1. Monte Carlo Physics-informed neural networks for multiscale heat conduction via phonon Boltzmann transport equation
- Author
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Lin, Qingyi, Zhang, Chuang, Meng, Xuhui, and Guo, Zhaoli
- Subjects
Physics - Computational Physics ,Mathematics - Analysis of PDEs - Abstract
The phonon Boltzmann transport equation (BTE) is widely used for describing multiscale heat conduction (from nm to $\mu$m or mm) in solid materials. Developing numerical approaches to solve this equation is challenging since it is a 7-dimensional integral-differential equation. In this work, we propose Monte Carlo physics-informed neural networks (MC-PINNs), which do not suffer from the "curse of dimensionality", to solve the phonon BTE to model the multiscale heat conduction in solid materials. MC-PINNs use a deep neural network to approximate the solution to the BTE, and encode the BTE as well as the corresponding boundary/initial conditions using the automatic differentiation. In addition, we propose a novel two-step sampling approach to address inefficiency and inaccuracy issues in the widely used sampling methods in PINNs. In particular, we first randomly sample a certain number of points in the temporal-spatial space (Step I), and then draw another number of points randomly in the solid angular space (Step II). The training points are constructed using the tensor product of the data drawn from the above two steps. The two-step sampling strategy enables MC-PINNs (1) to model the heat conduction from ballistic to diffusive regimes, and (2) is more memory-efficient compared to conventional numerical solvers or existing PINNs for BTE. A series of numerical examples including quasi-one-dimensional (quasi-1D) steady/unsteady heat conduction in a film, and the heat conduction in a quasi-two- and three-dimensional square domain, are conducted to justify the effectiveness of the MC-PINN for heat conduction spanning diffusive and ballistic regimes. Finally, we compare the computational time and memory usage of the MC-PINN and a state-of-art numerical methods to demonstrate the potential of the MC-PINN for large scale problems in real-world applications.
- Published
- 2024