Physics-informed neural networks for solving multiscale mode-resolved phonon Boltzmann transport equation

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Abstract

Boltzmann transport equation (BTE) is an ideal tool to describe the multiscale phonon transport phenomena, which are critical to applications like microelectronics cooling. Numerically solving phonon BTE is extremely computationally challenging due to the high dimensionality of such problems, especially when mode-resolved properties are considered. In this work, we demonstrate the use of physics-informed neural networks (PINNs) to efficiently solve phonon BTE for multiscale thermal transport problems with the consideration of phonon dispersion and polarization. In particular, a PINN framework is devised to predict the phonon energy distribution by minimizing the residuals of governing equations and boundary conditions, without the need for any labeled training data. Moreover, geometric parameters, such as the characteristic length scale, are included as a part of the input to PINN, which enables learning BTE solutions in a parametric setting. The effectiveness of the present scheme is demonstrated by solving a number of phonon transport problems in different spatial dimensions (from 1D to 3D). Compared to existing numerical BTE solvers, the proposed method exhibits superiority in efficiency and accuracy, showing great promises for practical applications, such as the thermal design of electronic devices.

Introduction

Effective thermal management has long been critical to the design and advancement of electronic devices. With the rapid development of micro/nano technology, the characteristic length of such devices becomes comparable or even smaller than the mean free path of phonons, which are the dominant thermal energy carriers in semiconductors. In these conditions, thermal transport is not purely diffusive, and thus it cannot be accurately described by the conventional Fourier's law [[1], [2], [3]]. To describe the multiscale thermal transport process from ballistic to diffusive regions, phonon Boltzmann transport equation (BTE) has been widely used when phase coherence effects are not important [4,5]. Since atomistic level simulation techniques, such as molecular dynamics and first-principles calculations, are computationally impractical for device-level thermal analysis, mesoscopic approaches based on the solution of phonon BTE provide a balance between computational complexity and accuracy.

However, solving the phonon BTE is a challenging task because of its high dimensionality. The BTE for phonons is a nonlinear integro-differential equation with eight independent variables: phonon frequency, phonon polarization, time, three spatial coordinates (x, y, z directions), and two directional angles (polar angle and azimuthal angle). To make the solution to the phonon BTE more computationally tractable, simplifications such as assuming all phonon modes with the same properties (i.e., gray model) are commonly used [[6], [7], [8], [9]]. However, such treatment can lead to significant inaccuracy in the solutions [10,11]. This is because different phonon modes can have a large span of mean free paths, and they behave differently at a given physical length scale. To address this issue, many numerical BTE solvers have been proposed, such as the Monte Carlo (MC) method [[12], [13], [14], [15], [16]], the lattice Boltzmann method [17,18], and deterministic discretization-based methods [[19], [20], [21], [22]]. While the MC method has achieved great success in accounting for dispersion, polarization and various scattering mechanisms, it suffers from stochastic statistical uncertainty and becomes prohibitively inefficient in or near the diffusive regime due to the restrictions on time step and grid size [13,14]. Improvements in MC methods have recently been made for solving 3D systems based on variation reduction techniques [15,23]. Different lattice Boltzmann methods have also been used in recent years, but there are still unphysical predictions reported for cases in the ballistic regime [24,25]. The discrete ordinate method (DOM), which solves the BTE directly using finite element or finite difference methods, discretizes the angular domain into small solid angles to capture the highly non-equilibrium phonon distribution. However, DOM and its variants usually show slow convergence near the diffusive limit and require large memory for solutions [26,27]. To solve the phonon BTE consistently over a wide length scale range, characterized by the Knudsen number (Kn, the ratio of the phonon mean free path to the characteristic length of the system), the hybrid ballistic-diffusive method [28,29] introduces a cutoff Knudsen number to separate frequency intervals for which the DOM or the Fourier's Law should be used. Although it can be applied to problems at different length scales, the proper selection of the cutoff Knudsen number remains a question and the improved accuracy comes at the cost of increased computational time. Recently, the discrete unified gas kinetics scheme (DUGKS) [30,31] and implicit kinetic scheme (IKS) [32] have been shown to work efficiently and accurately for low-dimensional thermal transport problems, but they have not been used for solving phonon BTE in 3D geometries.

It can be seen that all the aforementioned numerical schemes have some deficiencies, and few of them can be universally accurate for a wide range of length scales. Moreover, most methods have been only demonstrated for simple 1D or 2D systems, because they usually require massive computational resources for 3D geometries. For instance, while deterministic numerical methods such as IKS can solve 1D and 2D problems accurately within a few minutes [32], it can take several hours to simulate a 3D system for a single time step on a parallel machine with 400 processors [22]. Therefore, there is a pressing need to develop a numerical method that is accurate, efficient, easy to use and able to deal with high-dimensional cases.

Recently, machine learning (ML) methods have been leveraged to help predict thermal properties at the atomistic scale (1–10 nm), such as the development of ML potentials for molecular dynamics simulations [[33], [34], [35], [36]]. However, these efforts are not sufficient to model thermal transport phenomena in the meso- and micro-scales (e.g., 10 nm - 100 μm), which are more relevant to device applications. In the meantime, as deep neural networks (DNNs) possess the capability of accurately approximating any continuous function [[37], [38], [39]], they can be used to approximate solutions of partial differential equations (PDEs). Leveraging this property, physics-informed neural networks (PINNs) have emerged recently [[40], [41], [42], [43]], which incorporate the PDE residuals into the cost function and train the solutions using fully-connected DNNs. When the governing PDEs are known, the solutions can be learned in a physics-constrained manner without the need for any labeled training data, which is known as a data-free ML method. PINNs have been successfully employed to simulate forward and inverse problems for a variety of PDEs [[44], [45], [46], [47], [48], [49]]. Compared to the conventional mesh-based methods (e.g., finite element method), PINNs avoid discretizing the PDE by taking advantage of the automatic differentiation of DNNs [50]. It has also been shown that in many problems DNN as a universal function approximator can overcome the curse of dimensionality, since the number of parameters of a DNN grows at most polynomially in both the reciprocal of the prescribed accuracy ε and the dimension d [51,52]. In addition, PINNs have been shown capable of learning solutions of PDEs in parameterized spaces (e.g., variable initial/boundary conditions, geometry, equation parameters) [48,53]. This feature is especially valuable for optimization applications since a numerical procedure for solving PDE has to be performed each time any parameter is changed using conventional solvers – a problem also applicable to BTE solvers. In multiscale thermal transport problems, one such variable parameter could be the characteristic length. However, research efforts for solving PDEs with PINNs have been mainly in the fluid dynamics field aiming to solve Navier-Stokes equations, which usually have three input parameters. The mode-resolved phonon BTE, which can have eight variables in 3D problems, may benefit even more from the PINN scheme, but it has never been studied.

In this work, we demonstrate the use of PINNs to efficiently solve mode-resolved phonon BTE for multiscale thermal transport problems. In particular, a PINN framework is devised to predict the phonon energy distribution by minimizing the residuals of governing equations and boundary conditions, without the need for any labeled training data. Moreover, PINN can be trained to parameterize the solutions with varying geometric parameters, such as the characteristic length scale. This enables our method to handle structures over a wide range of length scales after a single training. The effectiveness of the present scheme is demonstrated by solving a number of phonon transport problems in different spatial dimensions (from 1D to 3D). Compared to conventional numerical BTE solvers, the proposed method exhibits superiority in efficiency and accuracy, showing great promises for practical applications, such as the thermal design of electronic devices.

Section snippets

PINNs for stationary phonon BTE

It can be shown that for a system without internal heat source at steady state, the multiscale thermal transport problem can be described by the BTE with the single mode relaxation time approximation (see the Methods section for derivation):Rex,s,k,p,μ=0:={v·eeeqeτ=v·e+eneqτ=0·q=·p0ωmax,p4πvedΩdω=0,x,s,k,pΓ,μd,where e represents phonon energy distribution function of variables (spatial coordinates x, directions s, wave number k, and phonon polarization p) in domain Γ and parameters μ

Impact of input size on the learning performance

During the training, although we discretize the angle and frequency domains for numerical integration, the collocation points are sampled in the spatial x and parameter μ spaces, without the need for mesh generation. We note that there have also been mesh-free (in spatial domain only) deterministic numerical methods for phonon BTE [66], but it was only validated for the gray model in 1D or 2D systems. As for the impact of input size on the prediction accuracy, the BTE residuals are usually

Energy-based phonon BTE

In general, the energy-based phonon BTE [4,21,30,32,74,75], under the single mode relaxation time approximation, can be written aset+v·e=eeqeτ,where e(x,s,k,p,t)=ωD(ω,p)[ffBE(Tref)] is the phonon energy deviational distribution function, eeq is the associated equilibrium distribution function, v is the group velocity, and τ is the effective relaxation time.

The phonon distribution function f=f(x,s,k,p,t) (or f(x,s,ω,p,t)) is a function dependent on the spatial vector x, directional unit

Credit author statement

Ruiyang Li: Conceptualization, Investigation, Coding, Writing – original draft. Eungkyu Lee: Conceptualization, Writing – reviewing & editing. Tengfei Luo: Conceptualization, Supervision, Writing – reviewing & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank ONR MURI (N00014-18-1-2429) for the financial support. The simulations are supported by the Notre Dame Center for Research Computing, and NSF through the eXtreme Science and Engineering Discovery Environment (XSEDE) computing resources provided by Texas Advanced Computing Center (TACC) Stampede II under grant number TG-CTS100078. This work is also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.

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