Efficient interatomic descriptors for accurate machine learning force fields of extended molecules.

Machine learning force fields (MLFFs) are gradually evolving towards enabling molecular dynamics simulations of molecules and materials with ab initio accuracy but at a small fraction of the computational cost. However, several challenges remain to be addressed to enable predictive MLFF simulations of realistic molecules, including: (1) developing efficient descriptors for non-local interatomic interactions, which are essential to capture long-range molecular fluctuations, and (2) reducing the dimensionality of the descriptors to enhance the applicability and interpretability of MLFFs. Here we propose an automatized approach to substantially reduce the number of interatomic descriptor features while preserving the accuracy and increasing the efficiency of MLFFs. To simultaneously address the two stated challenges, we illustrate our approach on the example of the global GDML MLFF. We found that non-local features (atoms separated by as far as 15 Å in studied systems) are crucial to retain the overall accuracy of the MLFF for peptides, DNA base pairs, fatty acids, and supramolecular complexes. Interestingly, the number of required non-local features in the reduced descriptors becomes comparable to the number of local interatomic features (those below 5 Å). These results pave the way to constructing global molecular MLFFs whose cost increases linearly, instead of quadratically, with system size.

Reconstructing Kernel-Based Machine Learning Force Fields with Superlinear Convergence.

Kernel machines have sustained continuous progress in the field of quantum chemistry. In particular, they have proven to be successful in the low-data regime of force field reconstruction. This is because many equivariances and invariances due to physical symmetries can be incorporated into the kernel function to compensate for much larger data sets. So far, the scalability of kernel machines has however been hindered by its quadratic memory and cubical runtime complexity in the number of training points. While it is known that iterative Krylov subspace solvers can overcome these burdens, their convergence crucially relies on effective preconditioners, which are elusive in practice. Effective preconditioners need to partially presolve the learning problem in a computationally cheap and numerically robust manner. Here, we consider the broad class of Nyström-type methods to construct preconditioners based on successively more sophisticated low-rank approximations of the original kernel matrix, each of which provides a different set of computational trade-offs. All considered methods aim to identify a representative subset of inducing (kernel) columns to approximate the dominant kernel spectrum.

Accurate global machine learning force fields for molecules with hundreds of atoms.

Global machine learning force fields, with the capacity to capture collective interactions in molecular systems, now scale up to a few dozen atoms due to considerable growth of model complexity with system size. For larger molecules, locality assumptions are introduced, with the consequence that nonlocal interactions are not described. Here, we develop an exact iterative approach to train global symmetric gradient domain machine learning (sGDML) force fields (FFs) for several hundred atoms, without resorting to any potentially uncontrolled approximations. All atomic degrees of freedom remain correlated in the global sGDML FF, allowing the accurate description of complex molecules and materials that present phenomena with far-reaching characteristic correlation lengths. We assess the accuracy and efficiency of sGDML on a newly developed MD22 benchmark dataset containing molecules from 42 to 370 atoms. The robustness of our approach is demonstrated in nanosecond path-integral molecular dynamics simulations for supramolecular complexes in the MD22 dataset.

Algorithmic Differentiation for Automated Modeling of Machine Learned Force Fields.

Reconstructing force fields (FFs) from atomistic simulation data is a challenge since accurate data can be highly expensive. Here, machine learning (ML) models can help to be data economic as they can be successfully constrained using the underlying symmetry and conservation laws of physics. However, so far, every descriptor newly proposed for an ML model has required a cumbersome and mathematically tedious remodeling. We therefore propose using modern techniques from algorithmic differentiation within the ML modeling process, effectively enabling the usage of novel descriptors or models fully automatically at an order of magnitude higher computational efficiency. This paradigmatic approach enables not only a versatile usage of novel representations and the efficient computation of larger systems─all of high value to the FF community─but also the simple inclusion of further physical knowledge, such as higher-order information (e.g., Hessians, more complex partial differential equations constraints etc.), even beyond the presented FF domain.

BIGDML-Towards accurate quantum machine learning force fields for materials.

Machine-learning force fields (MLFF) should be accurate, computationally and data efficient, and applicable to molecules, materials, and interfaces thereof. Currently, MLFFs often introduce tradeoffs that restrict their practical applicability to small subsets of chemical space or require exhaustive datasets for training. Here, we introduce the Bravais-Inspired Gradient-Domain Machine Learning (BIGDML) approach and demonstrate its ability to construct reliable force fields using a training set with just 10-200 geometries for materials including pristine and defect-containing 2D and 3D semiconductors and metals, as well as chemisorbed and physisorbed atomic and molecular adsorbates on surfaces. The BIGDML model employs the full relevant symmetry group for a given material, does not assume artificial atom types or localization of atomic interactions and exhibits high data efficiency and state-of-the-art energy accuracies (errors substantially below 1 meV per atom) for an extended set of materials. Extensive path-integral molecular dynamics carried out with BIGDML models demonstrate the counterintuitive localization of benzene-graphene dynamics induced by nuclear quantum effects and their strong contributions to the hydrogen diffusion coefficient in a Pd crystal for a wide range of temperatures.

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

Machine-learned force fields combine the accuracy of ab initio methods with the efficiency of conventional force fields. However, current machine-learned force fields typically ignore electronic degrees of freedom, such as the total charge or spin state, and assume chemical locality, which is problematic when molecules have inconsistent electronic states, or when nonlocal effects play a significant role. This work introduces SpookyNet, a deep neural network for constructing machine-learned force fields with explicit treatment of electronic degrees of freedom and nonlocality, modeled via self-attention in a transformer architecture. Chemically meaningful inductive biases and analytical corrections built into the network architecture allow it to properly model physical limits. SpookyNet improves upon the current state-of-the-art (or achieves similar performance) on popular quantum chemistry data sets. Notably, it is able to generalize across chemical and conformational space and can leverage the learned chemical insights, e.g. by predicting unknown spin states, thus helping to close a further important remaining gap for today's machine learning models in quantum chemistry.

Combining Machine Learning and Computational Chemistry for Predictive Insights Into Chemical Systems.

Machine learning models are poised to make a transformative impact on chemical sciences by dramatically accelerating computational algorithms and amplifying insights available from computational chemistry methods. However, achieving this requires a confluence and coaction of expertise in computer science and physical sciences. This Review is written for new and experienced researchers working at the intersection of both fields. We first provide concise tutorials of computational chemistry and machine learning methods, showing how insights involving both can be achieved. We follow with a critical review of noteworthy applications that demonstrate how computational chemistry and machine learning can be used together to provide insightful (and useful) predictions in molecular and materials modeling, retrosyntheses, catalysis, and drug design.

Machine Learning Force Fields.

In recent years, the use of machine learning (ML) in computational chemistry has enabled numerous advances previously out of reach due to the computational complexity of traditional electronic-structure methods. One of the most promising applications is the construction of ML-based force fields (FFs), with the aim to narrow the gap between the accuracy of ab initio methods and the efficiency of classical FFs. The key idea is to learn the statistical relation between chemical structure and potential energy without relying on a preconceived notion of fixed chemical bonds or knowledge about the relevant interactions. Such universal ML approximations are in principle only limited by the quality and quantity of the reference data used to train them. This review gives an overview of applications of ML-FFs and the chemical insights that can be obtained from them. The core concepts underlying ML-FFs are described in detail, and a step-by-step guide for constructing and testing them from scratch is given. The text concludes with a discussion of the challenges that remain to be overcome by the next generation of ML-FFs.

Dynamical strengthening of covalent and non-covalent molecular interactions by nuclear quantum effects at finite temperature.

Nuclear quantum effects (NQE) tend to generate delocalized molecular dynamics due to the inclusion of the zero point energy and its coupling with the anharmonicities in interatomic interactions. Here, we present evidence that NQE often enhance electronic interactions and, in turn, can result in dynamical molecular stabilization at finite temperature. The underlying physical mechanism promoted by NQE depends on the particular interaction under consideration. First, the effective reduction of interatomic distances between functional groups within a molecule can enhance the n → π* interaction by increasing the overlap between molecular orbitals or by strengthening electrostatic interactions between neighboring charge densities. Second, NQE can localize methyl rotors by temporarily changing molecular bond orders and leading to the emergence of localized transient rotor states. Third, for noncovalent van der Waals interactions the strengthening comes from the increase of the polarizability given the expanded average interatomic distances induced by NQE. The implications of these boosted interactions include counterintuitive hydroxyl-hydroxyl bonding, hindered methyl rotor dynamics, and molecular stiffening which generates smoother free-energy surfaces. Our findings yield new insights into the versatile role of nuclear quantum fluctuations in molecules and materials.

Molecular force fields with gradient-domain machine learning (GDML): Comparison and synergies with classical force fields.

Modern machine learning force fields (ML-FF) are able to yield energy and force predictions at the accuracy of high-level ab initio methods, but at a much lower computational cost. On the other hand, classical molecular mechanics force fields (MM-FF) employ fixed functional forms and tend to be less accurate, but considerably faster and transferable between molecules of the same class. In this work, we investigate how both approaches can complement each other. We contrast the ability of ML-FF for reconstructing dynamic and thermodynamic observables to MM-FFs in order to gain a qualitative understanding of the differences between the two approaches. This analysis enables us to modify the generalized AMBER force field by reparametrizing short-range and bonded interactions with more expressive terms to make them more accurate, without sacrificing the key properties that make MM-FFs so successful.

Ensemble learning of coarse-grained molecular dynamics force fields with a kernel approach.

Gradient-domain machine learning (GDML) is an accurate and efficient approach to learn a molecular potential and associated force field based on the kernel ridge regression algorithm. Here, we demonstrate its application to learn an effective coarse-grained (CG) model from all-atom simulation data in a sample efficient manner. The CG force field is learned by following the thermodynamic consistency principle, here by minimizing the error between the predicted CG force and the all-atom mean force in the CG coordinates. Solving this problem by GDML directly is impossible because coarse-graining requires averaging over many training data points, resulting in impractical memory requirements for storing the kernel matrices. In this work, we propose a data-efficient and memory-saving alternative. Using ensemble learning and stratified sampling, we propose a 2-layer training scheme that enables GDML to learn an effective CG model. We illustrate our method on a simple biomolecular system, alanine dipeptide, by reconstructing the free energy landscape of a CG variant of this molecule. Our novel GDML training scheme yields a smaller free energy error than neural networks when the training set is small, and a comparably high accuracy when the training set is sufficiently large.

Molecular force fields with gradient-domain machine learning: Construction and application to dynamics of small molecules with coupled cluster forces.

We present the construction of molecular force fields for small molecules (less than 25 atoms) using the recently developed symmetrized gradient-domain machine learning (sGDML) approach [Chmiela et al., Nat. Commun. 9, 3887 (2018) and Chmiela et al., Sci. Adv. 3, e1603015 (2017)]. This approach is able to accurately reconstruct complex high-dimensional potential-energy surfaces from just a few 100s of molecular conformations extracted from ab initio molecular dynamics trajectories. The data efficiency of the sGDML approach implies that atomic forces for these conformations can be computed with high-level wavefunction-based approaches, such as the "gold standard" coupled-cluster theory with single, double and perturbative triple excitations [CCSD(T)]. We demonstrate that the flexible nature of the sGDML model recovers local and non-local electronic interactions (e.g., H-bonding, proton transfer, lone pairs, changes in hybridization states, steric repulsion, and n → π* interactions) without imposing any restriction on the nature of interatomic potentials. The analysis of sGDML molecular dynamics trajectories yields new qualitative insights into dynamics and spectroscopy of small molecules close to spectroscopic accuracy.

Towards exact molecular dynamics simulations with machine-learned force fields.

Molecular dynamics (MD) simulations employing classical force fields constitute the cornerstone of contemporary atomistic modeling in chemistry, biology, and materials science. However, the predictive power of these simulations is only as good as the underlying interatomic potential. Classical potentials often fail to faithfully capture key quantum effects in molecules and materials. Here we enable the direct construction of flexible molecular force fields from high-level ab initio calculations by incorporating spatial and temporal physical symmetries into a gradient-domain machine learning (sGDML) model in an automatic data-driven way. The developed sGDML approach faithfully reproduces global force fields at quantum-chemical CCSD(T) level of accuracy and allows converged molecular dynamics simulations with fully quantized electrons and nuclei. We present MD simulations, for flexible molecules with up to a few dozen atoms and provide insights into the dynamical behavior of these molecules. Our approach provides the key missing ingredient for achieving spectroscopic accuracy in molecular simulations.

Machine learning of accurate energy-conserving molecular force fields.

Using conservation of energy-a fundamental property of closed classical and quantum mechanical systems-we develop an efficient gradient-domain machine learning (GDML) approach to construct accurate molecular force fields using a restricted number of samples from ab initio molecular dynamics (AIMD) trajectories. The GDML implementation is able to reproduce global potential energy surfaces of intermediate-sized molecules with an accuracy of 0.3 kcal mol-1 for energies and 1 kcal mol-1 Å̊-1 for atomic forces using only 1000 conformational geometries for training. We demonstrate this accuracy for AIMD trajectories of molecules, including benzene, toluene, naphthalene, ethanol, uracil, and aspirin. The challenge of constructing conservative force fields is accomplished in our work by learning in a Hilbert space of vector-valued functions that obey the law of energy conservation. The GDML approach enables quantitative molecular dynamics simulations for molecules at a fraction of cost of explicit AIMD calculations, thereby allowing the construction of efficient force fields with the accuracy and transferability of high-level ab initio methods.

Quantum-chemical insights from deep tensor neural networks.

Learning from data has led to paradigm shifts in a multitude of disciplines, including web, text and image search, speech recognition, as well as bioinformatics. Can machine learning enable similar breakthroughs in understanding quantum many-body systems? Here we develop an efficient deep learning approach that enables spatially and chemically resolved insights into quantum-mechanical observables of molecular systems. We unify concepts from many-body Hamiltonians with purpose-designed deep tensor neural networks, which leads to size-extensive and uniformly accurate (1 kcal mol-1) predictions in compositional and configurational chemical space for molecules of intermediate size. As an example of chemical relevance, the model reveals a classification of aromatic rings with respect to their stability. Further applications of our model for predicting atomic energies and local chemical potentials in molecules, reliable isomer energies, and molecules with peculiar electronic structure demonstrate the potential of machine learning for revealing insights into complex quantum-chemical systems.