# 1.1. Introduction

Over the last few decades, molecular dynamics (MD) simulations have attracted much attention due to their wide range of applications in many fields such as condensed matter physics, materials science, polymer chemistry, and molecular biology. They provide researchers access to examine the behavior of atoms or molecules, which is valuable and has the potential to enrich our knowledge, especially when experimentation is difficult, expensive, or even impossible.

It is well recognized that the quality of MD simulations is ultimately limited by the accuracy of the PES and accurately representing the PES is an important challenge in the field of MD simulations. The empirical atomic potential models and the quantum mechanical models have long been two types of models commonly used. The empirical atomic potential models consist of simple low-dimensional terms. They often show excellent computational efficiency, but have limited accuracy. The quantum mechanical models determine the energies and forces on atoms by approximately solving the Schrödinger equation for electronic structure and exhibit higher accuracy. However, quantum mechanical models are computationally demanding, and it is not quite practical for either large-scale or long-time calculations. Overall, a dilemma exists between the choice of an empirical atomic potential model for high efficiency and that of a quantum mechanics model for high accuracy.

Recently, machine-learning (ML) models are emerging as useful tools to address this dilemma. Descriptors and ML algorithms are two main components of the current ML models. The former is used to guarantee the natural symmetries of the system and the latter is used to establish a direct functional relationship between atomic configurations and potential energy by training on reference data generated by quantum mechanics. Once trained, ML models can provide the same accuracy as the quantum mechanical method used to generate the reference data, such as the density functional theory (DFT) accuracy. Meanwhile, the computational cost of DFT scales cubic and that of ML models scale linearly with the size of the system.

So far, different types of ML models have been reported in literatures, such as Behler-Parrinello neural network potentials (BPNNP)[1], Gaussian approximation potentials (GAP)[2], spectral neighbor analysis potentials (SNAP)[3], ANI-1[4], SchNet[5] and Deep Potentials (DP) [6,7,8]. It is worth pointing out that, despite great successes, there are still many challenging issues remaining to be tackled[9]. For example, the neglect of interactions beyond the cut-off radius may lead to systematic prediction errors[10].

This chapter focuses on the DP models. In addition to enabling quantum mechanical accuracy, current DP models also have the following characteristics: (i) ease to preserve the symmetry of the system, especially when there are multiple elemental species; (ii) high computational efficiency, being at least five orders of magnitude faster than DFT; (iii) the model is end-to-end and therefore has little human intervention; (iv) support for MPI and GPU, making it highly efficient on modern heterogeneous high performance supercomputers. Thanks to this, the DP models have been successfully employed in studies of water and water-containing systems[11,12,13,14], metals and alloys[15,16,17,18], phase diagrams[19,20,21], high-entropy ceramics[22,23], chemical reaction[24,25,26], solid-state electrolytes[27], ionic liquids[28], etc. We refer to Ref.[29] for a recent review of DP for materials systems.