[Paper Review] Material-Property-Field-based Deep Neural Network in Hopfield Framework

Yanxiao Hu, Ye Sheng+7 more|arXiv (Cornell University)|Mar 10, 2026

Machine Learning in Materials Science0 citations

TL;DR

The paper integrates Material Property Fields (MPF) with a Hopfield network to form an analytically structured deep neural network (mPFDNN) that models material systems with physics-inspired interactions and reveals a connection between linear expansions and nonlinear DNNs.

ABSTRACT

The current Deep Neural Networks (DNNs) lack the necessary physical priors and a clear formulation specifically designed for material systems. making them non-analytical and non-interpretable 'black boxes'. In this work, we integrate Material Property Fields (MPF) with the Hopfield network architecture and propose an analytical 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 initially developed for Ising-like systems, we prove that its dynamical evolution strategy for DNN design is equally well suited to MPF. By mathematically reformatting interatomic nonlinear interactions as 'hidden neurons', the MPF can be naturally evolved into a deep yet analytically tractable DNN architecture that approximates a fully connected interaction landscape. This framework also unifies nonlinear DNNs and linear approaches within a single cohesive model. Extensive validation across diverse systems (inorganic crystals, organic molecules, and aqueous solutions) and multiple properties (diffusion coefficients, adsorption energy, etc.) confirm that mPFDNN not only achieves accurate predictions but also provides a principled and universal framework for structure-property mapping for physical, chemical and materials science.

Motivation & Objective

• Develop a DNN framework (mPFDNN) that embeds Material Property Fields within a Hopfield network architecture.
• Ensure MPF representations respect fundamental symmetries and enable atomic-level decomposition of property distributions.
• Extend Hopfield dynamics to MPF to capture increasingly connected interatomic interactions in a deep, tractable manner.

Proposed method

• Represent material properties as an analytical MPF built on pairwise interactions.
• Map nonlinear interatomic interactions to hidden neurons within a Hopfield-like dynamics.
• Reformulate the MPF framework to a deep architecture that progressively captures broader interaction landscapes.
• Unify linear expansion and nonlinear DNN perspectives under an interaction-based formulation.

Experimental results

Research questions

• RQ1 Can MPF be integrated into a Hopfield framework to produce an analytically tractable DNN for materials modeling?
• RQ2 How does the mPFDNN handle interatomic interactions and symmetry-preserving representations across materials?
• RQ3 What is the relationship between linear expansions and nonlinear DNNs within the MPF-Hopfield formulation?
• RQ4 How does the mPFDNN perform across inorganic crystals, organic molecules, and aqueous solutions for various properties?

Key findings

• mPFDNN achieves competitive predictive accuracy across diverse materials systems.
• The framework provides a physically motivated perspective for structure-property mapping in chemistry, physics, and materials science.