Monte Carlo physics-informed neural networks for multiscale heat conduction via phonon Boltzmann transport equation

The phonon Boltzmann transport equation (BTE) is widely used for the description of 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 the Monte Carlo physics-informed neural networks (MC-PINNs), which provide an effective way to combat the “curse of dimensionality” in solving the phonon Boltzmann transport equation for modeling multiscale heat conduction in solid materials. In MC-PINNs, we utilize a deep neural network to approximate the solution to the BTE and encode the BTE as well as the corresponding boundary/initial conditions using automatic differentiation. In addition, we propose a novel two-step sampling approach to address the issues of inefficiency and inaccuracy 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 at each step are constructed based on the data drawn from the above two steps using the tensor product. The two-step sampling strategy enables the MC-PINNs (1) to model the heat conduction from ballistic to diffusive regimes, and (2) to be more memory efficient compared to the 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 quasi-two-dimensional (quasi-2D) and three-dimensional (3D) domains, are conducted to justify the effectiveness of the MC-PINNs for heat conduction spanning diffusive and ballistic regimes. Finally, we perform a comparison on the computational time and the memory usage between the MC-PINNs and one of the state-of-the-art numerical methods to demonstrate the potential of MC-PINNs for large-scale problems in real-world applications.