Hong-Kun Zhang

Ru Geng, Panayotis Kevrekidis, Yixian Gao et al.

Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous. Existing graph-based approaches either assume the graph is given or, as in $α$-separable graph Hamiltonian network, infer it only for separable Hamiltonians with homogeneous node dynamics. We introduce the Hamiltonian Graph Inference Network (HGIN), which jointly recovers the interaction graph and predicts long-time trajectories from state data alone, for both separable and non-separable Hamiltonians and under heterogeneous node dynamics. HGIN couples a structure-learning module -- a learnable weighted adjacency matrix trained under a Hamilton's-equations loss -- with a trajectory-prediction module that partitions edges into physically distinct subgraphs via $k$-means clustering, assigning each subgraph its own encoder and thereby breaking the parameter-sharing bottleneck of conventional GNNs. On three benchmarks -- a Klein--Gordon lattice with long-range interactions and two discrete nonlinear Schrödinger lattices (homogeneous and heterogeneous) -- HGIN reduces long-time energy prediction error and trajectory prediction error by six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss further shows that the learned weights encode the parity of the underlying pair potential, yielding an interpretable readout of the system's interaction structure.

Ru Geng, Yixian Gao, Jian Zu et al.

A deep understanding of the intricate interactions between particles within a system is a key approach to revealing the essential characteristics of the system, whether it is an in-depth analysis of molecular properties in the field of chemistry or the design of new materials for specific performance requirements in materials science. To this end, we propose Graph Attention Hamiltonian Neural Network (GAHN), a neural network method that can understand the underlying structure of lattice Hamiltonian systems solely through the dynamic trajectories of particles. We can determine which particles in the system interact with each other, the proportion of interactions between different particles, and whether the potential energy of interactions between particles exhibits even symmetry or not. The obtained structure helps the neural network model to continue predicting the trajectory of the system and further understand the dynamic properties of the system. In addition to understanding the underlying structure of the system, it can be used for detecting lattice structural abnormalities, such as link defects, abnormal interactions, etc. These insights benefit system optimization, design, and detection of aging or damage. Moreover, this approach can integrate other components to deduce the link structure needed for specific parts, showcasing its scalability and potential. We tested it on a challenging molecular dynamics dataset, and the results proved its ability to accurately infer molecular bond connectivity, highlighting its scientific research potential.

Hong-Kun Zhang, Xin Li, Sikun Yang et al.

A novel neural network inspired by Cauchy's integral formula, is proposed for function approximation tasks that include time series forecasting, missing data imputation, etc. Hence, the novel neural network is named CauchyNet. By embedding real-valued data into the complex plane, CauchyNet efficiently captures complex temporal dependencies, surpassing traditional real-valued models in both predictive performance and computational efficiency. Grounded in Cauchy's integral formula and supported by the universal approximation theorem, CauchyNet offers strong theoretical guarantees for function approximation. The architecture incorporates complex-valued activation functions, enabling robust learning from incomplete data while maintaining a compact parameter footprint and reducing computational overhead. Through extensive experiments in diverse domains, including transportation, energy consumption, and epidemiological data, CauchyNet consistently outperforms state-of-the-art models in predictive accuracy, often achieving a 50% lower mean absolute error with fewer parameters. These findings highlight CauchyNet's potential as an effective and efficient tool for data-driven predictive modeling, particularly in resource-constrained and data-scarce environments.

Weiguo Lu, Gangnan Yuan, Hong-kun Zhang et al.

Neural networks in general, from MLPs and CNNs to attention-based Transformers, are constructed from layers of linear combinations followed by nonlinear operations such as ReLU, Sigmoid, or Softmax. Despite their strength, these conventional designs are often limited in introducing non-linearity by the choice of activation functions. In this work, we introduce Gaussian Mixture-Inspired Nonlinear Modules (GMNM), a new class of differentiable modules that draw on the universal density approximation Gaussian mixture models (GMMs) and distance properties (metric space) of Gaussian kernal. By relaxing probabilistic constraints and adopting a flexible parameterization of Gaussian projections, GMNM can be seamlessly integrated into diverse neural architectures and trained end-to-end with gradient-based methods. Our experiments demonstrate that incorporating GMNM into architectures such as MLPs, CNNs, attention mechanisms, and LSTMs consistently improves performance over standard baselines. These results highlight GMNM's potential as a powerful and flexible module for enhancing efficiency and accuracy across a wide range of machine learning applications.