# Learnable & Interpretable Model Combination in Dynamical Systems Modeling

arXiv: https://arxiv.org/abs/2406.08093 · HTML: https://mlsanity.com/paper/2406.08093/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 12 Jun 2024.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [ModelingToolkit: A Composable Graph Transformation System For Equation-Based Modeling](https://mlsanity.com/paper/2103.05244/) (2021)
- [NeuralFMU: Presenting a workflow for integrating hybrid NeuralODEs into real world applications](https://mlsanity.com/paper/2209.03933/) (2022)
- [Hybrid modeling of the human cardiovascular system using NeuralFMUs](https://mlsanity.com/paper/2109.04880/) (2021)
- [Hybrid Modeling Design Patterns](https://mlsanity.com/paper/2401.00033/) (2023)

## Built on by (most cited first, 1 of 1)

- [Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximations](https://mlsanity.com/paper/2410.10174/) (2024)

## Abstract (arXiv)

During modeling of dynamical systems, often two or more model architectures are combined to obtain a more powerful or efficient model regarding a specific application area. This covers the combination of multiple machine learning architectures, as well as hybrid models, i.e., the combination of physical simulation models and machine learning. In this work, we briefly discuss which types of model are usually combined in dynamical systems modeling and propose a class of models that is capable of expressing mixed algebraic, discrete, and differential equation-based models. Further, we examine different established, as well as new ways of combining these models from the point of view of system theory and highlight two challenges - algebraic loops and local event functions in discontinuous models - that require a special approach. Finally, we propose a new wildcard architecture that is capable of describing arbitrary combinations of models in an easy-to-interpret fashion that can be learned as part of a gradient-based optimization procedure. In a final experiment, different combination architectures between two models are learned, interpreted, and compared using the methodology and software implementation provided.