Physics-informed neural networks with hard-encoded angle-dependent boundary conditions for phonon Boltzmann transport equation

Accurate solutions of the phonon Boltzmann transport equation (BTE), a high-dimensional partial differential equation (PDE), are critical for modeling thermal transport at the nanoscale. However, solving the phonon BTE remains computationally intensive, especially when accounting for complex, angle-dependent boundary conditions (BCs). Traditional numerical methods often struggle with high computational cost and limited flexibility in handling high dimensionality and complex BCs. Physics-informed neural networks (PINNs) have emerged as a promising framework for solving high-dimensional PDEs owing to the mesh-free formulation and efficient inference capabilities. Yet, PINNs typically suffer from slow convergence, largely due to the multi-objective non-convex optimization nature of model training, particularly when BCs are enforced via soft penalty terms. In this study, a novel PINN-based framework is proposed that, for the first time, hard-encodes thermalizing and scattering boundary conditions directly into the neural network architecture. This removes the need for auxiliary BC loss terms, substantially accelerating convergence and improving solution accuracy. Our method is especially advantageous for encoding angle-dependent BCs, which are essential for high-fidelity phonon transport modeling. Extensive numerical experiments demonstrate that the proposed PINN model outperforms traditional PINNs in both accuracy and convergence for forward and inverse BTE problems. This framework offers a robust and efficient approach for solving complex transport equations, opening avenues for applications across computational physics and thermal engineering.