Reconstructing the thermal phonon transmission coefficient at solid interfaces in the phonon transport equation

TL;DR

This paper develops an optimization-based method using stochastic gradient descent to reconstruct the frequency-dependent phonon transmission coefficient at solid interfaces, supported by mathematical proofs of its properties.

Contribution

It formulates the inverse problem of determining phonon transmission coefficients as an optimization problem and justifies the use of SGD through mathematical analysis.

Findings

Successfully applied SGD to reconstruct transmission coefficients.

Proved maximum principle and Lipschitz continuity of the Fréchet derivative.

Validated the mathematical framework for inverse phonon transport problems.

Abstract

The ab initio model for heat propagation is the phonon transport equation, a Boltzmann-like kinetic equation. When two materials are put side by side, the heat that propagates from one material to the other experiences thermal boundary resistance. Mathematically, it is represented by the reflection coefficient of the phonon transport equation on the interface of the two materials. This coefficient takes different values at different phonon frequencies, between different materials. In experiments scientists measure the surface temperature of one material to infer the reflection coefficient as a function of phonon frequency. In this article, we formulate this inverse problem in an optimization framework and apply the stochastic gradient descent (SGD) method for finding the optimal solution. We furthermore prove the maximum principle and show the Lipschitz continuity of the Fr\'echet derivative. These properties allow us to justify the application of SGD in this setup.