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We present the Materials Learning Algorithms (MALA) package, a scalable machine learning framework designed to accelerate density functional theory (DFT) calculations suitable for large-scale atomistic simulations. Using local descriptors of the atomic environment, MALA models efficiently predict key electronic observables, including local density of states, electronic density, density of states, and total energy. The package integrates data sampling, model training and scalable inference into a unified library, while ensuring compatibility with standard DFT and molecular dynamics codes. We demonstrate MALA's capabilities with examples including boron clusters, aluminum across its solid-liquid phase boundary, and predicting the electronic structure of a stacking fault in a large beryllium slab. Scaling analyses reveal MALA's computational efficiency and identify bottlenecks for future optimization. With its ability to model electronic structures at scales far beyond standard DFT, MALA is well suited for modeling complex material systems, making it a versatile tool for advanced materials research.

The evolution of molecular dynamics (MD) simulations has been intimately linked to that of computing hardware. For decades following the creation of MD, simulations have improved with computing power along the three principal dimensions of accuracy, atom count (spatial scale), and duration (temporal scale). Since the mid-2000s, computer platforms have however failed to provide strong scaling for MD as scale-out CPU and GPU platforms that provide substantial increases to spatial scale do not lead to proportional increases in temporal scale. Important scientific problems therefore remained inaccessible to direct simulation, prompting the development of increasingly sophisticated algorithms that present significant complexity, accuracy, and efficiency challenges. While bespoke MD-only hardware solutions have provided a path to longer timescales for specific physical systems, their impact on the broader community has been mitigated by their limited adaptability to new methods and potentials. In this work, we show that a novel computing architecture, the Cerebras Wafer Scale Engine, completely alters the scaling path by delivering unprecedentedly high simulation rates up to 1.144M steps/second for 200,000 atoms whose interactions are described by an Embedded Atom Method potential. This enables direct simulations of the evolution of materials using general-purpose programmable hardware over millisecond timescales, dramatically increasing the space of direct MD simulations that can be carried out.

Molecular dynamics (MD) simulations have transformed our understanding of the nanoscale, driving breakthroughs in materials science, computational chemistry, and several other fields, including biophysics and drug design. Even on exascale supercomputers, however, runtimes are excessive for systems and timescales of scientific interest. Here, we demonstrate strong scaling of MD simulations on the Cerebras Wafer-Scale Engine. By dedicating a processor core for each simulated atom, we demonstrate a 179-fold improvement in timesteps per second versus the Frontier GPU-based Exascale platform, along with a large improvement in timesteps per unit energy. Reducing every year of runtime to two days unlocks currently inaccessible timescales of slow microstructure transformation processes that are critical for understanding material behavior and function. Our dataflow algorithm runs Embedded Atom Method (EAM) simulations at rates over 270,000 timesteps per second for problems with up to 800k atoms. This demonstrated performance is unprecedented for general-purpose processing cores., Comment: 10 pages, 10 figures, 5 tables

Diamond possesses exceptional physical properties due to its remarkably strong carbon-carbon bonding, leading to significant resilience to structural transformations at very high pressures and temperatures. Despite several experimental attempts, synthesis and recovery of the theoretically predicted post-diamond BC8 phase remains elusive. Through quantum accurate, multi-million atom molecular dynamics (MD) simulations, we have uncovered the extreme metastability of diamond at very high pressures, significantly exceeding its range of thermodynamic stability. We predict the post-diamond BC8 phase to be experimentally accessible only within a narrow high pressure-temperature region of the carbon phase diagram. The diamond to BC8 transformation proceeds through pre-melting followed by BC8 nucleation and growth in the metastable carbon liquid. We propose a double-shock compression pathway to achieve BC8 synthesis, which is currently being explored in theory-inspired experiments at the National Ignition Facility.

Machine learning (ML) enables the development of interatomic potentials that promise the accuracy of first principles methods while retaining the low cost and parallel efficiency of empirical potentials. While ML potentials traditionally use atom-centered descriptors as inputs, different models such as linear regression and neural networks can map these descriptors to atomic energies and forces. This begs the question: what is the improvement in accuracy due to model complexity irrespective of choice of descriptors? We curate three datasets to investigate this question in terms of ab initio energy and force errors: (1) solid and liquid silicon, (2) gallium nitride, and (3) the superionic conductor LGPS. We further investigate how these errors affect simulated properties with these models and verify if the improvement in fitting errors corresponds to measurable improvement in property prediction. Since linear and nonlinear regression models have different advantages and disadvantages, the results presented herein help researchers choose models for their particular application. By assessing different models, we observe correlations between fitting quantity (e.g. atomic force) error and simulated property error with respect to ab initio values. Such observations can be repeated by other researchers to determine the level of accuracy, and hence model complexity, needed for their particular systems of interest.

The exceptional properties observed in complex concentrated alloys (CCAs) arise from the interplay between crystalline order and chemical disorder at the atomic scale, complicating a unique determination of properties. In contrast to conventional alloys, CCA properties emerge as distributions due to varying local chemical environments and the specific scale of measurement. Currently there are few ways to quantitatively define, track, and compare local alloy compositions (versus a global label, i.e. equiatomic) contained in a CCA. Molecular dynamics is used here to build descriptive metrics that connect a global alloy composition to the diverse local alloy compositions that define it. A machine-learned interatomic potential for MoNbTaTi is developed and we use these metrics to investigate how property distributions change with excursions in global-local composition space. Short-range order is examined through the lens of local chemistry for the equiatomic composition, demonstrating stark changes in vacancy formation energy with local chemistry evolution., Comment: Version 3: updated funding statement, fixed typos, otherwise unchanged. 15 pages, 6 main figures, 1 supplemental figure

The properties of electrons in matter are of fundamental importance. They give rise to virtually all molecular and material properties and determine the physics at play in objects ranging from semiconductor devices to the interior of giant gas planets. Modeling and simulation of such diverse applications rely primarily on density functional theory (DFT), which has become the principal method for predicting the electronic structure of matter. While DFT calculations have proven to be very useful to the point of being recognized with a Nobel prize in 1998, their computational scaling limits them to small systems. We have developed a machine learning framework for predicting the electronic structure on any length scale. It shows up to three orders of magnitude speedup on systems where DFT is tractable and, more importantly, enables predictions on scales where DFT calculations are infeasible. Our work demonstrates how machine learning circumvents a long-standing computational bottleneck and advances science to frontiers intractable with any current solutions. This unprecedented modeling capability opens up an inexhaustible range of applications in astrophysics, novel materials discovery, and energy solutions for a sustainable future.

Atomic cluster expansion (ACE) methods provide a systematic way to describe particle local environments of arbitrary body order. For practical applications it is often required that the basis of cluster functions be symmetrized with respect to rotations and permutations. Existing methodologies yield sets of symmetrized functions that are over-complete. These methodologies thus require an additional numerical procedure, such as singular value decomposition (SVD), to eliminate redundant functions. In this work, it is shown that analytical linear relationships for subsets of cluster functions may be derived using recursion and permutation properties of generalized Wigner symbols. From these relationships, subsets (blocks) of cluster functions can be selected such that, within each block, functions are guaranteed to be linearly independent. It is conjectured that this block-wise independent set of permutation-adapted rotation and permutation invariant (PA-RPI) functions forms a complete, independent basis for ACE. Along with the first analytical proofs of block-wise linear dependence of ACE cluster functions and other theoretical arguments, numerical results are offered to demonstrate this. The utility of the method is demonstrated in the development of an ACE interatomic potential for tantalum. Using the new basis functions in combination with Bayesian compressive sensing sparse regression, some high degree descriptors are observed to persist and help achieve high-accuracy models.

A Spectral Neighbor Analysis (SNAP) machine learning interatomic potential (MLIP) has been developed for simulations of carbon at extreme pressures (up to 5 TPa) and temperatures (up to 20,000 K). This was achieved using a large database of experimentally relevant quantum molecular dynamics (QMD) data, training the SNAP potential using a robust machine learning methodology, and performing extensive validation against QMD and experimental data. The resultant carbon MLIP demonstrates unprecedented accuracy and transferability in predicting the carbon phase diagram, melting curves of crystalline phases, and the shock Hugoniot, all within 3% of QMD. By achieving quantum accuracy and efficient implementation on leadership class high performance computing systems, SNAP advances frontiers of classical MD simulations by enabling atomic-scale insights at experimental time and length scales.

Advances in machine learning (ML) techniques have enabled the development of interatomic potentials that promise both the accuracy of first principles methods and the low-cost, linear scaling, and parallel efficiency of empirical potentials. Despite rapid progress in the last few years, ML-based potentials often struggle to achieve transferability, that is, to provide consistent accuracy across configurations that significantly differ from those used to train the model. In order to truly realize the promise of ML-based interatomic potentials, it is therefore imperative to develop systematic and scalable approaches for the generation of diverse training sets that ensure broad coverage of the space of atomic environments. This work explores a diverse-by-construction approach that leverages the optimization of the entropy of atomic descriptors to create a very large ($>2\cdot10^{5}$ configurations, $>7\cdot10^{6}$ atomic environments) training set for tungsten in an automated manner, i.e., without any human intervention. This dataset is used to train polynomial as well as multiple neural network potentials with different architectures. For comparison, a corresponding family of potentials were also trained on an expert-curated dataset for tungsten. The models trained to entropy-optimized data exhibited vastly superior transferability compared to the expert-curated models. Furthermore, while the models trained with heavy user input (i.e., domain expertise) yield the lowest errors when tested on similar configurations, out-sample predictions are dramatically more robust when the models are trained on a deliberately diverse set of training data. Herein we demonstrate the development of both accurate and transferable ML potentials using automated and data-driven approaches for generating large and diverse training sets., Comment: Main: 13 pages, 7 Figures. Supplemental: 17 pages, 16 Figures

The atomic cluster expansion is a general polynomial expansion of the atomic energy in multi-atom basis functions. Here we implement the atomic cluster expansion in the performant C++ code \verb+PACE+ that is suitable for use in large scale atomistic simulations. We briefly review the atomic cluster expansion and give detailed expressions for energies and forces as well as efficient algorithms for their evaluation. We demonstrate that the atomic cluster expansion as implemented in \verb+PACE+ shifts a previously established Pareto front for machine learning interatomic potentials towards faster and more accurate calculations. Moreover, general purpose parameterizations are presented for copper and silicon and evaluated in detail. We show that the new Cu and Si potentials significantly improve on the best available potentials for highly accurate large-scale atomistic simulations.

A data-driven framework is presented for building magneto-elastic machine-learning interatomic potentials (ML-IAPs) for large-scale spin-lattice dynamics simulations. The magneto-elastic ML-IAPs are constructed by coupling a collective atomic spin model with an ML-IAP. Together they represent a potential energy surface from which the mechanical forces on the atoms and the precession dynamics of the atomic spins are computed. Both the atomic spin model and the ML-IAP are parametrized on data from first-principles calculations. We demonstrate the efficacy of our data-driven framework across magneto-structural phase transitions by generating a magneto-elastic ML-IAP for {\alpha}-iron. The combined potential energy surface yields excellent agreement with first-principles magneto-elastic calculations and quantitative predictions of diverse materials properties including bulk modulus, magnetization, and specific heat across the ferromagnetic-paramagnetic phase transition., Comment: 15 pages, 6 figures, 2 tables

The exascale race is at an end with the announcement of the Aurora and Frontier machines. This next generation of supercomputers utilize diverse hardware architectures to achieve their compute performance, providing an added onus on the performance portability of applications. An expanding fragmentation of programming models would provide a compounding optimization challenge were it not for the evolution of performance-portable frameworks, providing unified models for mapping abstract hierarchies of parallelism to diverse architectures. A solution to this challenge is the evolution of performance-portable frameworks, providing unified models for mapping abstract hierarchies of parallelism to diverse architectures. Kokkos is one such performance portable programming model for C++ applications, providing back-end implementations for each major HPC platform. Even with a performance portable framework, restructuring algorithms to expose higher degrees of parallelism is non-trivial. The Spectral Neighbor Analysis Potential (SNAP) is a machine-learned inter-atomic potential utilized in cutting-edge molecular dynamics simulations. Previous implementations of the SNAP calculation showed a downward trend in their performance relative to peak on newer-generation CPUs and low performance on GPUs. In this paper we describe the restructuring and optimization of SNAP as implemented in the Kokkos CUDA backend of the LAMMPS molecular dynamics package, benchmarked on NVIDIA GPUs. We identify novel patterns of hierarchical parallelism, facilitating a minimization of memory access overheads and pushing the implementation into a compute-saturated regime. Our implementation via Kokkos enables recompile-and-run efficiency on upcoming architectures. We find a $\sim$22x time-to-solution improvement relative to an existing implementation as measured on an NVIDIA Tesla V100-16GB for an important benchmark., Comment: Submitted to IPDPS 2021, October 19, 2020

We present a numerical modeling workflow based on machine learning (ML) which reproduces the the total energies produced by Kohn-Sham density functional theory (DFT) at finite electronic temperature to within chemical accuracy at negligible computational cost. Based on deep neural networks, our workflow yields the local density of states (LDOS) for a given atomic configuration. From the LDOS, spatially-resolved, energy-resolved, and integrated quantities can be calculated, including the DFT total free energy, which serves as the Born-Oppenheimer potential energy surface for the atoms. We demonstrate the efficacy of this approach for both solid and liquid metals and compare results between independent and unified machine-learning models for solid and liquid aluminum. Our machine-learning density functional theory framework opens up the path towards multiscale materials modeling for matter under ambient and extreme conditions at a computational scale and cost that is unattainable with current algorithms.

Abstract Machine learning models, trained on data from ab initio quantum simulations, are yielding molecular dynamics potentials with unprecedented accuracy. One limiting factor is the quantity of available training data, which can be expensive to obtain. A quantum simulation often provides all atomic forces, in addition to the total energy of the system. These forces provide much more information than the energy alone. It may appear that training a model to this large quantity of force data would introduce significant computational costs. Actually, training to all available force data should only be a few times more expensive than training to energies alone. Here, we present a new algorithm for efficient force training, and benchmark its accuracy by training to forces from real-world datasets for organic chemistry and bulk aluminum.

We present a scale-bridging approach based on a multi-fidelity (MF) machine-learning (ML) framework leveraging Gaussian processes (GP) to fuse atomistic computational model predictions across multiple levels of fidelity. Through the posterior variance of the MFGP, our framework naturally enables uncertainty quantification, providing estimates of confidence in the predictions. We used Density Functional Theory as high-fidelity prediction, while a ML interatomic potential is used as the low-fidelity prediction. Practical materials design efficiency is demonstrated by reproducing the ternary composition dependence of a quantity of interest (bulk modulus) across the full aluminum-niobium-titanium ternary random alloy composition space. The MFGP is then coupled to a Bayesian optimization procedure and the computational efficiency of this approach is demonstrated by performing an on-the-fly search for the global optimum of bulk modulus in the ternary composition space. The framework presented in this manuscript is the first application of MFGP to atomistic materials simulations fusing predictions between Density Functional Theory and classical interatomic potential calculations.

A natural extension of the descriptors used in the Spectral Neighbor Analysis Potential (SNAP) method is derived to treat atomic interactions in chemically complex systems. Atomic environment descriptors within SNAP are obtained from a basis function expansion of the weighted density of neighboring atoms. This new formulation instead partitions the neighbor density into partial densities for each chemical element, thus leading to explicit multi-element descriptors. For $N_{elem}$ chemical elements, the number of descriptors increases as $\mathcal{O}(N_{elem}^3)$, while the computational cost of the force calculation as implemented in LAMMPS is limited to $\mathcal{O}(N_{elem}^2)$ and the favorable linear scaling in the number of atoms is retained. We demonstrate these chemically aware descriptors by producing an interatomic potential for indium phosphide capable of capturing high-energy defects that result from radiation damage cascades. This new explicit multi-element SNAP method reproduces the relaxed defect formation energies with substantially greater accuracy than weighted-density SNAP, while retaining accurate representation of the bulk indium phosphide properties., Comment: 11 pages, 4 figures

Machine learning of the quantitative relationship between local environment descriptors and the potential energy surface of a system of atoms has emerged as a new frontier in the development of interatomic potentials (IAPs). Here, we present a comprehensive evaluation of ML-IAPs based on four local environment descriptors --- Behler-Parrinello symmetry functions, smooth overlap of atomic positions (SOAP), the Spectral Neighbor Analysis Potential (SNAP) bispectrum components, and moment tensors --- using a diverse data set generated using high-throughput density functional theory (DFT) calculations. The data set comprising bcc (Li, Mo) and fcc (Cu, Ni) metals and diamond group IV semiconductors (Si, Ge) is chosen to span a range of crystal structures and bonding. All descriptors studied show excellent performance in predicting energies and forces far surpassing that of classical IAPs, as well as predicting properties such as elastic constants and phonon dispersion curves. We observe a general trade-off between accuracy and the degrees of freedom of each model, and consequently computational cost. We will discuss these trade-offs in the context of model selection for molecular dynamics and other applications.

The central approximation made in classical molecular dynamics simulation of materials is the interatomic potential used to calculate the forces on the atoms. Great effort and ingenuity is required to construct viable functional forms and find accurate parameterizations for potentials using traditional approaches. Machine-learning has emerged as an effective alternative approach to develop accurate and robust interatomic potentials. Starting with a very general model form, the potential is learned directly from a database of electronic structure calculations and therefore can be viewed as a multiscale link between quantum and classical atomistic simulations. Risk of inaccurate extrapolation exists outside the narrow range of time- and length-scales where the two methods can be directly compared. In this work, we use the Spectral Neighbor Analysis Potential (SNAP) and show how a fit can be produced with minimal interpolation errors which is also robust in extrapolating beyond training. To demonstrate the method, we have developed a new tungsten-beryllium potential suitable for the full range of binary compositions. Subsequently, large-scale molecular dynamics simulations were performed of high energy Be atom implantation onto the (001) surface of solid tungsten. The new machine learned W-Be potential generates a population of implantation structures consistent with quantum calculations of defect formation energies. A very shallow (<2nm) average Be implantation depth is predicted which may explain ITER diverter degradation in the presence of beryllium., Comment: 5 Figures, 2 Tables

The Spectral Neighbor Analysis Potential (SNAP) is a classical interatomic potential that expresses the energy of each atom as a linear function of selected bispectrum components of the neighbor atoms. An extension of the SNAP form is proposed that includes quadratic terms in the bispectrum components. The extension is shown to provide a large increase in accuracy relative to the linear form, while incurring only a modest increase in computational cost. The mathematical structure of the quadratic SNAP form is similar to the embedded atom method (EAM), with the SNAP bispectrum components serving as counterparts to the two-body density functions in EAM. The effectiveness of the new form is demonstrated using an extensive set of training data for tantalum structures. Similarly to artificial neural network potentials, the quadratic SNAP form requires substantially more training data in order to prevent overfitting. The quality of this new potential form is measured through a robust cross-validation analysis.

Shock wave interactions with defects, such as pores, are known to play a key role in the chemical initiation of energetic materials. The shock response of hexanitrostilbene is studied through a combination of large scale reactive molecular dynamics and mesoscale hydrodynamic simulations. In order to extend our simulation capability at the mesoscale to include weak shock conditions (< 6 GPa), atomistic simulations of pore collapse are used to define a strain rate dependent strength model. Comparing these simulation methods allows us to impose physically-reasonable constraints on the mesoscale model parameters. In doing so, we have been able to study shock waves interacting with pores as a function of this viscoplastic material response. We find that the pore collapse behavior of weak shocks is characteristically different to that of strong shocks., Comment: 9 Pages, 7 Figures + 2 pages, 2 Figures Supplemental

The purpose of this short contribution is to report on the development of a Spectral Neighbor Analysis Potential (SNAP) for tungsten. We have focused on the characterization of elastic and defect properties of the pure material in order to support molecular dynamics simulations of plasma-facing materials in fusion reactors. A parallel genetic algorithm approach was used to efficiently search for fitting parameters optimized against a large number of objective functions. In addition, we have shown that this many-body tungsten potential can be used in conjunction with a simple helium pair potential to produce accurate defect formation energies for the W-He binary system., Comment: 7 Pages, 3 Figures, 3 Tables

We present a new interatomic potential for solids and liquids called Spectral Neighbor Analysis Potential (SNAP). The SNAP potential has a very general form and uses machine-learning techniques to reproduce the energies, forces, and stress tensors of a large set of small configurations of atoms, which are obtained using high-accuracy quantum electronic structure (QM) calculations. The local environment of each atom is characterized by a set of bispectrum components of the local neighbor density projected on to a basis of hyperspherical harmonics in four dimensions. The bispectrum components are the same bond-orientational order parameters employed by the GAP potential [arXiv:0910.1019]. The SNAP potential, unlike GAP, assumes a linear relationship between atom energy and bispectrum components. The linear SNAP coefficients are determined using weighted least-squares linear regression against the full QM training set. This allows the SNAP potential to be fit in a robust, automated manner to large QM data sets using many bispectrum coefficients.