# DiffEqFlux.jl - A Julia Library for Neural Differential Equations

arXiv: https://arxiv.org/abs/1902.02376 · HTML: https://mlsanity.com/paper/1902.02376/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Feb 2019.

## Built on by (most cited first, 30 of 33)

- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Opening the Blackbox: Accelerating Neural Differential Equations by Regularizing Internal Solver Heuristics](https://mlsanity.com/paper/2105.03918/) (2021)
- [Differentiable Multiple Shooting Layers](https://mlsanity.com/paper/2106.03885/) (2021)
- [Multiple shooting for training neural differential equations on time series](https://mlsanity.com/paper/2109.06786/) (2021)
- [Neural Ordinary Differential Equations for Nonlinear System Identification](https://mlsanity.com/paper/2203.00120/) (2022)
- [Homotopy-based training of NeuralODEs for accurate dynamics discovery](https://mlsanity.com/paper/2210.01407/) (2022)
- [torchode: A Parallel ODE Solver for PyTorch](https://mlsanity.com/paper/2210.12375/) (2022)
- [Closed-Form Diffeomorphic Transformations for Time Series Alignment](https://mlsanity.com/paper/2206.08107/) (2022)
- [Scientific Machine Learning for Modeling and Simulating Complex Fluids](https://mlsanity.com/paper/2210.04431/) (2022)
- [Deep learning approaches to surrogates for solving the diffusion equation for mechanistic real-world simulations](https://mlsanity.com/paper/2102.05527/) (2021)
- [Control of Stochastic Quantum Dynamics by Differentiable Programming](https://mlsanity.com/paper/2101.01190/) (2021)
- [Neural ODEs as Feedback Policies for Nonlinear Optimal Control](https://mlsanity.com/paper/2210.11245/) (2022)
- [Analyzing the Performance of Deep Encoder-Decoder Networks as Surrogates for a Diffusion Equation](https://mlsanity.com/paper/2302.03786/) (2023)
- [Locally Regularized Neural Differential Equations: Some Black Boxes Were Meant to Remain Closed!](https://mlsanity.com/paper/2303.02262/) (2023)
- [Physics-Informed Neural Networks for Satellite State Estimation](https://mlsanity.com/paper/2403.19736/) (2024)
- [NeuralFMU: Towards Structural Integration of FMUs into Neural Networks](https://mlsanity.com/paper/2109.04351/) (2021)
- [NeuralFMU: Presenting a workflow for integrating hybrid NeuralODEs into real world applications](https://mlsanity.com/paper/2209.03933/) (2022)
- [Parameter Inference with Bifurcation Diagrams](https://mlsanity.com/paper/2106.04243/) (2021)
- [A research framework for writing differentiable PDE discretizations in JAX](https://mlsanity.com/paper/2111.05218/) (2021)
- [Eigen-informed NeuralODEs: Dealing with stability and convergence issues of NeuralODEs](https://mlsanity.com/paper/2302.10892/) (2023)
- [Effective Latent Differential Equation Models via Attention and Multiple Shooting](https://mlsanity.com/paper/2307.05735/) (2023)
- [Deep learning and differential equations for modeling changes in individual-level latent dynamics between observation periods](https://mlsanity.com/paper/2202.07403/) (2022)
- [Optimizing differential equations to fit data and predict outcomes](https://mlsanity.com/paper/2204.07833/) (2022)
- [Automated differential equation solver based on the parametric approximation optimization](https://mlsanity.com/paper/2205.05383/) (2022)
- [Gradients of Functions of Large Matrices](https://mlsanity.com/paper/2405.17277/) (2024)
- [Modeling chaotic Lorenz ODE System using Scientific Machine Learning](https://mlsanity.com/paper/2410.06452/) (2024)
- [An Overview of Healthcare Data Analytics With Applications to the COVID-19 Pandemic](https://mlsanity.com/paper/2111.14623/) (2021)
- [Incremental Correction in Dynamic Systems Modelled with Neural Networks for Constraint Satisfaction](https://mlsanity.com/paper/2209.03698/) (2022)
- [A Deep Neural Network -- Mechanistic Hybrid Model to Predict Pharmacokinetics in Rat](https://mlsanity.com/paper/2310.09167/) (2023)
- [Graph Neural Ordinary Differential Equations for Coarse-Grained Socioeconomic Dynamics](https://mlsanity.com/paper/2407.18108/) (2024)

## Abstract (arXiv)

DiffEqFlux.jl is a library for fusing neural networks and differential equations. In this work we describe differential equations from the viewpoint of data science and discuss the complementary nature between machine learning models and differential equations. We demonstrate the ability to incorporate DifferentialEquations.jl-defined differential equation problems into a Flux-defined neural network, and vice versa. The advantages of being able to use the entire DifferentialEquations.jl suite for this purpose is demonstrated by counter examples where simple integration strategies fail, but the sophisticated integration strategies provided by the DifferentialEquations.jl library succeed. This is followed by a demonstration of delay differential equations and stochastic differential equations inside of neural networks. We show high-level functionality for defining neural ordinary differential equations (neural networks embedded into the differential equation) and describe the extra models in the Flux model zoo which includes neural stochastic differential equations. We conclude by discussing the various adjoint methods used for backpropogation of the differential equation solvers. DiffEqFlux.jl is an important contribution to the area, as it allows the full weight of the differential equation solvers developed from decades of research in the scientific computing field to be readily applied to the challenges posed by machine learning and data science.