Lixin He

Shi Yin, Xinyang Pan, Xudong Zhu et al.

Deep learning for predicting the electronic-structure Hamiltonian of quantum systems necessitates satisfying the covariance laws, among which achieving SO(3)-equivariance without sacrificing the non-linear expressive capability of networks remains unsolved. To navigate the harmonization between equivariance and expressiveness, we propose a deep learning method synergizing two distinct categories of neural mechanisms as a two-stage encoding and regression framework. The first stage corresponds to group theory-based neural mechanisms with inherent SO(3)-equivariant properties prior to the parameter learning process, while the second stage is characterized by a non-linear 3D graph Transformer network we propose, featuring high capability on non-linear expressiveness. The novel combination lies in the point that, the first stage predicts baseline Hamiltonians with abundant SO(3)-equivariant features extracted, assisting the second stage in empirical learning of equivariance; and in turn, the second stage refines the first stage's output as a fine-grained prediction of Hamiltonians using powerful non-linear neural mappings, compensating for the intrinsic weakness on non-linear expressiveness capability of mechanisms in the first stage. Our method enables precise, generalizable predictions while capturing SO(3)-equivariance under rotational transformations, and achieves state-of-the-art performance in Hamiltonian prediction on six benchmark databases.

Shi Yin, Jinming Mu, Xudong Zhu et al.

Crystal structure prediction (CSP), which aims to predict the three-dimensional atomic arrangement of a crystal from its composition, is central to materials discovery and mechanistic understanding. Existing deep learning models often treat crystallographic symmetry only as a soft heuristic or rely on space group and Wyckoff templates retrieved from known structures, which limits both physical fidelity and the ability to discover genuinely new material structures. In contrast to retrieval-based methods, our approach leverages large language models to encode chemical semantics and directly generate fine-grained Wyckoff patterns from composition, effectively circumventing the limitations inherent to database lookups. Crucially, we incorporate domain knowledge into the generative process through an efficient constrained-optimization search that rigorously enforces algebraic consistency between site multiplicities and atomic stoichiometry. By integrating this symmetry-consistent template into a diffusion backbone, our approach constrains the stochastic generative trajectory to a physically valid geometric manifold. This framework achieves state-of-the-art performance across stability, uniqueness, and novelty (SUN) benchmarks, alongside superior matching performance, thereby establishing a new paradigm for the rigorous exploration of targeted crystallographic space. This framework enables efficient expansion into previously uncharted materials space, eliminating reliance on existing databases or a priori structural knowledge.

Shi Yin, Zujian Dai, Xinyang Pan et al.

Deep learning methods for electronic-structure Hamiltonian prediction has offered significant computational efficiency advantages over traditional DFT methods, yet the diversity of atomic types, structural patterns, and the high-dimensional complexity of Hamiltonians pose substantial challenges to the generalization performance. In this work, we contribute on both the methodology and dataset sides to advance universal deep learning paradigm for Hamiltonian prediction. On the method side, we propose NextHAM, a neural E(3)-symmetry and expressive correction method for efficient and generalizable materials electronic-structure Hamiltonian prediction. First, we introduce the zeroth-step Hamiltonians, which can be efficiently constructed by the initial charge density of DFT, as informative descriptors of neural regression model in the input level and initial estimates of the target Hamiltonian in the output level, so that the regression model directly predicts the correction terms to the target ground truths, thereby significantly simplifying the input-output mapping for learning. Second, we present a neural Transformer architecture with strict E(3)-Symmetry and high non-linear expressiveness for Hamiltonian prediction. Third, we propose a novel training objective to ensure the accuracy performance of Hamiltonians in both real space and reciprocal space, preventing error amplification and the occurrence of "ghost states" caused by the large condition number of the overlap matrix. On the dataset side, we curate a high-quality broad-coverage large benchmark, namely Materials-HAM-SOC, comprising 17,000 material structures spanning 68 elements from six rows of the periodic table and explicitly incorporating SOC effects. Experimental results on Materials-HAM-SOC demonstrate that NextHAM achieves excellent accuracy and efficiency in predicting Hamiltonians and band structures.

Meng Zhang, Chao Wang, Hao Zhang et al.

Design of neural networks that incorporate symmetry is crucial for geometric deep learning. Central to this effort is the development of invariant and equivariant operations. This works presents a systematic method for constructing valid invariant and equivariant operations. It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types. In addition, our method features a graphical representation utilizing the symmetric tensor network, which simplifies both the proofs and constructions related to invariant and equivariant functions. We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the constitutive law of materials.

Shi Yin, Xinyang Pan, Fengyan Wang et al.

We propose a framework to combine strong non-linear expressiveness with strict SO(3)-equivariance in prediction of the electronic-structure Hamiltonian, by exploring the mathematical relationships between SO(3)-invariant and SO(3)-equivariant quantities and their representations. The proposed framework, called TraceGrad, first constructs theoretical SO(3)-invariant trace quantities derived from the Hamiltonian targets, and use these invariant quantities as supervisory labels to guide the learning of high-quality SO(3)-invariant features. Given that SO(3)-invariance is preserved under non-linear operations, the learning of invariant features can extensively utilize non-linear mappings, thereby fully capturing the non-linear patterns inherent in physical systems. Building on this, we propose a gradient-based mechanism to induce SO(3)-equivariant encodings of various degrees from the learned SO(3)-invariant features. This mechanism can incorporate powerful non-linear expressive capabilities into SO(3)-equivariant features with consistency of physical dimensions to the regression targets, while theoretically preserving equivariant properties, establishing a strong foundation for predicting Hamiltonian. Our method achieves state-of-the-art performance in prediction accuracy across eight challenging benchmark databases on Hamiltonian prediction. Experimental results demonstrate that this approach not only improves the accuracy of Hamiltonian prediction but also significantly enhances the prediction for downstream physical quantities, and also markedly improves the acceleration performance for the traditional Density Functional Theory algorithms.

Chengqiang Lu, Qi Liu, Chao Wang et al.

Predicting molecular properties (e.g., atomization energy) is an essential issue in quantum chemistry, which could speed up much research progress, such as drug designing and substance discovery. Traditional studies based on density functional theory (DFT) in physics are proved to be time-consuming for predicting large number of molecules. Recently, the machine learning methods, which consider much rule-based information, have also shown potentials for this issue. However, the complex inherent quantum interactions of molecules are still largely underexplored by existing solutions. In this paper, we propose a generalizable and transferable Multilevel Graph Convolutional neural Network (MGCN) for molecular property prediction. Specifically, we represent each molecule as a graph to preserve its internal structure. Moreover, the well-designed hierarchical graph neural network directly extracts features from the conformation and spatial information followed by the multilevel interactions. As a consequence, the multilevel overall representations can be utilized to make the prediction. Extensive experiments on both datasets of equilibrium and off-equilibrium molecules demonstrate the effectiveness of our model. Furthermore, the detailed results also prove that MGCN is generalizable and transferable for the prediction.