Title: Synergistic integration of physical embedding and machine learning enabling precise and reliable force field

Abstract: The machine learning force field has achieved significant strides in accurately reproducing the potential energy surface with quantum chemical accuracy. However, it still faces significant challenges, e.g., extrapolating to uncharted chemical spaces, interpreting long-range electrostatics, and mapping complex macroscopic properties. To address these issues, we advocate for a synergistic integration of physical principles and machine learning techniques within the framework of a physically informed neural network (PINN). This innovative approach involves the incorporation of physical constraints directly into the parameters of the neural network, coupled with the implementation of a global optimization strategy. We choose the AMOEBA+ force field as the physics-based model for embedding, and then train and test it using the diethylene glycol dimethyl ether (DEGDME) dataset as a case study. The results reveal a significant breakthrough in constructing a precise and noise-robust machine learning force field. Utilizing two training sets with hundreds of samples, our model exhibits remarkable generalization and DFT accuracy in describing molecular interactions and enables a precise prediction of the macroscopic properties such as diffusion coefficient with minimal cost. This work provides a crucial insight into establishing a fundamental framework of PINN.