2025-07-25 –, Main Room 3
Universal Differential Equations (or UDEs for short) have emerged as a novel way to integrate information
from experimental data into mechanistic models. One of the most important questions when using UDEs
is what equations to modify in order to add the effects of a neural network. In this talk we will
explore what kind of corrections we can make based on the architecture of the UDE.
In this talk we will discuss about how to augment acasual models in the ModelingToolkit framework and about how does the architecture impact the kind of corrections we can make.
The initial paper on UDEs [1], a methodology is described for systems of differential equations, but in the context of acasual models we start with sets of differential algebraic equations (DAEs) for each component and then a simplification algorithm condenses them in a final system of DAEs that will be solved.
This two stage nature of the process naturally divides the possible UDE architectures in two: before structural simplification, where we have we deal with individual components and after simplification where we only have the final simplified system without a direct link to the individual components. We will call these component level UDEs and system level UDEs.
In the case of component level UDEs, we can describe the neural network correction via components too. The ModelingToolkitNeuralNets.jl package provides a block component that abstracts a Lux.jl neural network. This block componet can be then used as any other component from the ModelingToolkitStandardLibrary.
In the case of system level UDEs we only have access to the final system of DAEs generated by structural
simplification and thus the corrections cannot be expressed at the symbolic level. As such, it can be harder to decide what equations to change with the neural network. This talk will present some approaches based on sensitivity analysis that can help us decide what equations to target.
The presentation will also showcase how JuliaSim can be used to easily build UDEs.
[1] C. Rackauckas et al., “Universal Differential Equations for Scientific Machine Learning,” arXiv:2001.04385 [cs, math, q-bio, stat], Jan. 2020, Accessed: Feb. 09, 2020. [Online]. Available: http://arxiv.org/abs/2001.04385
Fredrik received his MSc and Ph.D. from Dept. Automatic Control at Lund University. He has a background in robotics and an interest in developing software tools for control, identification, and simulation, and is a co author of the JuliaControl suite of software for analysis and design of control systems in the Julia programming language. He is currently leading the control-systems team at JuliaHub, developing tools for model-based design and deployment of control systems.
Software Eng. at JuliaHub & PhD student at University of Bucharest.