Monte Carlo Physics-informed Neural Networks for Inverse Multiscale Heat Conduction Problems via the Phonon Boltzmann Transport Equation

Abstract

Inferring thermal fields and thermophysical properties from limited measurements is a fundamental challenge in micro- and nanoscale heat conduction, where the classical Fourier law breaks down and the phonon Boltzmann transport equation (BTE) is needed to capture non-diffusive transport effects. In this work, we extend Monte Carlo physics-informed neural networks (MC-PINNs), originally developed for forward phonon BTE problems [J. Comput. Phys. 542, 114364, 2025], to inverse multiscale heat conduction problems. Two representative classes of inverse problems are considered: (i) reconstructing the full thermal field from sparse interior temperature measurements when boundary conditions are unknown, and (ii) simultaneously inferring the unknown relaxation time together with the thermal field. Problem-specific MC-PINN architectures and training strategies are designed for each class. The mesh-free Monte Carlo sampling strategy enables a unified treatment across diffusive, transitional, and ballistic transport regimes without requiring a priori knowledge of the relaxation time. The proposed method is evaluated on quasi-one-dimensional, quasi-two-dimensional, and three-dimensional benchmark problems covering a wide range of Knudsen numbers, as well as on a realistic 3D fin field-effect transistor (FinFET) structure. Results demonstrate that MC-PINNs consistently outperform purely data-driven deep neural networks, particularly in the sparse-data regime, and can accurately infer spatially uniform relaxation times. For spatially varying relaxation times, the inferred distributions capture the dominant thermal response, and numerical simulations using the recovered parameters reproduce the macroscopic fields with good accuracy. These findings establish MC-PINNs as an effective and physically consistent framework for inverse thermal analysis at micro- and nanoscales.