A Unified Multiscale Auxiliary PINN Framework for Generalized Phonon Transport

Abstract: Nanoscale thermal transport is governed by the phonon Boltzmann transport equation (BTE). However, simulating the sub-continuum dynamics remains computationally prohibitive due to the high dimensionality of the phase space and the intrinsic nonlinearity of the scattering collision operator. Traditional numerical solvers and standard physics-informed neural networks (PINNs) inherently struggle with these integro-differential equations due to deterministic quadrature limitations, artificial thermalization introduced by the relaxation time approximation (RTA), and multiscale spectral bias. This work introduces a multiscale auxiliary physics-informed neural network (MTNet) to solve the generalized equation of phonon radiative transfer (GEPRT). By leveraging an auxiliary formulation, this mesh-free framework recasts the GEPRT into a fully differential system, enabling the analytical evaluation of scattering operators via automatic differentiation and facilitating scalable multi-GPU parallelization. To circumvent optimization stiffness, the architecture employs a decoupled, shallow neural network explicitly constrained by radiative equilibrium. MTNet is validated by simulating steady-state cross-plane transport in a silicon thin film, successfully capturing ballistic-diffusive regimes and characteristic boundary slips across extreme temperature gradients ($ΔT = 100$ K) beyond the standard linearization approach. Furthermore, we show that our framework successfully solves a geometric inverse problem in a slab geometry, retrieving the unknown slab thickness based only on interface temperature constraints in the mesoscopic regime. Ultimately, MTNet establishes a robust, fully differentiable foundation for predicting high-fidelity kinetic transport and extracting material properties in next-generation nanostructures.