# NeuralFMU: Presenting a workflow for integrating hybrid NeuralODEs into real world applications

arXiv: https://arxiv.org/abs/2209.03933 · HTML: https://mlsanity.com/paper/2209.03933/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 36 references that are in the map.
- First posted on arXiv 8 Sep 2022.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A Differentiable Programming System to Bridge Machine Learning and Scientific Computing](https://mlsanity.com/paper/1907.07587/) (2019)
- [Deep AutoRegressive Networks](https://mlsanity.com/paper/1310.8499/) (2013)
- [Don't Unroll Adjoint: Differentiating SSA-Form Programs](https://mlsanity.com/paper/1810.07951/) (2018)
- [DiffEqFlux.jl - A Julia Library for Neural Differential Equations](https://mlsanity.com/paper/1902.02376/) (2019)
- [Multiple shooting for training neural differential equations on time series](https://mlsanity.com/paper/2109.06786/) (2021)
- [Learning Partially Known Stochastic Dynamics with Empirical PAC Bayes](https://mlsanity.com/paper/2006.09914/) (2020)
- [NeuralFMU: Towards Structural Integration of FMUs into Neural Networks](https://mlsanity.com/paper/2109.04351/) (2021)
- [Hybrid modeling of the human cardiovascular system using NeuralFMUs](https://mlsanity.com/paper/2109.04880/) (2021)

## Built on by (most cited first, 3 of 3)

- [Eigen-informed NeuralODEs: Dealing with stability and convergence issues of NeuralODEs](https://mlsanity.com/paper/2302.10892/) (2023)
- [Learnable & Interpretable Model Combination in Dynamical Systems Modeling](https://mlsanity.com/paper/2406.08093/) (2024)
- [Efficient Training of Physics-enhanced Neural ODEs via Direct Collocation and Nonlinear Programming](https://mlsanity.com/paper/2505.03552/) (2025)

## Abstract (arXiv)

The term NeuralODE describes the structural combination of an Artifical Neural Network (ANN) and a numerical solver for Ordinary Differential Equations (ODEs), the former acts as the right-hand side of the ODE to be solved. This concept was further extended by a black-box model in the form of a Functional Mock-up Unit (FMU) to obtain a subclass of NeuralODEs, named NeuralFMUs. The resulting structure features the advantages of first-principle and data-driven modeling approaches in one single simulation model: A higher prediction accuracy compared to conventional First Principle Models (FPMs), while also a lower training effort compared to purely data-driven models. We present an intuitive workflow to setup and use NeuralFMUs, enabling the encapsulation and reuse of existing conventional models exported from common modeling tools. Moreover, we exemplify this concept by deploying a NeuralFMU for a consumption simulation based on a Vehicle Longitudinal Dynamics Model (VLDM), which is a typical use case in automotive industry. Related challenges that are often neglected in scientific use cases, like real measurements (e.g. noise), an unknown system state or high-frequent discontinuities, are handled in this contribution. For the aim to build a hybrid model with a higher prediction quality than the original FPM, we briefly highlight two open-source libraries: FMI.jl for integrating FMUs into the Julia programming environment, as well as an extension to this library called FMIFlux.jl, that allows for the integration of FMUs into a neural network topology to finally obtain a NeuralFMU.