# Physics aware machine learning for micromagnetic energy minimization: recent algorithmic developments

arXiv: https://arxiv.org/abs/2409.12877 · HTML: https://mlsanity.com/paper/2409.12877/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 62 references that are in the map.
- First posted on arXiv 19 Sep 2024.

## Builds on (most cited first, 11 of 11)

- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries](https://mlsanity.com/paper/2004.06490/) (2020)
- [Conditional physics informed neural networks](https://mlsanity.com/paper/2104.02741/) (2021)

## Abstract (arXiv)

In this work, we explore advanced machine learning techniques for minimizing Gibbs free energy in full 3D micromagnetic simulations. Building on Brown's bounds for magnetostatic self-energy, we revisit their application in the context of variational formulations of the transmission problems for the scalar and vector potential. To overcome the computational challenges posed by whole-space integrals, we reformulate these bounds on a finite domain, making the method more efficient and scalable for numerical simulation. Our approach utilizes an alternating optimization scheme for joint minimization of Brown's energy bounds and the Gibbs free energy. The Cayley transform is employed to rigorously enforce the unit norm constraint, while R-functions are used to impose essential boundary conditions in the computation of magnetostatic fields. Our results highlight the potential of mesh-free Physics-Informed Neural Networks (PINNs) and Extreme Learning Machines (ELMs) when integrated with hard constraints, providing highly accurate approximations. These methods exhibit competitive performance compared to traditional numerical approaches, showing significant promise in computing magnetostatic fields and the application for energy minimization, such as the computation of hysteresis curves. This work opens the path for future directions of research on more complex geometries, such as grain structure models, and the application to large scale problem settings which are intractable with traditional numerical methods.