Accelerating Electronic Stopping Power Predictions by 10 Million Times with a Combination of Time-Dependent Density Functional Theory and Machine Learning

TL;DR

A method that combines time-dependent density functional theory (TDDFT) and machine learning to reduce the time to assess new materials to hours on a supercomputer and provide valuable data on how atomic details influence electronic stopping is established.

Abstract

Knowing the rate at which particle radiation releases energy in a material, the stopping power, is key to designing nuclear reactors, medical treatments, semiconductor and quantum materials, and many other technologies. While the nuclear contribution to stopping power, i.e., elastic scattering between atoms, is well understood in the literature, the route for gathering data on the electronic contribution has for decades remained costly and reliant on many simplifying assumptions, including that materials are isotropic. We establish a method that combines time-dependent density functional theory (TDDFT) and machine learning to reduce the time to assess new materials to mere hours on a supercomputer and provides valuable data on how atomic details influence electronic stopping. Our approach uses TDDFT to compute the electronic stopping contributions to stopping power from first principles in several directions and then machine learning to interpolate to other directions at a cost of 10 million times fewer core-hours. We demonstrate the combined approach in a study of proton irradiation in aluminum and employ it to predict how the depth of maximum energy deposition, the "Bragg Peak," varies depending on incident angle -- a quantity otherwise inaccessible to modelers. The lack of any experimental information requirement makes our method applicable to most materials, and its speed makes it a prime candidate for enabling quantum-to-continuum models of radiation damage. The prospect of reusing valuable TDDFT data for training the model make our approach appealing for applications in the age of materials data science.

Figures (7)

• Figure 1: We extend the capability of Time-Dependent Density Functional Theory (TD-DFT) by building a surrogate model that predicts the drag force on a projectile based on its atomic environment and local electron density. The machine learning surrogate operates 107$\times$ faster than TD-DFT which enables first-principles assessments of direction-dependent stopping forces and full computations of stopping distances.
• Figure 2: Force acting on a proton projectile passing through aluminum as calculated by RT-TDDFT (black, dotted line) and predicted by a ML model (red, solid line) trained on progressively larger training sets. The data from timesteps in the gray region were used to train the ML model in each frame. The mean absolute error (MAE) is measured across the entire trajectory. The inset in the bottom left panel shows the performance of a model selected based on MAE (blue, dashed line), Bayesian linear regression without feature selection, compared to our model which was chosen based on Spearman's correlation coefficient $\rho$, Bayesian linear regression with LASSO-based feature selection. Predictions from the model selected by using MAE are less smooth than either RT-TDDFT or the model based on $\rho$.
• Figure 3: The group of features that are most influential in predicting the stopping force as a function of projectile displacement in a random trajectory with $\left|v\right|$ = 1.0 a.u. Features are grouped into those that include Coulomb repulsion between the projectile and nearby nuclei, and those related to the charge density ($\rho$) ahead of or behind the projectile. The color of the background indicates which group of features is most important over a certain range of displacements. The distance to the closest atom is shown as a black line.
• Figure 4: Predicted stopping force acting on a proton moving along the $\left<\textrm{100}\right>$ channel of FCC Aluminum as computed with RT-TDDFT (black, solid), as predicted by an ML model trained by using RT-TDDFT data from a randomly-oriented trajectory (red, solid), and an ML model trained on both the channel and random data (blue, dashed). Vertical gray lines indicate the boundaries of the unit cells. We show the second and third unit cells of the six available in our training data.
• Figure 5: Average stopping powers at $v=$ 1.0 a.u. computed by RT-TDDFT for six different trajectories in FCC aluminium compared to an ML model (red squares) and the Firsov model (purple circles). Both ML and Firsov models are parameterized by using RT-TDDFT data from the trajectories marked with white fill. Trajectories used to validate model performance are shown with colored fill. Points which lie closer to the black, dashed line ($y=x$) are better predictions. The two random trajectories start at the same position as the $\left<100\right>$ trajectory but travel on different directions, and the diverse trajectory was selected to sample as many atomic environments as possible.