Neural Network Matrix Product Operator: A Multi-Dimensionally Integrable Machine Learning Potential

The developments of machine learning potential energy surfaces (PESs) in quantum chemistry and condensed matter physics have been accelerating alongside the advancements in a wide range of computational technologies [1-6]. Neural network potentials, in particular, have garnered significant attention for their ability to reproduce the accurate but expensive ab initio electronic structure calculations. For universal interatomic potentials, the model input is often replaced by user-defined descriptors that are augmented with atomic numbers and symmetry-consistent features, such as interatomic distances [7-11]. The incorporation of atomic descriptors enables the model to estimate individual atomic energies. Consequently, by summating these atomic contributions, the model can predict the total potential energy of systems comprising arbitrary numbers of atoms. Neural network potentials commonly employ multi-layer perceptrons (MLPs) as their foundational architecture. These structures are characterized by their extensive parameter spaces, which allow them to approximate highly complex functions. Notably, despite the substantial number of parameters involved, recent research has demonstrated that overparameterized neural networks, when trained using stochastic optimization techniques, exhibit the ability to mitigate complexity-induced errors and achieve robust generalization performance [12-14].