E3x: $\mathrm{E}(3)$-Equivariant Deep Learning Made Easy

This work introduces E3x, a software package for building neural networks that are equivariant with respect to the Euclidean group $\mathrm{E}(3)$, consisting of translations, rotations, and reflections of three-dimensional space. Compared to ordinary neural networks, $\mathrm{E}(3)$-equivariant models promise benefits whenever input and/or output data are quantities associated with three-dimensional objects. This is because the numeric values of such quantities (e.g. positions) typically depend on the chosen coordinate system. Under transformations of the reference frame, the values change predictably, but the underlying rules can be difficult to learn for ordinary machine learning models. With built-in $\mathrm{E}(3)$-equivariance, neural networks are guaranteed to satisfy the relevant transformation rules exactly, resulting in superior data efficiency and accuracy. The code for E3x is available from https://github.com/google-research/e3x, detailed documentation and usage examples can be found on https://e3x.readthedocs.io.

From Peptides to Nanostructures: 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 suitability of MLFFs in molecular dynamics (MD) simulations is being increasingly scrutinized due to concerns about instability. Our findings suggest a potential connection between MD simulation stability and the presence of equivariant representations in MLFFs, but their computational cost can limit practical advantages they would otherwise bring. To address this, we propose a transformer architecture called SO3krates that combines sparse equivariant representations (Euclidean variables) with a self-attention mechanism that can separate 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 unprecedented 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.

So3krates -- Self-attention for higher-order geometric interactions on arbitrary length-scales

The application of machine learning methods in quantum chemistry has enabled the study of numerous chemical phenomena, which are computationally intractable with traditional ab-initio methods. However, some quantum mechanical properties of molecules and materials depend on non-local electronic effects, which are often neglected due to the difficulty of modeling them efficiently. This work proposes a modified attention mechanism adapted to the underlying physics, which allows to recover the relevant non-local effects. Namely, we introduce spherical harmonic coordinates (SPHCs) to reflect higher-order geometric information for each atom in a molecule, enabling a non-local formulation of attention in the SPHC space. Our proposed model So3krates -- a self-attention based message passing neural network -- uncouples geometric information from atomic features, making them independently amenable to attention mechanisms. We show that in contrast to other published methods, So3krates is able to describe non-local quantum mechanical effects over arbitrary length scales. Further, we find evidence that the inclusion of higher-order geometric correlations increases data efficiency and improves generalization. So3krates matches or exceeds state-of-the-art performance on popular benchmarks, notably, requiring a significantly lower number of parameters (0.25--0.4x) while at the same time giving a substantial speedup (6--14x for training and 2--11x for inference) compared to other models.

Accurate Machine Learned Quantum-Mechanical Force Fields for Biomolecular Simulations

Molecular dynamics (MD) simulations allow atomistic insights into chemical and biological processes. Accurate MD simulations require computationally demanding quantum-mechanical calculations, being practically limited to short timescales and few atoms. For larger systems, efficient, but much less reliable empirical force fields are used. Recently, machine learned force fields (MLFFs) emerged as an alternative means to execute MD simulations, offering similar accuracy as ab initio methods at orders-of-magnitude speedup. Until now, MLFFs mainly capture short-range interactions in small molecules or periodic materials, due to the increased complexity of constructing models and obtaining reliable reference data for large molecules, where long-ranged many-body effects become important. This work proposes a general approach to constructing accurate MLFFs for large-scale molecular simulations (GEMS) by training on "bottom-up" and "top-down" molecular fragments of varying size, from which the relevant physicochemical interactions can be learned. GEMS is applied to study the dynamics of alanine-based peptides and the 46-residue protein crambin in aqueous solution, allowing nanosecond-scale MD simulations of >25k atoms at essentially ab initio quality. Our findings suggest that structural motifs in peptides and proteins are more flexible than previously thought, indicating that simulations at ab initio accuracy might be necessary to understand dynamic biomolecular processes such as protein (mis)folding, drug-protein binding, or allosteric regulation.

Automatic Identification of Chemical Moieties

In recent years, the prediction of quantum mechanical observables with machine learning methods has become increasingly popular. Message-passing neural networks (MPNNs) solve this task by constructing atomic representations, from which the properties of interest are predicted. Here, we introduce a method to automatically identify chemical moieties (molecular building blocks) from such representations, enabling a variety of applications beyond property prediction, which otherwise rely on expert knowledge. The required representation can either be provided by a pretrained MPNN, or learned from scratch using only structural information. Beyond the data-driven design of molecular fingerprints, the versatility of our approach is demonstrated by enabling the selection of representative entries in chemical databases, the automatic construction of coarse-grained force fields, as well as the identification of reaction coordinates.

SE(3)-equivariant prediction of molecular wavefunctions and electronic densities

Machine learning has enabled the prediction of quantum chemical properties with high accuracy and efficiency, allowing to bypass computationally costly ab initio calculations. Instead of training on a fixed set of properties, more recent approaches attempt to learn the electronic wavefunction (or density) as a central quantity of atomistic systems, from which all other observables can be derived. This is complicated by the fact that wavefunctions transform non-trivially under molecular rotations, which makes them a challenging prediction target. To solve this issue, we introduce general SE(3)-equivariant operations and building blocks for constructing deep learning architectures for geometric point cloud data and apply them to reconstruct wavefunctions of atomistic systems with unprecedented accuracy. Our model reduces prediction errors by up to two orders of magnitude compared to the previous state-of-the-art and makes it possible to derive properties such as energies and forces directly from the wavefunction in an end-to-end manner. We demonstrate the potential of our approach in a transfer learning application, where a model trained on low accuracy reference wavefunctions implicitly learns to correct for electronic many-body interactions from observables computed at a higher level of theory. Such machine-learned wavefunction surrogates pave the way towards novel semi-empirical methods, offering resolution at an electronic level while drastically decreasing computational cost. While we focus on physics applications in this contribution, the proposed equivariant framework for deep learning on point clouds is promising also beyond, say, in computer vision or graphics.

SpookyNet: Learning Force Fields with Electronic Degrees of Freedom and Nonlocal Effects

In recent years, machine-learned force fields (ML-FFs) have gained increasing popularity in the field of computational chemistry. Provided they are trained on appropriate reference data, ML-FFs combine the accuracy of ab initio methods with the efficiency of conventional force fields. However, current ML-FFs typically ignore electronic degrees of freedom, such as the total charge or spin, when forming their prediction. In addition, they often assume chemical locality, which can be problematic in cases where nonlocal effects play a significant role. This work introduces SpookyNet, a deep neural network for constructing ML-FFs with explicit treatment of electronic degrees of freedom and quantum nonlocality. Its predictions are further augmented with physically-motivated corrections to improve the description of long-ranged interactions and nuclear repulsion. SpookyNet improves upon the current state-of-the-art (or achieves similar performance) on popular quantum chemistry data sets. Notably, it can leverage the learned chemical insights, e.g. by predicting unknown spin states or by properly modeling physical limits. Moreover, it is able to generalize across chemical and conformational space and thus close an important remaining gap for today's machine learning models in quantum chemistry.

Equivariant message passing for the prediction of tensorial properties and molecular spectra

Message passing neural networks have become a method of choice for learning on graphs, in particular the prediction of chemical properties and the acceleration of molecular dynamics studies. While they readily scale to large training data sets, previous approaches have proven to be less data efficient than kernel methods. We identify limitations of invariant representations as a major reason and extend the message passing formulation to rotationally equivariant representations. On this basis, we propose the polarizable atom interaction neural network (PaiNN) and improve on common molecule benchmarks over previous networks, while reducing model size and inference time. We leverage the equivariant atomwise representations obtained by PaiNN for the prediction of tensorial properties. Finally, we apply this to the simulation of molecular spectra, achieving speedups of 4-5 orders of magnitude compared to the electronic structure reference.

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.