Potential Score Matching: Debiasing Molecular Structure Sampling with Potential Energy Guidance

The ensemble average of physical properties of molecules is closely related to the distribution of molecular conformations, and sampling such distributions is a fundamental challenge in physics and chemistry. Traditional methods like molecular dynamics (MD) simulations and Markov chain Monte Carlo (MCMC) sampling are commonly used but can be time-consuming and costly. Recently, diffusion models have emerged as efficient alternatives by learning the distribution of training data. Obtaining an unbiased target distribution is still an expensive task, primarily because it requires satisfying ergodicity. To tackle these challenges, we propose Potential Score Matching (PSM), an approach that utilizes the potential energy gradient to guide generative models. PSM does not require exact energy functions and can debias sample distributions even when trained on limited and biased data. Our method outperforms existing state-of-the-art (SOTA) models on the Lennard-Jones (LJ) potential, a commonly used toy model. Furthermore, we extend the evaluation of PSM to high-dimensional problems using the MD17 and MD22 datasets. The results demonstrate that molecular distributions generated by PSM more closely approximate the Boltzmann distribution compared to traditional diffusion models.

Graph Fourier Neural ODEs: Bridging Spatial and Temporal Multiscales in Molecular Dynamics

Molecular dynamics simulations are crucial for understanding complex physical, chemical, and biological processes at the atomic level. However, accurately capturing interactions across multiple spatial and temporal scales remains a significant challenge. We present a novel framework that jointly models spatial and temporal multiscale interactions in molecular dynamics. Our approach leverages Graph Fourier Transforms to decompose molecular structures into different spatial scales and employs Neural Ordinary Differential Equations to model the temporal dynamics in a curated manner influenced by the spatial modes. This unified framework links spatial structures with temporal evolution in a flexible manner, enabling more accurate and comprehensive simulations of molecular systems. We evaluate our model on the MD17 dataset, demonstrating consistent performance improvements over state-of-the-art baselines across multiple molecules, particularly under challenging conditions such as irregular timestep sampling and long-term prediction horizons. Ablation studies confirm the significant contributions of both spatial and temporal multiscale modeling components. Our method advances the simulation of complex molecular systems, potentially accelerating research in computational chemistry, drug discovery, and materials science.

Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?

Equivariant Graph Neural Networks (GNNs) that incorporate E(3) symmetry have achieved significant success in various scientific applications. As one of the most successful models, EGNN leverages a simple scalarization technique to perform equivariant message passing over only Cartesian vectors (i.e., 1st-degree steerable vectors), enjoying greater efficiency and efficacy compared to equivariant GNNs using higher-degree steerable vectors. This success suggests that higher-degree representations might be unnecessary. In this paper, we disprove this hypothesis by exploring the expressivity of equivariant GNNs on symmetric structures, including $k$-fold rotations and regular polyhedra. We theoretically demonstrate that equivariant GNNs will always degenerate to a zero function if the degree of the output representations is fixed to 1 or other specific values. Based on this theoretical insight, we propose HEGNN, a high-degree version of EGNN to increase the expressivity by incorporating high-degree steerable vectors while maintaining EGNN's efficiency through the scalarization trick. Our extensive experiments demonstrate that HEGNN not only aligns with our theoretical analyses on toy datasets consisting of symmetric structures, but also shows substantial improvements on more complicated datasets such as $N$-body and MD17. Our theoretical findings and empirical results potentially open up new possibilities for the research of equivariant GNNs.

Learning Equivariant Non-Local Electron Density Functionals

The accuracy of density functional theory hinges on the approximation of non-local contributions to the exchange-correlation (XC) functional. To date, machine-learned and human-designed approximations suffer from insufficient accuracy, limited scalability, or dependence on costly reference data. To address these issues, we introduce Equivariant Graph Exchange Correlation (EG-XC), a novel non-local XC functional based on equivariant graph neural networks. EG-XC combines semi-local functionals with a non-local feature density parametrized by an equivariant nuclei-centered point cloud representation of the electron density to capture long-range interactions. By differentiating through a self-consistent field solver, we train EG-XC requiring only energy targets. In our empirical evaluation, we find EG-XC to accurately reconstruct `gold-standard' CCSD(T) energies on MD17. On out-of-distribution conformations of 3BPA, EG-XC reduces the relative MAE by 35% to 50%. Remarkably, EG-XC excels in data efficiency and molecular size extrapolation on QM9, matching force fields trained on 5 times more and larger molecules. On identical training sets, EG-XC yields on average 51% lower MAEs.

A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks

Designing neural network architectures that can handle data symmetry is crucial. This is especially important for geometric graphs whose properties are equivariance under Euclidean transformations. Current equivariant graph neural networks (EGNNs), particularly those using message passing, have a limitation in expressive power. Recent high-order graph neural networks can overcome this limitation, yet they lack equivariance properties, representing a notable drawback in certain applications in chemistry and physical sciences. In this paper, we introduce the Clifford Group Equivariant Graph Neural Networks (CG-EGNNs), a novel EGNN that enhances high-order message passing by integrating high-order local structures in the context of Clifford algebras. As a key benefit of using Clifford algebras, CG-EGNN can learn functions that capture equivariance from positional features. By adopting the high-order message passing mechanism, CG-EGNN gains richer information from neighbors, thus improving model performance. Furthermore, we establish the universality property of the $k$-hop message passing framework, showcasing greater expressive power of CG-EGNNs with additional $k$-hop message passing mechanism. We empirically validate that CG-EGNNs outperform previous methods on various benchmarks including n-body, CMU motion capture, and MD17, highlighting their effectiveness in geometric deep learning.

Distribution Learning for Molecular Regression

Using "soft" targets to improve model performance has been shown to be effective in classification settings, but the usage of soft targets for regression is a much less studied topic in machine learning. The existing literature on the usage of soft targets for regression fails to properly assess the method's limitations, and empirical evaluation is quite limited. In this work, we assess the strengths and drawbacks of existing methods when applied to molecular property regression tasks. Our assessment outlines key biases present in existing methods and proposes methods to address them, evaluated through careful ablation studies. We leverage these insights to propose Distributional Mixture of Experts (DMoE): A model-independent, and data-independent method for regression which trains a model to predict probability distributions of its targets. Our proposed loss function combines the cross entropy between predicted and target distributions and the L1 distance between their expected values to produce a loss function that is robust to the outlined biases. We evaluate the performance of DMoE on different molecular property prediction datasets -- Open Catalyst (OC20), MD17, and QM9 -- across different backbone model architectures -- SchNet, GemNet, and Graphormer. Our results demonstrate that the proposed method is a promising alternative to classical regression for molecular property prediction tasks, showing improvements over baselines on all datasets and architectures.

FreeCG: Free the Design Space of Clebsch-Gordan Transform for machine learning force field

The Clebsch-Gordan Transform (CG transform) effectively encodes many-body interactions. Many studies have proven its accuracy in depicting atomic environments, although this comes with high computational needs. The computational burden of this challenge is hard to reduce due to the need for permutation equivariance, which limits the design space of the CG transform layer. We show that, implementing the CG transform layer on permutation-invariant inputs allows complete freedom in the design of this layer without affecting symmetry. Developing further on this premise, our idea is to create a CG transform layer that operates on permutation-invariant abstract edges generated from real edge information. We bring in group CG transform with sparse path, abstract edges shuffling, and attention enhancer to form a powerful and efficient CG transform layer. Our method, known as FreeCG, achieves State-of-The-Art (SoTA) results in force prediction for MD17, rMD17, MD22, and property prediction in QM9 datasets with notable enhancement. It introduces a novel paradigm for carrying out efficient and expressive CG transform in future geometric neural network designs.

CheMFi: A Multifidelity Dataset of Quantum Chemical Properties of Diverse Molecules

Progress in both Machine Learning (ML) and conventional Quantum Chemistry (QC) computational methods have resulted in high accuracy ML models for QC properties ranging from atomization energies to excitation energies. Various datasets such as MD17, MD22, and WS22, which consist of properties calculated at some level of QC method, or fidelity, have been generated to benchmark such ML models. The term fidelity refers to the accuracy of the chosen QC method to the actual real value of the property. The higher the fidelity, the more accurate the calculated property, albeit at a higher computational cost. Research in multifidelity ML (MFML) methods, where ML models are trained on data from more than one numerical QC method, has shown the effectiveness of such models over single fidelity methods. Much research is progressing in this direction for diverse applications ranging from energy band gaps to excitation energies. A major hurdle for effective research in this field of research in the community is the lack of a diverse multifidelity dataset for benchmarking. Here, we present a comprehensive multifidelity dataset drawn from the WS22 molecular conformations. We provide the quantum Chemistry MultiFidelity (CheMFi) dataset consisting of five fidelities calculated with the TD-DFT formalism. The fidelities differ in their basis set choice and are namely: STO-3G, 3-21G, 6-31G, def2-SVP, and def2-TZVP. CheMFi offers to the community a variety of QC properties including vertical excitation energies, oscillator strengths, molecular dipole moments, and ground state energies. In addition to the dataset, multifidelity benchmarks are set with state-of-the-art MFML and optimized-MFML

Molecule Graph Networks with Many-body Equivariant Interactions

Message passing neural networks have demonstrated significant efficacy in predicting molecular interactions. Introducing equivariant vectorial representations augments expressivity by capturing geometric data symmetries, thereby improving model accuracy. However, two-body bond vectors in opposition may cancel each other out during message passing, leading to the loss of directional information on their shared node. In this study, we develop Equivariant N-body Interaction Networks (ENINet) that explicitly integrates equivariant many-body interactions to preserve directional information in the message passing scheme. Experiments indicate that integrating many-body equivariant representations enhances prediction accuracy across diverse scalar and tensorial quantum chemical properties. Ablation studies show an average performance improvement of 7.9% across 11 out of 12 properties in QM9, 27.9% in forces in MD17, and 11.3% in polarizabilities (CCSD) in QM7b.

Infusing Self-Consistency into Density Functional Theory Hamiltonian Prediction via Deep Equilibrium Models

In this study, we introduce a unified neural network architecture, the Deep Equilibrium Density Functional Theory Hamiltonian (DEQH) model, which incorporates Deep Equilibrium Models (DEQs) for predicting Density Functional Theory (DFT) Hamiltonians. The DEQH model inherently captures the self-consistency nature of Hamiltonian, a critical aspect often overlooked by traditional machine learning approaches for Hamiltonian prediction. By employing DEQ within our model architecture, we circumvent the need for DFT calculations during the training phase to introduce the Hamiltonian's self-consistency, thus addressing computational bottlenecks associated with large or complex systems. We propose a versatile framework that combines DEQ with off-the-shelf machine learning models for predicting Hamiltonians. When benchmarked on the MD17 and QH9 datasets, DEQHNet, an instantiation of the DEQH framework, has demonstrated a significant improvement in prediction accuracy. Beyond a predictor, the DEQH model is a Hamiltonian solver, in the sense that it uses the fixed-point solving capability of the deep equilibrium model to iteratively solve for the Hamiltonian. Ablation studies of DEQHNet further elucidate the network's effectiveness, offering insights into the potential of DEQ-integrated networks for Hamiltonian learning.