the sum of the PINN loss function, namely the meta loss function, is obtained. Then, the meta network updates the weights by minimizing the meta loss function in the outer loop.Finally, the updated weights are used to improve the training efficiency when dealing with new tasks of solving Boltzmann equations.
To validate the performance of Meta-PINN, the Boltzmann equations under different reduced electric fields and gas mixing ratios are solved under the framework of Meta-PINN. The results show that, when dealing with new tasks, the loss function values and L2 errors of Meta-PINN are reduced faster than that of PINN. Specifically, the minimum and maximum acceleration speeds increase by 75% and 22 times respectively, which indicates that the Meta-PINN outperforms the PINN in solution efficiency. Additionally, the effects of network capacities and inner steps on the computational efficiency are investigated in multi-tasks of solving Boltzmann equation. The result reveals that the computational efficiency does not improve with the increase of network capacities and is less affected by the inner steps.
The following conclusions could be drawn from the numerical experiments: (1) Meta-PINN can improve the efficiency of solving Boltzmann equation describing electron transport in gas discharge plasmas. Moreover, in some cases, the loss function value and L2 error of Meta-PINN can be reduced to a lower order of magnitude than PINN, indicating that solution accuracy of Meta-PINN is also better than that of PINN. (2) In solving the Boltzmann equation, the computation efficiency is not improved with the increase of network capacity. The most suitable network for solving the Boltzmann equation under different reduced electric fields in argon plasma is the neural network with 3 hidden layers and 300 neurons per layer. (3) The computation efficiency is either not improved with the increase of inner steps. The most suitable inner steps for solving the Boltzmann equation under different reduced electric fields in argon plasma is 5 steps. Generally, the Meta-PINN can be extended easily to other numerical solution of plasma governing equations.