Lattice thermal conductivity (κ) is a critical property influencing the performance of numerous technologies, including thermal insulation, semiconductor device thermal management, thermoelectrics, and thermal barrier coatings. Accurate determination of κ typically relies on either experimental measurements or computationally intensive first-principles calculations that incorporate three-phonon (3ph) and four-phonon (4ph) scattering processes. While the Peierls-Boltzmann transport equation (BTE) approach, coupled with advancements in 3ph and 4ph scattering theories and ab initio force constants, provides a path for accurate κ prediction, the computational cost remains substantial, particularly for complex materials and 4ph scattering. For instance, calculating κ for silicon with 3ph+4ph scattering using a 16x16x16 q-point mesh can require approximately 7000 CPU hours. This computational burden severely limits the application of these methods to a small fraction of existing materials, hindering material discovery and design. Machine learning (ML) has emerged as a potential solution, but existing end-to-end surrogate models relying solely on structural information as descriptors have fallen short of achieving the desired accuracy. This paper introduces a novel ML approach that bridges this gap, achieving accuracy comparable to experimental and first-principles methods while significantly reducing computational time.

Previous research has explored the prediction of lattice thermal conductivity using machine learning. However, these efforts have primarily focused on end-to-end models that predict κ directly from material structural information, without explicitly modeling phonon scattering processes. While these methods offer speed advantages, their accuracy is considerably lower than that of experiments or detailed first-principles calculations. The lack of a physics-based framework limits their predictive capabilities and prevents the extraction of valuable insights into underlying thermal transport mechanisms. This work addresses this limitation by focusing on machine learning models that directly predict phonon scattering rates, incorporating the underlying physics into the ML framework.

The methodology consists of several key steps. First, a computational time analysis of the phonon BTE workflow is performed, highlighting the computationally intensive nature of calculating 3ph and 4ph scattering rates (Γ³ᵖʰ and Γ⁴ᵖʰ). This analysis motivates the use of deep neural networks (DNNs) as surrogate models to accelerate this process. Three representative materials – Si, MgO, and LiCoO₂ – are selected, representing a range of thermal conductivities and crystal structures. For each material, two separate DNN models are trained: one for 3ph scattering and another for 4ph scattering. The input features (descriptors) for each model are the properties of the phonons involved in the scattering process (frequency, wave vector, eigenvector, and group velocity). A crucial aspect of the methodology is addressing the high skewness of the phonon scattering rates. A negative logarithm transform is applied to the target labels to mitigate this issue. Furthermore, a target-value-based loss function weight is introduced to ensure accurate prediction of high scattering rate processes, which are crucial contributors to thermal conductivity. The DNNs are trained using a small subset of scattering processes calculated using the analytical ShengBTE method with the FourPhonon module. The trained DNN models are then used to predict the scattering rates for the remaining processes in the phonon phase space, significantly reducing computational time. For 4ph scattering, a mode-by-mode prediction strategy is employed to manage memory constraints. Finally, transfer learning is employed to leverage the knowledge gained from training the 3ph model to improve the performance of the 4ph model. This is accomplished by adding a ‘virtual phonon’ to the 3ph model, making its input dimension compatible with the 4ph model. The weights and biases of the modified 3ph model are then used to initialize the 4ph model, accelerating training and enhancing prediction accuracy.

The study demonstrates that the developed machine learning models accurately predict phonon scattering rates and thermal conductivities for Si, MgO, and LiCoO₂. The 3ph scattering surrogate models exhibit high accuracy (R² > 0.89), with slightly lower accuracy observed for LiCoO₂ due to its more complex phonon scattering. The 4ph scattering surrogate models also show excellent accuracy (R² > 0.97). The models' prediction of relaxation times (τ) satisfy the physical scaling law at low frequencies, indicating that they capture the underlying physics. The cumulative κ values obtained from the surrogate models are in close agreement with the analytical results, with mean absolute percentage errors (MAPEs) of less than 3% for κ³ᵖʰ and less than 5% for κ³ᵖʰ⁺⁴ᵖʰ. Furthermore, transfer learning from the 3ph to the 4ph model significantly improves the accuracy of the 4ph predictions, demonstrating that the DNNs capture the fundamental physics of phonon scattering. The surrogate models deliver substantial computational speedup, achieving up to four times faster predictions for κ³ᵖʰ and up to seventy times faster predictions for κ³ᵖʰ⁺⁴ᵖʰ compared to traditional analytical methods. The speedup is further enhanced by using transfer learning, enabling significant reductions in computational time for materials like LiCoO₂.

The results show that the proposed ML approach offers a significant advancement in predicting phonon scattering rates and thermal conductivity. Unlike previous black-box end-to-end models, this method incorporates fundamental physics, leading to significantly higher accuracy and enabling the determination of crucial quantities like relaxation times, which are essential for various applications. The improved accuracy, up to an order of magnitude better than existing end-to-end models, makes this approach highly suitable for quantitative materials design. The computational speed-up demonstrates the potential for large-scale thermal transport informatics, enabling studies of far more materials than previously possible. The successful transfer learning further underscores the ability of the model to capture the inherent physics of phonon scattering, facilitating broader applicability across different materials and scattering orders.

This work successfully develops machine learning models for predicting phonon scattering rates at an unprecedented level of accuracy. The models show excellent agreement with analytical results, providing accurate predictions of lattice thermal conductivities with minimal errors. The substantial computational speedup achieved makes large-scale thermal transport informatics feasible. Future work could explore extending this framework to incorporate phonon renormalization, optimizing model performance through coding language changes, and developing descriptors suitable for materials with diverse crystal structures. The approach offers significant potential for accelerating materials discovery and design in thermal applications.

The current models are based on the relaxation time approximation (RTA), which is valid for materials where Umklapp scattering dominates. For materials where other scattering mechanisms are significant, the accuracy may be affected. The use of Python, while convenient for development, may lead to slightly lower computational efficiency compared to compiled languages like C or Fortran. The current descriptor approach might not be optimal for materials with drastically different crystal structures. Transfer learning between materials with vastly different structures requires further investigation and improved descriptor design.