# Machine learning materials physics: Integrable deep neural networks enable scale bridging by learning free energy functions

## TL;DR

This paper introduces Integrable Deep Neural Networks (IDNNs) that learn free energy functions from derivative data, enabling accurate phase field simulations and bridging scales in materials physics.

## Contribution

The work presents a novel IDNN approach that can be trained on derivative data and analytically integrated to accurately represent free energy functions in materials modeling.

## Key findings

- IDNN accurately predicts free energy and phase boundaries.
- IDNN outperforms B-spline in representing complex free energy landscapes.
- Enables scale bridging in materials simulations.

## Abstract

The free energy of a system is central to many material models. Although free energy data is not generally found directly, its derivatives can be observed or calculated. In this work, we present an Integrable Deep Neural Network (IDNN) that can be trained to derivative data, then analytically integrated to recover an accurate representation of the free energy. The IDNN is demonstrated by training to the chemical potential values of a binary alloy with B2 ordering. The resulting DNN representation of the free energy is used in a phase field simulation and found to predict the appropriate formation of antiphase boundaries in the material. In contrast, a B-spline representation of the same data failed to represent the physics of the system with sufficient fidelity to resolve the antiphase boundaries.