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.

Figures (5)

• Figure 1: Diagram of the MTNet architecture and schematics of the distributed training routine. The collocation points are passed to the three independent neural networks that comprise MTNet, namely the Intensity-Network, the Temperature-Network, and the Auxiliary Network. MTNet detects the available computing hardware and replicates its architecture across all available GPU devices using TensorFlow's MirroredStrategy. Then, by means of automatic differentiation (AD), the derivatives of $\hat{I}(\omega_p,x,\mu)$, $\hat{f}(\omega_p,\tilde{\mu},x)$, $\hat{v}(\tilde{\omega},x, \mu)$, $\hat{g}(\omega_p,\tilde{\mu},x)$, and $\hat{T}(x)$ along with the PDE and boundary conditions. During each training step, the global batch of collocation points is partitioned into sub-batches, and each GPU independently evaluates the network and computes the loss gradients for its designated sub-batch. The gradients are then appropriately combined and the loss functions are evaluated globally. This process is repeated until convergence.
• Figure 2: (a-b) Solution of the EPRT in the relaxation-time approximation for the diffusive and mesoscopic regimes, respectively, in the silicon thin film geometry schematically shown over the plots. The black arrows are meant to schematically represent a few phonon scattering lengths $\Lambda$, which are much smaller than $L$ for diffusive transport, and of the order of $L$ for mesoscopic transport. The blue and orange circles correspond to the respective boxes in panels (c), (f), where the phonon intensity has been plotted at a single spatial point as a function of frequency and angle. It can be noticed that, in the intensity plots for the mesoscopic regime shown in panels (d) and (f), the intensity displays a sharp asymmetry about $\mu=0$. This asymmetry is almost absent in the intensity plots in panels (c) and (e), which is expected in the diffusive regime.
• Figure 3: (a) Agreement between the temperature profiles of the GEPRT and RTA-based EPRT at small temperature differences. (b) GEPRT prediction of the temperature profile in the weakly scattering regime for silicon. The blue and orange dots correspond to the colored boxes in panels (c) and (d), where a representative phonon intensity for the TA polarization has been plotted as a function of frequency and angle. The two panels show the precision of the multiscale architecture in capturing the discontinuity at $\mu=0$.
• Figure 4: (a) Large $\Delta T$ temperature solutions for the GEPRT in the diffusive regime. As expected due to explicit temperature dependence in the scattering rates, the large thermal gradient introduces a noticeable asymmetry in the temperature solution. (b) Temperature solution at large $\Delta T$ in the mesoscopic regime. The solution exhibits the characteristic boundary temperature slips and S-shaped" curvature indicative of the mesoscopic transport regime.
• Figure 5: (a) Diagram of the inverse problem, consisting of a fabricated quasi-2D silicon thin film. To infer the effective thermal transport length $L$, the inverse solver utilizes incident blackbody intensities as the driving potentials, constrained by the instantaneous lattice temperatures at the boundaries. This setup directly simulates a non-destructive metrology environment where a laboratory instrument can only probe the surface temperature of the silicon lattice. (b) Asymptotic convergence of MTNet to the effective thermal transport length $L$ that corresponds to the measured temperature slip.