SpookyNet: Learning force fields with electronic degrees of freedom and nonlocal effects

Neural Network Potential with Multiresolution Approach Enables Accurate Prediction of Reaction Free Energies in Solution

We present the design and implementation of a novel neural network potential (NNP) and its combination with an electrostatic embedding scheme, commonly used within the context of hybrid quantum-mechanical/molecular-mechanical (QM/MM) simulations. Substitution of a computationally expensive QM Hamiltonian by an NNP with the same accuracy largely reduces the computational cost and enables efficient sampling in prospective MD simulations, the main limitation faced by traditional QM/MM setups. The model relies on the recently introduced anisotropic message passing (AMP) formalism to compute atomic interactions and encode symmetries found in QM systems. AMP is shown to be highly efficient in terms of both data and computational costs and can be readily scaled to sample systems involving more than 350 solute and 40,000 solvent atoms for hundreds of nanoseconds using umbrella sampling. Most deviations of AMP predictions from the underlying DFT ground truth lie within chemical accuracy (4.184 kJ mol-1). The performance and broad applicability of our approach are showcased by calculating the free-energy surface of alanine dipeptide, the preferred ligation states of nickel phosphine complexes, and dissociation free energies of charged pyridine and quinoline dimers. Results with this ML/MM approach show excellent agreement with experimental data and reach chemical accuracy in most cases. In contrast, free energies calculated with static DFT calculations paired with implicit solvent models or QM/MM MD simulations using cheaper semiempirical methods show up to ten times higher deviation from the experimental ground truth and sometimes even fail to reproduce qualitative trends.

Broadening the Scope of Neural Network Potentials through Direct Inclusion of Additional Molecular Attributes

Most state-of-the-art neural network potentials do not account for molecular attributes other than atomic numbers and positions, which limits its range of applicability by design. In this work, we demonstrate the importance of including additional electronic attributes in neural network potential representations with a minimal architectural change to TensorNet, a state-of-the-art equivariant model based on Cartesian rank-2 tensor representations. By performing experiments on both custom-made and public benchmarking data sets, we show that this modification resolves input degeneracy issues stemming from the use of atomic numbers and positions alone, while enhancing the model's predictive accuracy across diverse chemical systems with different charge or spin states. This is accomplished without tailored strategies or the inclusion of physics-based energy terms, while maintaining efficiency and accuracy. These findings should furthermore encourage researchers to train and use models incorporating these additional representations.

Crash testing machine learning force fields for molecules, materials, and interfaces: molecular dynamics in the TEA challenge 2023

We present the second part of the rigorous evaluation of modern machine learning force fields (MLFFs) within the TEA Challenge 2023. This study provides an in-depth analysis of the performance of MACE, SO3krates, sGDML, SOAP/GAP, and FCHL19* in modeling molecules, molecule-surface interfaces, and periodic materials. We compare observables obtained from molecular dynamics (MD) simulations using different MLFFs under identical conditions. Where applicable, density-functional theory (DFT) or experiment serves as a reference to reliably assess the performance of the ML models. In the absence of DFT benchmarks, we conduct a comparative analysis based on results from various MLFF architectures. Our findings indicate that, at the current stage of MLFF development, the choice of ML model is in the hands of the practitioner. When a problem falls within the scope of a given MLFF architecture, the resulting simulations exhibit weak dependency on the specific architecture used. Instead, emphasis should be placed on developing complete, reliable, and representative training datasets. Nonetheless, long-range noncovalent interactions remain challenging for all MLFF models, necessitating special caution in simulations of physical systems where such interactions are prominent, such as molecule-surface interfaces. The findings presented here reflect the state of MLFF models as of October 2023.

Crash testing machine learning force fields for molecules, materials, and interfaces: model analysis in the TEA Challenge 2023

Atomistic simulations are routinely employed in academia and industry to study the behavior of molecules, materials, and their interfaces. Central to these simulations are force fields (FFs), whose development is challenged by intricate interatomic interactions at different spatio-temporal scales and the vast expanse of chemical space. Machine learning (ML) FFs, trained on quantum-mechanical energies and forces, have shown the capacity to achieve sub-kcal (mol-1 Å-1) accuracy while maintaining computational efficiency. The TEA Challenge 2023 rigorously evaluated commonly used MLFFs across diverse applications, highlighting their strengths and weaknesses. Participants trained their models using provided datasets, and the results were systematically analyzed to assess the ability of MLFFs to reproduce potential energy surfaces, handle incomplete reference data, manage multi-component systems, and model complex periodic structures. This publication describes the datasets, outlines the proposed challenges, and presents a detailed analysis of the accuracy, stability, and efficiency of the MACE, SO3krates, sGDML, SOAP/GAP, and FCHL19* architectures in molecular dynamics simulations. The models represent the MLFF developers who participated in the TEA Challenge 2023. All results presented correspond to the state of the ML architectures as of October 2023. A comprehensive analysis of the molecular dynamics results obtained with different MLFFs will be presented in the second part of this manuscript.

Application of modern artificial intelligence techniques in the development of organic molecular force fields

The molecular force field (FF) determines the accuracy of molecular dynamics (MD) and is one of the major bottlenecks that limits the application of MD in molecular design. Recently, artificial intelligence (AI) techniques, such as machine-learning potentials (MLPs), have been rapidly reshaping the landscape of MD. Meanwhile, organic molecular systems feature unique characteristics, and require more careful treatment in both model construction, optimization, and validation. While an accurate and generic organic molecular force field is still missing, significant progress has been made with the facilitation of AI, warranting a promising future. In this review, we provide an overview of the various types of AI techniques used in molecular FF development and discuss both the advantages and weaknesses of these methodologies. We show how AI methods provide unprecedented capabilities in many tasks such as potential fitting, atom typification, and automatic optimization. Meanwhile, it is also worth noting that more efforts are needed to improve the transferability of the model, develop a more comprehensive database, and establish more standardized validation procedures. With these discussions, we hope to inspire more efforts to solve the existing problems, eventually leading to the birth of next-generation generic organic FFs.

How graph neural network interatomic potentials extrapolate: Role of the message-passing algorithm

Graph neural network interatomic potentials (GNN-IPs) are gaining significant attention due to their capability of learning from large datasets. Specifically, universal interatomic potentials based on GNN, usually trained with crystalline geometries, often exhibit remarkable extrapolative behavior toward untrained domains, such as surfaces and amorphous configurations. However, the origin of this extrapolation capability is not well understood. This work provides a theoretical explanation of how GNN-IPs extrapolate to untrained geometries. First, we demonstrate that GNN-IPs can capture non-local electrostatic interactions through the message-passing algorithm, as evidenced by tests on toy models and density-functional theory data. We find that GNN-IP models, SevenNet and MACE, accurately predict electrostatic forces in untrained domains, indicating that they have learned the exact functional form of the Coulomb interaction. Based on these results, we suggest that the ability to learn non-local electrostatic interactions, coupled with the embedding nature of GNN-IPs, explains their extrapolation ability. We find that the universal GNN-IP, SevenNet-0, effectively infers non-local Coulomb interactions in untrained domains but fails to extrapolate the non-local forces arising from the kinetic term, which supports the suggested theory. Finally, we address the impact of hyperparameters on the extrapolation performance of universal potentials, such as SevenNet-0 and MACE-MP-0, and discuss the limitations of the extrapolation capabilities.

Complete and Efficient Covariants for Three-Dimensional Point Configurations with Application to Learning Molecular Quantum Properties

When physical properties of molecules are being modeled with machine learning, it is desirable to incorporate SO(3)-covariance. While such models based on low body order features are not complete, we formulate and prove general completeness properties for higher order methods and show that 6k - 5 of these features are enough for up to k atoms. We also find that the Clebsch-Gordan operations commonly used in these methods can be replaced by matrix multiplications without sacrificing completeness, lowering the scaling from O(l6) to O(l3) in the degree of the features. We apply this to quantum chemistry, but the proposed methods are generally applicable for problems involving three-dimensional point configurations.

Simple and Efficient Equivariant Message-Passing Neural Network Model for Non-local Potential Energy Surfaces

Machine learning potential has become increasingly successful in atomistic simulations. Many of these potentials are based on an atomistic representation in a local environment, but an efficient description of nonlocal interactions that exceed a common local environment remains a challenge. Herein, we propose a simple and efficient equivariant model, EquiREANN, to effectively represent a nonlocal potential energy surface. It relies on a physically inspired message-passing framework, where the fundamental descriptors are linear combinations of atomic orbitals, while both invariant orbital coefficients and the equivariant orbital functions are iteratively updated. We demonstrate that this EquiREANN model is able to describe the subtle potential energy variation due to the nonlocal structural change with high accuracy and little extra computational cost than an invariant message passing model. Our work offers a generalized approach to create equivariant message-passing adaptations of other advanced local many-body descriptors.

Data Generation for Machine Learning Interatomic Potentials and Beyond

The field of data-driven chemistry is undergoing an evolution, driven by innovations in machine learning models for predicting molecular properties and behavior. Recent strides in ML-based interatomic potentials have paved the way for accurate modeling of diverse chemical and structural properties at the atomic level. The key determinant defining MLIP reliability remains the quality of the training data. A paramount challenge lies in constructing training sets that capture specific domains in the vast chemical and structural space. This Review navigates the intricate landscape of essential components and integrity of training data that ensure the extensibility and transferability of the resulting models. We delve into the details of active learning, discussing its various facets and implementations. We outline different types of uncertainty quantification applied to atomistic data acquisition and the correlations between estimated uncertainty and true error. The role of atomistic data samplers in generating diverse and informative structures is highlighted. Furthermore, we discuss data acquisition via modified and surrogate potential energy surfaces as an innovative approach to diversify training data. The Review also provides a list of publicly available data sets that cover essential domains of chemical space.

Predicting solvation free energies with an implicit solvent machine learning potential

Machine learning (ML) potentials are a powerful tool in molecular modeling, enabling ab initio accuracy for comparably small computational costs. Nevertheless, all-atom simulations employing best-performing graph neural network architectures are still too expensive for applications requiring extensive sampling, such as free energy computations. Implicit solvent models could provide the necessary speed-up due to reduced degrees of freedom and faster dynamics. Here, we introduce a Solvation Free Energy Path Reweighting (ReSolv) framework to parameterize an implicit solvent ML potential for small organic molecules that accurately predicts the hydration free energy, an essential parameter in drug design and pollutant modeling. Learning on a combination of experimental hydration free energy data and ab initio data of molecules in vacuum, ReSolv bypasses the need for intractable ab initio data of molecules in an explicit bulk solvent and does not have to resort to less accurate data-generating models. On the FreeSolv dataset, ReSolv achieves a mean absolute error close to average experimental uncertainty, significantly outperforming standard explicit solvent force fields. Compared to the explicit solvent ML potential, ReSolv offers a computational speedup of four orders of magnitude and attains closer agreement with experiments. The presented framework paves the way for deep molecular models that are more accurate yet computationally more cost-effective than classical atomistic models.

ConfRank: Improving GFN-FF Conformer Ranking with Pairwise Training

Conformer ranking is a crucial task for drug discovery, with methods for generating conformers often based on molecular (meta)dynamics or sophisticated sampling techniques. These methods are constrained by the underlying force computation regarding runtime and energy ranking accuracy, limiting their effectiveness for large-scale screening applications. To address these ranking limitations, we introduce ConfRank, a machine learning-based approach that enhances conformer ranking using pairwise training. We demonstrate its performance using GFN-FF-generated conformer ensembles, leveraging the DimeNet++ architecture trained on pairs of 159 760 uncharged organic compounds from the GEOM data set with r2SCAN-3c reference level. Instead of predicting only on single molecules, this approach captures relative energy differences between conformers, leading to a significant improvement of the overall conformational ranking, outperforming GFN-FF and GFN2-xTB. Thereby, the pairwise RMSD of the relative energy difference of two conformers can be reduced from 5.65 to 0.71 kcal mol-1 on the test data set, allowing to correctly identify up to 81% of all lowest lying conformers correctly (GFN-FF: 10%, GFN2-xTB: 47%). The ConfRank approach is cost-effective, allowing for scalable deployment on both CPU and GPU, achieving runtime accelerations by up to 2 orders of magnitude compared to GFN2-xTB. Out-of-sample investigations on CREST-generated conformer ensembles from the QM9 data set and conformers taken from an extended GMTKN55 data set show promising results for the robustness of this approach. Thereby, ranking correlation coefficient such as Spearman can be improved to 0.90 (GFN-FF: 0.39, GFN2-xTB: 0.84) reducing the probability of an incorrect sign flip in pairwise energy comparison from 32 to 7%. On the extended GMTKN55 subsets the pairwise MAD (RMSD) could be reduced on almost all subsets by up to 62% (58%) with an average improvement of 30% (29%). Moreover, an exemplary case study on vancomycin shows similar performance, indicating applicability to larger (bio)molecular structures. Furthermore, we motivate the usage of the pairwise training approach from a theoretical perspective, highlighting that while pairwise training can lead to a decline in single sample prediction of absolute energies for ML models, it significantly enhances conformer ranking performance. The data and models used in this study are available at https://github.com/grimme-lab/confrank.

Machine Learned Potential Enables Molecular Dynamics Simulation to Predict the Experimental Branching Ratios in the NO Release Channel of Nitroaromatic Compounds

This study employs a machine learning (ML) model using the Gaussian process regression algorithm to generate potential energy surfaces (PES) from density functional theory calculations, facilitating the investigation of photodissociation dynamics of nitroaromatic compounds, resulting in NO release. The experimentally observed trends in the slow-to-fast branching ratios of the NO moiety were captured by estimating the branching ratio between the two distinct reaction pathways, viz., roaming and oxaziridine mechanisms, calculated from molecular dynamics simulations performed on a reduced two-dimensional T1 surface. The qualitative agreement between the calculated and experimental results suggests that the mechanism dictating NO release is primarily governed by the dynamics on the T1 surface.

Leveraging Machine Learning Potentials for In-Situ Searching of Active sites in Heterogeneous Catalysis

This Perspective explores the integration of machine learning potentials (MLPs) in the research of heterogeneous catalysis, focusing on their role in identifying in situ active sites and enhancing the understanding of catalytic processes. MLPs utilize extensive databases from high-throughput density functional theory (DFT) calculations to train models that predict atomic configurations, energies, and forces with near-DFT accuracy. These capabilities allow MLPs to handle significantly larger systems and extend simulation times beyond the limitations of traditional ab initio methods. Coupled with global optimization algorithms, MLPs enable systematic investigations across vast structural spaces, making substantial contributions to the modeling of catalyst surface structures under reactive conditions. The review aims to provide a broad introduction to recent advancements and practical guidance on employing MLPs and also showcases several exemplary cases of MLP-driven discoveries related to surface structure changes under reactive conditions and the nature of active sites in heterogeneous catalysis. The prevailing challenges faced by this approach are also discussed.

Constructing Accurate and Efficient General-Purpose Atomistic Machine Learning Model with Transferable Accuracy for Quantum Chemistry

Density functional theory (DFT) has been a cornerstone in computational science, providing powerful insights into structure-property relationships for molecules and materials through first-principles quantum-mechanical (QM) calculations. However, the advent of atomistic machine learning (ML) is reshaping the landscape by enabling large-scale dynamics simulations and high-throughput screening at DFT-equivalent accuracy with drastically reduced computational cost. Yet, the development of general-purpose atomistic ML models as surrogates for QM calculations faces several challenges, particularly in terms of model capacity, data efficiency, and transferability across chemically diverse systems. This work introduces a novel extension of the polarizable atom interaction neural network (namely, XPaiNN) to address these challenges. Two distinct training strategies have been employed, one direct-learning and the other Δ-ML on top of a semiempirical QM method. These methodologies have been implemented within the same framework, allowing for a detailed comparison of their results. The XPaiNN models, in particular the one using Δ-ML, not only demonstrate competitive performance on standard benchmarks, but also demonstrate the effectiveness against other ML models and QM methods on comprehensive downstream tasks, including noncovalent interactions, reaction energetics, barrier heights, geometry optimization and reaction thermodynamics, etc. This work represents a significant step forward in the pursuit of accurate and efficient atomistic ML models of general-purpose, capable of handling complex chemical systems with transferable accuracy.

A deep equivariant neural network approach for efficient hybrid density functional calculations

Hybrid density functional calculations are essential for accurate description of electronic structure, yet their widespread use is restricted by the substantial computational cost. Here we develop DeepH-hybrid, a deep equivariant neural network method for learning the hybrid-functional Hamiltonian as a function of material structure, which circumvents the time-consuming self-consistent field iterations and enables the study of large-scale materials with hybrid-functional accuracy. Our extensive experiments demonstrate good reliability as well as effective transferability and efficiency of the method. As a notable application, DeepH-hybrid is applied to study large-supercell Moiré-twisted materials, offering the first case study on how the inclusion of exact exchange affects flat bands in magic-angle twisted bilayer graphene. The work generalizes deep-learning electronic structure methods to beyond conventional density functional theory, facilitating the development of deep-learning-based ab initio methods.

Nutmeg and SPICE: Models and Data for Biomolecular Machine Learning

We describe version 2 of the SPICE data set, a collection of quantum chemistry calculations for training machine learning potentials. It expands on the original data set by adding much more sampling of chemical space and more data on noncovalent interactions. We train a set of potential energy functions called Nutmeg on it. They are based on the TensorNet architecture. They use a novel mechanism to improve performance on charged and polar molecules, injecting precomputed partial charges into the model to provide a reference for the large-scale charge distribution. Evaluation of the new models shows that they do an excellent job of reproducing energy differences between conformations even on highly charged molecules or ones that are significantly larger than the molecules in the training set. They also produce stable molecular dynamics trajectories and are fast enough to be useful for routine simulation of small molecules.

Accurate nuclear quantum statistics on machine-learned classical effective potentials

The contribution of nuclear quantum effects (NQEs) to the properties of various hydrogen-bound systems, including biomolecules, is increasingly recognized. Despite the development of many acceleration techniques, the computational overhead of incorporating NQEs in complex systems is sizable, particularly at low temperatures. In this work, we leverage deep learning and multiscale coarse-graining techniques to mitigate the computational burden of path integral molecular dynamics (PIMD). In particular, we employ a machine-learned potential to accurately represent corrections to classical potentials, thereby significantly reducing the computational cost of simulating NQEs. We validate our approach using four distinct systems: Morse potential, Zundel cation, single water molecule, and bulk water. Our framework allows us to accurately compute position-dependent static properties, as demonstrated by the excellent agreement obtained between the machine-learned potential and computationally intensive PIMD calculations, even in the presence of strong NQEs. This approach opens the way to the development of transferable machine-learned potentials capable of accurately reproducing NQEs in a wide range of molecular systems.

Accurate modeling of the potential energy surface of atmospheric molecular clusters boosted by neural networks

The computational cost of accurate quantum chemistry (QC) calculations of large molecular systems can often be unbearably high. Machine learning offers a lower computational cost compared to QC methods while maintaining their accuracy. In this study, we employ the polarizable atom interaction neural network (PaiNN) architecture to train and model the potential energy surface of molecular clusters relevant to atmospheric new particle formation, such as sulfuric acid-ammonia clusters. We compare the differences between PaiNN and previous kernel ridge regression modeling for the Clusteromics I-V data sets. We showcase three models capable of predicting electronic binding energies and interatomic forces with mean absolute errors of <0.3 kcal mol-1 and <0.2 kcal mol-1 Å-1, respectively. Furthermore, we demonstrate that the error of the modeled properties remains below the chemical accuracy of 1 kcal mol-1 even for clusters vastly larger than those in the training database (up to (H2SO4)15(NH3)15 clusters, containing 30 molecules). Consequently, we emphasize the potential applications of these models for faster and more thorough configurational sampling and for boosting molecular dynamics studies of large atmospheric molecular clusters.

Unsupervised Machine Learning in the Analysis of Nonadiabatic Molecular Dynamics Simulation

The all-atomic full-dimensional-level simulations of nonadiabatic molecular dynamics (NAMD) in large realistic systems has received high research interest in recent years. However, such NAMD simulations normally generate an enormous amount of time-dependent high-dimensional data, leading to a significant challenge in result analyses. Based on unsupervised machine learning (ML) methods, considerable efforts were devoted to developing novel and easy-to-use analysis tools for the identification of photoinduced reaction channels and the comprehensive understanding of complicated molecular motions in NAMD simulations. Here, we tried to survey recent advances in this field, particularly to focus on how to use unsupervised ML methods to analyze the trajectory-based NAMD simulation results. Our purpose is to offer a comprehensive discussion on several essential components of this analysis protocol, including the selection of ML methods, the construction of molecular descriptors, the establishment of analytical frameworks, their advantages and limitations, and persistent challenges.

Machine learning the electric field response of condensed phase systems using perturbed neural network potentials

The interaction of condensed phase systems with external electric fields is of major importance in a myriad of processes in nature and technology, ranging from the field-directed motion of cells (galvanotaxis), to geochemistry and the formation of ice phases on planets, to field-directed chemical catalysis and energy storage and conversion systems including supercapacitors, batteries and solar cells. Molecular simulation in the presence of electric fields would give important atomistic insight into these processes but applications of the most accurate methods such as ab-initio molecular dynamics (AIMD) are limited in scope by their computational expense. Here we introduce Perturbed Neural Network Potential Molecular Dynamics (PNNP MD) to push back the accessible time and length scales of such simulations. We demonstrate that important dielectric properties of liquid water including the field-induced relaxation dynamics, the dielectric constant and the field-dependent IR spectrum can be machine learned up to surprisingly high field strengths of about 0.2 V Å-1 without loss in accuracy when compared to ab-initio molecular dynamics. This is remarkable because, in contrast to most previous approaches, the two neural networks on which PNNP MD is based are exclusively trained on molecular configurations sampled from zero-field MD simulations, demonstrating that the networks not only interpolate but also reliably extrapolate the field response. PNNP MD is based on rigorous theory yet it is simple, general, modular, and systematically improvable allowing us to obtain atomistic insight into the interaction of a wide range of condensed phase systems with external electric fields.

Scalable crystal structure relaxation using an iteration-free deep generative model with uncertainty quantification

In computational molecular and materials science, determining equilibrium structures is the crucial first step for accurate subsequent property calculations. However, the recent discovery of millions of new crystals and super large twisted structures has challenged traditional computational methods, both ab initio and machine-learning-based, due to their computationally intensive iterative processes. To address these scalability issues, here we introduce DeepRelax, a deep generative model capable of performing geometric crystal structure relaxation rapidly and without iterations. DeepRelax learns the equilibrium structural distribution, enabling it to predict relaxed structures directly from their unrelaxed ones. The ability to perform structural relaxation at the millisecond level per structure, combined with the scalability of parallel processing, makes DeepRelax particularly useful for large-scale virtual screening. We demonstrate DeepRelax's reliability and robustness by applying it to five diverse databases, including oxides, Materials Project, two-dimensional materials, van der Waals crystals, and crystals with point defects. DeepRelax consistently shows high accuracy and efficiency, validated by density functional theory calculations. Finally, we enhance its trustworthiness by integrating uncertainty quantification. This work significantly accelerates computational workflows, offering a robust and trustworthy machine-learning method for material discovery and advancing the application of AI for science.

Improving Molecular-Dynamics Simulations for Solid-Liquid Interfaces with Machine-Learning Interatomic Potentials

Emerging developments in artificial intelligence have opened infinite possibilities for material simulation. Depending on the powerful fitting of machine learning algorithms to first-principles data, machine learning interatomic potentials (MLIPs) can effectively balance the accuracy and efficiency problems in molecular dynamics (MD) simulations, serving as powerful tools in various complex physicochemical systems. Consequently, this brings unprecedented enthusiasm for researchers to apply such novel technology in multiple fields to revisit the major scientific problems that have remained controversial owing to the limitations of previous computational methods. Herein, we introduce the evolution of MLIPs, provide valuable application examples for solid-liquid interfaces, and present current challenges. Driven by solving multitudinous difficulties in terms of the accuracy, efficiency, and versatility of MLIPs, this booming technique, combined with molecular simulation methods, will provide an underlying and valuable understanding of interdisciplinary scientific challenges, including materials, physics, and chemistry.

A physics-aware neural network for protein-ligand interactions with quantum chemical accuracy

Quantifying intermolecular interactions with quantum chemistry (QC) is useful for many chemical problems, including understanding the nature of protein-ligand interactions. Unfortunately, QC computations on protein-ligand systems are too computationally expensive for most use cases. The flourishing field of machine-learned (ML) potentials is a promising solution, but it is limited by an inability to easily capture long range, non-local interactions. In this work we develop an atomic-pairwise neural network (AP-Net) specialized for modeling intermolecular interactions. This model benefits from a number of physical constraints, including a two-component equivariant message passing neural network architecture that predicts interaction energies via an intermediate prediction of monomer electron densities. The AP-Net model is trained on a comprehensive dataset composed of paired ligand and protein fragments. This model accurately predicts QC-quality interaction energies of protein-ligand systems at a computational cost reduced by orders of magnitude. Applications of the AP-Net model to molecular crystal structure prediction are explored, as well as limitations in modeling highly polarizable systems.

3DReact: Geometric Deep Learning for Chemical Reactions

Geometric deep learning models, which incorporate the relevant molecular symmetries within the neural network architecture, have considerably improved the accuracy and data efficiency of predictions of molecular properties. Building on this success, we introduce 3DReact, a geometric deep learning model to predict reaction properties from three-dimensional structures of reactants and products. We demonstrate that the invariant version of the model is sufficient for existing reaction data sets. We illustrate its competitive performance on the prediction of activation barriers on the GDB7-22-TS, Cyclo-23-TS, and Proparg-21-TS data sets in different atom-mapping regimes. We show that, compared to existing models for reaction property prediction, 3DReact offers a flexible framework that exploits atom-mapping information, if available, as well as geometries of reactants and products (in an invariant or equivariant fashion). Accordingly, it performs systematically well across different data sets, atom-mapping regimes, as well as both interpolation and extrapolation tasks.

A Euclidean transformer for fast and stable machine learned force fields

Recent years have seen vast progress in the development of machine learned force fields (MLFFs) based on ab-initio reference calculations. Despite achieving low test errors, the reliability of MLFFs in molecular dynamics (MD) simulations is facing growing scrutiny due to concerns about instability over extended simulation timescales. Our findings suggest a potential connection between robustness to cumulative inaccuracies and the use of equivariant representations in MLFFs, but the computational cost associated with these representations can limit this advantage in practice. To address this, we propose a transformer architecture called SO3KRATES that combines sparse equivariant representations (Euclidean variables) with a self-attention mechanism that separates invariant and equivariant information, eliminating the need for expensive tensor products. SO3KRATES achieves a unique combination of accuracy, stability, and speed that enables insightful analysis of quantum properties of matter on extended time and system size scales. To showcase this capability, we generate stable MD trajectories for flexible peptides and supra-molecular structures with hundreds of atoms. Furthermore, we investigate the PES topology for medium-sized chainlike molecules (e.g., small peptides) by exploring thousands of minima. Remarkably, SO3KRATES demonstrates the ability to strike a balance between the conflicting demands of stability and the emergence of new minimum-energy conformations beyond the training data, which is crucial for realistic exploration tasks in the field of biochemistry.

FeNNol: An efficient and flexible library for building force-field-enhanced neural network potentials

Neural network interatomic potentials (NNPs) have recently proven to be powerful tools to accurately model complex molecular systems while bypassing the high numerical cost of ab initio molecular dynamics simulations. In recent years, numerous advances in model architectures as well as the development of hybrid models combining machine-learning (ML) with more traditional, physically motivated, force-field interactions have considerably increased the design space of ML potentials. In this paper, we present FeNNol, a new library for building, training, and running force-field-enhanced neural network potentials. It provides a flexible and modular system for building hybrid models, allowing us to easily combine state-of-the-art embeddings with ML-parameterized physical interaction terms without the need for explicit programming. Furthermore, FeNNol leverages the automatic differentiation and just-in-time compilation features of the Jax Python library to enable fast evaluation of NNPs, shrinking the performance gap between ML potentials and standard force-fields. This is demonstrated with the popular ANI-2x model reaching simulation speeds nearly on par with the AMOEBA polarizable force-field on commodity GPUs (graphics processing units). We hope that FeNNol will facilitate the development and application of new hybrid NNP architectures for a wide range of molecular simulation problems.

Pathways to a Shiny Future: Building the Foundation for Computational Physical Chemistry and Biophysics in 2050

In the last quarter-century, the field of molecular dynamics (MD) has undergone a remarkable transformation, propelled by substantial enhancements in software, hardware, and underlying methodologies. In this Perspective, we contemplate the future trajectory of MD simulations and their possible look at the year 2050. We spotlight the pivotal role of artificial intelligence (AI) in shaping the future of MD and the broader field of computational physical chemistry. We outline critical strategies and initiatives that are essential for the seamless integration of such technologies. Our discussion delves into topics like multiscale modeling, adept management of ever-increasing data deluge, the establishment of centralized simulation databases, and the autonomous refinement, cross-validation, and self-expansion of these repositories. The successful implementation of these advancements requires scientific transparency, a cautiously optimistic approach to interpreting AI-driven simulations and their analysis, and a mindset that prioritizes knowledge-motivated research alongside AI-enhanced big data exploration. While history reminds us that the trajectory of technological progress can be unpredictable, this Perspective offers guidance on preparedness and proactive measures, aiming to steer future advancements in the most beneficial and successful direction.

Accurate Quantum Monte Carlo Forces for Machine-Learned Force Fields: Ethanol as a Benchmark

Quantum Monte Carlo (QMC) is a powerful method to calculate accurate energies and forces for molecular systems. In this work, we demonstrate how we can obtain accurate QMC forces for the fluxional ethanol molecule at room temperature by using either multideterminant Jastrow-Slater wave functions in variational Monte Carlo or just a single determinant in diffusion Monte Carlo. The excellent performance of our protocols is assessed against high-level coupled cluster calculations on a diverse set of representative configurations of the system. Finally, we train machine-learning force fields on the QMC forces and compare them to models trained on coupled cluster reference data, showing that a force field based on the diffusion Monte Carlo forces with a single determinant can faithfully reproduce coupled cluster power spectra in molecular dynamics simulations.

Inverse mapping of quantum properties to structures for chemical space of small organic molecules

Computer-driven molecular design combines the principles of chemistry, physics, and artificial intelligence to identify chemical compounds with tailored properties. While quantum-mechanical (QM) methods, coupled with machine learning, already offer a direct mapping from 3D molecular structures to their properties, effective methodologies for the inverse mapping in chemical space remain elusive. We address this challenge by demonstrating the possibility of parametrizing a chemical space with a finite set of QM properties. Our proof-of-concept implementation achieves an approximate property-to-structure mapping, the QIM model (which stands for "Quantum Inverse Mapping"), by forcing a variational auto-encoder with a property encoder to obtain a common internal representation for both structures and properties. After validating this mapping for small drug-like molecules, we illustrate its capabilities with an explainability study as well as by the generation of de novo molecular structures with targeted properties and transition pathways between conformational isomers. Our findings thus provide a proof-of-principle demonstration aiming to enable the inverse property-to-structure design in diverse chemical spaces.

Geometric deep learning-guided Suzuki reaction conditions assessment for applications in medicinal chemistry

Suzuki cross-coupling reactions are considered a valuable tool for constructing carbon-carbon bonds in small molecule drug discovery. However, the synthesis of chemical matter often represents a time-consuming and labour-intensive bottleneck. We demonstrate how machine learning methods trained on high-throughput experimentation (HTE) data can be leveraged to enable fast reaction condition selection for novel coupling partners. We show that the trained models support chemists in determining suitable catalyst-solvent-base combinations for individual transformations including an evaluation of the need for HTE screening. We introduce an algorithm for designing 96-well plates optimized towards reaction yields and discuss the model performance of zero- and few-shot machine learning. The best-performing machine learning model achieved a three-category classification accuracy of 76.3% (±0.2%) and an F 1-score for a binary classification of 79.1% (±0.9%). Validation on eight reactions revealed a receiver operating characteristic (ROC) curve (AUC) value of 0.82 (±0.07) for few-shot machine learning. On the other hand, zero-shot machine learning models achieved a mean ROC-AUC value of 0.63 (±0.16). This study positively advocates the application of few-shot machine learning-guided reaction condition selection for HTE campaigns in medicinal chemistry and highlights practical applications as well as challenges associated with zero-shot machine learning.

Dataset for quantum-mechanical exploration of conformers and solvent effects in large drug-like molecules

We here introduce the Aquamarine (AQM) dataset, an extensive quantum-mechanical (QM) dataset that contains the structural and electronic information of 59,783 low-and high-energy conformers of 1,653 molecules with a total number of atoms ranging from 2 to 92 (mean: 50.9), and containing up to 54 (mean: 28.2) non-hydrogen atoms. To gain insights into the solvent effects as well as collective dispersion interactions for drug-like molecules, we have performed QM calculations supplemented with a treatment of many-body dispersion (MBD) interactions of structures and properties in the gas phase and implicit water. Thus, AQM contains over 40 global and local physicochemical properties (including ground-state and response properties) per conformer computed at the tightly converged PBE0+MBD level of theory for gas-phase molecules, whereas PBE0+MBD with the modified Poisson-Boltzmann (MPB) model of water was used for solvated molecules. By addressing both molecule-solvent and dispersion interactions, AQM dataset can serve as a challenging benchmark for state-of-the-art machine learning methods for property modeling and de novo generation of large (solvated) molecules with pharmaceutical and biological relevance.

In Silico Chemical Experiments in the Age of AI: From Quantum Chemistry to Machine Learning and Back

Computational chemistry is an indispensable tool for understanding molecules and predicting chemical properties. However, traditional computational methods face significant challenges due to the difficulty of solving the Schrödinger equations and the increasing computational cost with the size of the molecular system. In response, there has been a surge of interest in leveraging artificial intelligence (AI) and machine learning (ML) techniques to in silico experiments. Integrating AI and ML into computational chemistry increases the scalability and speed of the exploration of chemical space. However, challenges remain, particularly regarding the reproducibility and transferability of ML models. This review highlights the evolution of ML in learning from, complementing, or replacing traditional computational chemistry for energy and property predictions. Starting from models trained entirely on numerical data, a journey set forth toward the ideal model incorporating or learning the physical laws of quantum mechanics. This paper also reviews existing computational methods and ML models and their intertwining, outlines a roadmap for future research, and identifies areas for improvement and innovation. Ultimately, the goal is to develop AI architectures capable of predicting accurate and transferable solutions to the Schrödinger equation, thereby revolutionizing in silico experiments within chemistry and materials science.

Machine Learning of Reactive Potentials

In the past two decades, machine learning potentials (MLPs) have driven significant developments in chemical, biological, and material sciences. The construction and training of MLPs enable fast and accurate simulations and analysis of thermodynamic and kinetic properties. This review focuses on the application of MLPs to reaction systems with consideration of bond breaking and formation. We review the development of MLP models, primarily with neural network and kernel-based algorithms, and recent applications of reactive MLPs (RMLPs) to systems at different scales. We show how RMLPs are constructed, how they speed up the calculation of reactive dynamics, and how they facilitate the study of reaction trajectories, reaction rates, free energy calculations, and many other calculations. Different data sampling strategies applied in building RMLPs are also discussed with a focus on how to collect structures for rare events and how to further improve their performance with active learning.

TorchMD-Net 2.0: Fast Neural Network Potentials for Molecular Simulations

Achieving a balance between computational speed, prediction accuracy, and universal applicability in molecular simulations has been a persistent challenge. This paper presents substantial advancements in TorchMD-Net software, a pivotal step forward in the shift from conventional force fields to neural network-based potentials. The evolution of TorchMD-Net into a more comprehensive and versatile framework is highlighted, incorporating cutting-edge architectures such as TensorNet. This transformation is achieved through a modular design approach, encouraging customized applications within the scientific community. The most notable enhancement is a significant improvement in computational efficiency, achieving a very remarkable acceleration in the computation of energy and forces for TensorNet models, with performance gains ranging from 2× to 10× over previous, nonoptimized, iterations. Other enhancements include highly optimized neighbor search algorithms that support periodic boundary conditions and smooth integration with existing molecular dynamics frameworks. Additionally, the updated version introduces the capability to integrate physical priors, further enriching its application spectrum and utility in research. The software is available at https://github.com/torchmd/torchmd-net.

TorchMD-Net 2.0: Fast Neural Network Potentials for Molecular Simulations

Achieving a balance between computational speed, prediction accuracy, and universal applicability in molecular simulations has been a persistent challenge. This paper presents substantial advancements in the TorchMD-Net software, a pivotal step forward in the shift from conventional force fields to neural network-based potentials. The evolution of TorchMD-Net into a more comprehensive and versatile framework is highlighted, incorporating cutting-edge architectures such as TensorNet. This transformation is achieved through a modular design approach, encouraging customized applications within the scientific community. The most notable enhancement is a significant improvement in computational efficiency, achieving a very remarkable acceleration in the computation of energy and forces for Tensor-Net models, with performance gains ranging from 2x to 10x over previous, non-optimized, iterations. Other enhancements include highly optimized neighbor search algorithms that support periodic boundary conditions and smooth integration with existing molecular dynamics frameworks. Additionally, the updated version introduces the capability to integrate physical priors, further enriching its application spectrum and utility in research. The software is available at https://github.com/torchmd/torchmd-net.

Ultra-flat bands at large twist angles in group-V twisted bilayer materials

Flat bands in 2D twisted materials are key to the realization of correlation-related exotic phenomena. However, a flat band often was achieved in the large system with a very small twist angle, which enormously increases the computational and experimental complexity. In this work, we proposed group-V twisted bilayer materials, including P, As, and Sb in the β phase with large twist angles. The band structure of twisted bilayer materials up to 2524 atoms has been investigated by a deep learning method DeepH, which significantly reduces the computational time. Our results show that the bandgap and the flat bandwidth of twisted bilayer β-P, β-As, and β-Sb reduce gradually with the decreasing of twist angle, and the ultra-flat band with bandwidth approaching 0 eV is achieved. Interestingly, we found that a twist angle of 9.43° is sufficient to achieve the band flatness for β-As comparable to that of twist bilayer graphene at the magic angle of 1.08°. Moreover, we also find that the bandgap reduces with decreasing interlayer distance while the flat band is still preserved, which suggests interlayer distance as an effective routine to tune the bandgap of flat band systems. Our research provides a feasible platform for exploring physical phenomena related to flat bands in twisted layered 2D materials.

Machine learning interatomic potential: Bridge the gap between small-scale models and realistic device-scale simulations

Machine learning interatomic potential (MLIP) overcomes the challenges of high computational costs in density-functional theory and the relatively low accuracy in classical large-scale molecular dynamics, facilitating more efficient and precise simulations in materials research and design. In this review, the current state of the four essential stages of MLIP is discussed, including data generation methods, material structure descriptors, six unique machine learning algorithms, and available software. Furthermore, the applications of MLIP in various fields are investigated, notably in phase-change memory materials, structure searching, material properties predicting, and the pre-trained universal models. Eventually, the future perspectives, consisting of standard datasets, transferability, generalization, and trade-off between accuracy and complexity in MLIPs, are reported.

Diffeomorphic Counterfactuals With Generative Models

Counterfactuals can explain classification decisions of neural networks in a human interpretable way. We propose a simple but effective method to generate such counterfactuals. More specifically, we perform a suitable diffeomorphic coordinate transformation and then perform gradient ascent in these coordinates to find counterfactuals which are classified with great confidence as a specified target class. We propose two methods to leverage generative models to construct such suitable coordinate systems that are either exactly or approximately diffeomorphic. We analyze the generation process theoretically using Riemannian differential geometry and validate the quality of the generated counterfactuals using various qualitative and quantitative measures.

Exploring the frontiers of condensed-phase chemistry with a general reactive machine learning potential

Atomistic simulation has a broad range of applications from drug design to materials discovery. Machine learning interatomic potentials (MLIPs) have become an efficient alternative to computationally expensive ab initio simulations. For this reason, chemistry and materials science would greatly benefit from a general reactive MLIP, that is, an MLIP that is applicable to a broad range of reactive chemistry without the need for refitting. Here we develop a general reactive MLIP (ANI-1xnr) through automated sampling of condensed-phase reactions. ANI-1xnr is then applied to study five distinct systems: carbon solid-phase nucleation, graphene ring formation from acetylene, biofuel additives, combustion of methane and the spontaneous formation of glycine from early earth small molecules. In all studies, ANI-1xnr closely matches experiment (when available) and/or previous studies using traditional model chemistry methods. As such, ANI-1xnr proves to be a highly general reactive MLIP for C, H, N and O elements in the condensed phase, enabling high-throughput in silico reactive chemistry experimentation.

Construction of High Accuracy Machine Learning Interatomic Potential for Surface/Interface of Nanomaterials-A Review

The inherent discontinuity and unique dimensional attributes of nanomaterial surfaces and interfaces bestow them with various exceptional properties. These properties, however, also introduce difficulties for both experimental and computational studies. The advent of machine learning interatomic potential (MLIP) addresses some of the limitations associated with empirical force fields, presenting a valuable avenue for accurate simulations of these surfaces/interfaces of nanomaterials. Central to this approach is the idea of capturing the relationship between system configuration and potential energy, leveraging the proficiency of machine learning (ML) to precisely approximate high-dimensional functions. This review offers an in-depth examination of MLIP principles and their execution and elaborates on their applications in the realm of nanomaterial surface and interface systems. The prevailing challenges faced by this potent methodology are also discussed.

The emergence of machine learning force fields in drug design

In the field of molecular simulation for drug design, traditional molecular mechanic force fields and quantum chemical theories have been instrumental but limited in terms of scalability and computational efficiency. To overcome these limitations, machine learning force fields (MLFFs) have emerged as a powerful tool capable of balancing accuracy with efficiency. MLFFs rely on the relationship between molecular structures and potential energy, bypassing the need for a preconceived notion of interaction representations. Their accuracy depends on the machine learning models used, and the quality and volume of training data sets. With recent advances in equivariant neural networks and high-quality datasets, MLFFs have significantly improved their performance. This review explores MLFFs, emphasizing their potential in drug design. It elucidates MLFF principles, provides development and validation guidelines, and highlights successful MLFF implementations. It also addresses potential challenges in developing and applying MLFFs. The review concludes by illuminating the path ahead for MLFFs, outlining the challenges to be overcome and the opportunities to be harnessed. This inspires researchers to embrace MLFFs in their investigations as a new tool to perform molecular simulations in drug design.

Modeling Zinc Complexes Using Neural Networks

Understanding the energetic landscapes of large molecules is necessary for the study of chemical and biological systems. Recently, deep learning has greatly accelerated the development of models based on quantum chemistry, making it possible to build potential energy surfaces and explore chemical space. However, most of this work has focused on organic molecules due to the simplicity of their electronic structures as well as the availability of data sets. In this work, we build a deep learning architecture to model the energetics of zinc organometallic complexes. To achieve this, we have compiled a configurationally and conformationally diverse data set of zinc complexes using metadynamics to overcome the limitations of traditional sampling methods. In terms of the neural network potentials, our results indicate that for zinc complexes, partial charges play an important role in modeling the long-range interactions with a neural network. Our developed model outperforms semiempirical methods in predicting the relative energy of zinc conformers, yielding a mean absolute error (MAE) of 1.32 kcal/mol with reference to the double-hybrid PWPB95 method.

Collaboration on Machine-Learned Potentials with IPSuite: A Modular Framework for Learning-on-the-Fly

The field of machine learning potentials has experienced a rapid surge in progress, thanks to advances in machine learning theory, algorithms, and hardware capabilities. While the underlying methods are continuously evolving, the infrastructure for their deployment has lagged. The community, due to these rapid developments, frequently finds itself split into groups built around different implementations of machine-learned potentials. In this work, we introduce IPSuite, a Python-driven software package designed to connect different methods and algorithms from the comprehensive field of machine-learned potentials into a single platform while also providing a collaborative infrastructure, helping ensure reproducibility. Furthermore, the data management infrastructure of the IPSuite code enables simple model sharing and deployment in simulations. Currently, IPSuite supports six state-of-the-art machine learning approaches for the fitting of interatomic potentials as well as a variety of methods for the selection of training data, running of ab initio calculations, learning-on-the-fly strategies, model evaluation, and simulation deployment.

Biomolecular dynamics with machine-learned quantum-mechanical force fields trained on diverse chemical fragments

The GEMS method enables molecular dynamics simulations of large heterogeneous systems at ab initio quality.

Modeling Fe(II) Complexes Using Neural Networks

We report a Fe(II) data set of more than 23000 conformers in both low-spin (LS) and high-spin (HS) states. This data set was generated to develop a neural network model that is capable of predicting the energy and the energy splitting as a function of the conformation of a Fe(II) organometallic complex. In order to achieve this, we propose a type of scaled electronic embedding to cover the long-range interactions implicitly in our neural network describing the Fe(II) organometallic complexes. For the total energy prediction, the lowest MAE is 0.037 eV, while the lowest MAE of the splitting energy is 0.030 eV. Compared to baseline models, which only incorporate short-range interactions, our scaled electronic embeddings improve the accuracy by over 70% for the prediction of the total energy and the splitting energy. With regard to semiempirical methods, our proposed models reduce the MAE, with respect to these methods, by 2 orders of magnitude.

Numerical Accuracy Matters: Applications of Machine Learned Potential Energy Surfaces

The role of numerical accuracy in training and evaluating neural network-based potential energy surfaces is examined for different experimental observables. For observables that require third- and fourth-order derivatives of the potential energy with respect to Cartesian coordinates single-precision arithmetics as is typically used in ML-based approaches is insufficient and leads to roughness of the underlying PES as is explicitly demonstrated. Increasing the numerical accuracy to double-precision gives a smooth PES with higher-order derivatives that are numerically stable and yield meaningful anharmonic frequencies and tunneling splitting as is demonstrated for H2CO and malonaldehyde. For molecular dynamics simulations, which only require first-order derivatives, single-precision arithmetics appears to be sufficient, though.

Deep-Learning Density Functional Perturbation Theory

Calculating perturbation response properties of materials from first principles provides a vital link between theory and experiment, but is bottlenecked by the high computational cost. Here, a general framework is proposed to perform density functional perturbation theory (DFPT) calculations by neural networks, greatly improving the computational efficiency. Automatic differentiation is applied on neural networks, facilitating accurate computation of derivatives. High efficiency and good accuracy of the approach are demonstrated by studying electron-phonon coupling and related physical quantities. This work brings deep-learning density functional theory and DFPT into a unified framework, creating opportunities for developing ab initio artificial intelligence.

From Organic Fragments to Photoswitchable Catalysts: The OFF-ON Structural Repository for Transferable Kernel-Based Potentials

Structurally and conformationally diverse databases are needed to train accurate neural networks or kernel-based potentials capable of exploring the complex free energy landscape of flexible functional organic molecules. Curating such databases for species beyond "simple" drug-like compounds or molecules composed of well-defined building blocks (e.g., peptides) is challenging as it requires thorough chemical space mapping and evaluation of both chemical and conformational diversities. Here, we introduce the OFF-ON (organic fragments from organocatalysts that are non-modular) database, a repository of 7869 equilibrium and 67,457 nonequilibrium geometries of organic compounds and dimers aimed at describing conformationally flexible functional organic molecules, with an emphasis on photoswitchable organocatalysts. The relevance of this database is then demonstrated by training a local kernel regression model on a low-cost semiempirical baseline and comparing it with a PBE0-D3 reference for several known catalysts, notably the free energy surfaces of exemplary photoswitchable organocatalysts. Our results demonstrate that the OFF-ON data set offers reliable predictions for simulating the conformational behavior of virtually any (photoswitchable) organocatalyst or organic compound composed of H, C, N, O, F, and S atoms, thereby opening a computationally feasible route to explore complex free energy surfaces in order to rationalize and predict catalytic behavior.

A Primer on 2D Descriptors in Selectivity Modeling for Asymmetric Catalysis

Machine learning has permeated all fields of research, including chemistry, and is now an integral part of the design of novel compounds with desired properties. In the field of asymmetric catalysis, the preference still lies with models based on a physical understanding of the catalysis phenomenon and the electronic and steric properties of catalysts. However, such models require quantum chemical calculations and are thus limited by their computational cost. Here, we highlight the recent advances in modeling catalyst selectivity by using the 2D structures of catalysts and substrates. While these have a less explicit mechanistic connection to the modeled property, 2D descriptors, such as topological indices, molecular fingerprints, and fragments, offer the tremendous advantages of low cost and high speed of calculations. This makes them optimal for the in-silico screening of large amounts of data. We provide an overview of common quantitative structure-property relationship workflow, model building and validation techniques, applications of these methodologies in asymmetric catalysis design, and an outlook on improving the understanding of 2D-based models.

Phase Diagrams of Alloys and Their Hydrides via On-Lattice Graph Neural Networks and Limited Training Data

Efficient prediction of sampling-intensive thermodynamic properties is needed to evaluate material performance and permit high-throughput materials modeling for a diverse array of technology applications. To alleviate the prohibitive computational expense of high-throughput configurational sampling with density functional theory (DFT), surrogate modeling strategies like cluster expansion are many orders of magnitude more efficient but can be difficult to construct in systems with high compositional complexity. We therefore employ minimal-complexity graph neural network models that accurately predict and can even extrapolate to out-of-train distribution formation energies of DFT-relaxed structures from an ideal (unrelaxed) crystallographic representation. This enables the large-scale sampling necessary for various thermodynamic property predictions that may otherwise be intractable and can be achieved with small training data sets. Two exemplars, optimizing the thermodynamic stability of low-density high-entropy alloys and modulating the plateau pressure of hydrogen in metal alloys, demonstrate the power of this approach, which can be extended to a variety of materials discovery and modeling problems.