# Forecasting the outcome of spintronic experiments with Neural Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2108.02318 · HTML: https://mlsanity.com/paper/2108.02318/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 88 references that are in the map.
- First posted on arXiv 23 Jul 2021.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [A Dual-Stage Attention-Based Recurrent Neural Network for Time Series Prediction](https://mlsanity.com/paper/1704.02971/) (2017)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [On Second Order Behaviour in Augmented Neural ODEs](https://mlsanity.com/paper/2006.07220/) (2020)
- [Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems](https://mlsanity.com/paper/2010.14685/) (2020)
- [Learning Dynamical Systems from Partial Observations](https://mlsanity.com/paper/1902.11136/) (2019)
- [Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanics](https://mlsanity.com/paper/2005.14617/) (2020)
- [Vowel recognition with four coupled spin-torque nano-oscillators](https://mlsanity.com/paper/1711.02704/) (2017)
- [Spatio-Temporal Neural Networks for Space-Time Series Forecasting and Relations Discovery](https://mlsanity.com/paper/1804.08562/) (2018)
- [Stochasticity in Neural ODEs: An Empirical Study](https://mlsanity.com/paper/2002.09779/) (2020)
- [Constrained Neural Ordinary Differential Equations with Stability Guarantees](https://mlsanity.com/paper/2004.10883/) (2020)
- [Deep Neural Networks to Recover Unknown Physical Parameters from Oscillating Time Series](https://mlsanity.com/paper/2101.03850/) (2021)
- [Magnetic skyrmion-based synaptic devices](https://mlsanity.com/paper/1608.07955/) (2016)
- [Extracting local switching fields in permanent magnets using machine learning](https://mlsanity.com/paper/1910.09279/) (2019)

## Built on by (most cited first, 13 of 13)

- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [A perspective on physical reservoir computing with nanomagnetic devices](https://mlsanity.com/paper/2212.04851/) (2022)
- [On the Forward Invariance of Neural ODEs](https://mlsanity.com/paper/2210.04763/) (2022)
- [Efficient, Accurate and Stable Gradients for Neural ODEs](https://mlsanity.com/paper/2410.11648/) (2024)
- [Bayesian Gaussian Process ODEs via Double Normalizing Flows](https://mlsanity.com/paper/2309.09222/) (2023)
- [Noise-Aware Training of Neuromorphic Dynamic Device Networks](https://mlsanity.com/paper/2401.07387/) (2024)
- [Correcting Auto-Differentiation in Neural-ODE Training](https://mlsanity.com/paper/2306.02192/) (2023)
- [Discover physical concepts and equations with machine learning](https://mlsanity.com/paper/2412.12161/) (2024)
- [High-order expansion of Neural Ordinary Differential Equations flows](https://mlsanity.com/paper/2504.08769/) (2025)
- [Learning Nonlinear Heterogeneity in Physical Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2601.15340/) (2026)
- [Forecasting with an N-dimensional Langevin Equation and a Neural-Ordinary Differential Equation](https://mlsanity.com/paper/2405.07359/) (2024)
- [NeuralMAG: Fast and Generalizable Micromagnetic Simulation with Deep Neural Nets](https://mlsanity.com/paper/2410.14986/) (2024)
- [FLARE: Flow Matching with Local Axis-Angle Representations for Stochastic Micromagnetic Evolution](https://mlsanity.com/paper/2609.34070/) (2026)

## Abstract (arXiv)

Deep learning has an increasing impact to assist research, allowing, for example, the discovery of novel materials. Until now, however, these artificial intelligence techniques have fallen short of discovering the full differential equation of an experimental physical system. Here we show that a dynamical neural network, trained on a minimal amount of data, can predict the behavior of spintronic devices with high accuracy and an extremely efficient simulation time, compared to the micromagnetic simulations that are usually employed to model them. For this purpose, we re-frame the formalism of Neural Ordinary Differential Equations (ODEs) to the constraints of spintronics: few measured outputs, multiple inputs and internal parameters. We demonstrate with Spin-Neural ODEs an acceleration factor over 200 compared to micromagnetic simulations for a complex problem -- the simulation of a reservoir computer made of magnetic skyrmions (20 minutes compared to three days). In a second realization, we show that we can predict the noisy response of experimental spintronic nano-oscillators to varying inputs after training Spin-Neural ODEs on five milliseconds of their measured response to different excitations. Spin-Neural ODE is a disruptive tool for developing spintronic applications in complement to micromagnetic simulations, which are time-consuming and cannot fit experiments when noise or imperfections are present. Spin-Neural ODE can also be generalized to other electronic devices involving dynamics.