Material-Property-Field-based Deep Neural Network in Hopfield Framework
Current deep neural networks (DNNs) used in materials modeling often lack explicit physical structure and clear analytical formulations tailored to material systems, which can limit their interpretability. In this work, we integrate Material Property Fields (MPF) with the Hopfield network architecture and propose an analytically structured DNN framework named mPFDNN. MPF provides a unified framework that represents physical properties of materials as an analytical field built upon pairwise interactions, rigorously respecting fundamental symmetries, while also enabling a physically legitimate decomposition of property distributions at the atomic level. Although the Hopfield model was originally developed for Ising-like systems, we show that its dynamical evolution strategy can be naturally extended to the MPF framework. By reformulating nonlinear interatomic interactions as "hidden neurons", MPF can be extended into a deep yet analytically tractable DNN architecture that progressively captures an increasingly connected interaction landscape. This framework also provides a unified perspective that connects linear expansions and nonlinear DNN architectures within a common interaction-based formulation. Extensive validation across diverse systems, including inorganic crystals, organic molecules, and aqueous solutions, and across multiple target properties, shows that mPFDNN achieves competitive predictive accuracy while offering a physically motivated framework for structure-property mapping in chemistry, physics, and materials science.