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This talk was part of SciMLCon 2022! For more information, check out https://scimlcon.org/2022/. For more information on the SciML Open Source Software Organization for Scientific Machine Learning, check out https://sciml.ai/. Learning measured bifurcation diagrams with UDEs | Sandor Beregi | SciMLCon 2022 Abstract: Nonlinear dynamical systems are frequently characterised by their bifurcation diagrams indicating the steady-state response as certain system parameters vary. This often involves multiple solutions corresponding to a single parameter-combination. We use universal differential equations (UDEs) to capture the dynamics accurately in such challenging parameter-domains whereas we also propose a boundary-value-problem-based approach to improve the robustness of the training procedure. Description: Nonlinear dynamical systems are often used to model vibrational phenomena in engineering e.g. machine-tool vibrations, aeroelastic flutter or wheel-shimmy. These systems are frequently characterised by their bifurcation diagrams indicating the steady-state response as one or more system parameters, referred to as bifurcation parameters, vary. By having information about the stability of the steady-state solutions, the bifurcation diagrams can be used as an indicator for the transient behaviour too. The structure of the steady-state solutions however is often complex, potentially with multiple solutions corresponding to a single parameter-combination, often making it challenging to find an accurate numerical model. Universal differential equations (UDEs) are a promising approach to include measurement data from physical structures with machine-learnt structures inside differential equations alongside algebraic terms representing mechanistic models. By having a physics-based core, UDE models can incorporate the insight and expertise one has on the behaviour of the modelled structure. The purpose of using machine-learnable structures with the mechanistic model is to compensate the error between the observed and predicted behaviours resulting in not just a qualitatively but also quantitatively representative model of the physical structure. Most of the studies on UDEs however focus on identifying a well-fitting model with a constant set of system parameters while it is often also interesting to investigate the system's response to varying parameters, e.g., to allow for parameter-uncertainty, or to incorporate the change of external conditions. By focusing on nonlinear systems with limit cycles, we consider inherently varying parameter problems. Capturing the dynamics accurately in the challenging parameter-domains with multiple co-existing steady-state solutions is stretching the boundaries of the current training framework. On the examples of aeroelastic flutter and a forced nonlinear oscillator, we carry out an exploratory study using the UDE modelling approach to capture the observed bifurcation diagrams of these systems. To consider problems of different complexity, both numerical models and physical experiments are used to generate training data. Using neural networks and Gaussian processes as the machine-learnt part of the model using the DiffEqFlux.jl and GaussianProcesses.jl packages in Julia, we train UDE models of the observed nonlinear systems using trajectories corresponding to steady-state solutions as reference. We demonstrate that this modelling approach has a great potential in delivering accurate models of nonlinear dynamical systems whereas we also report on the challenges encountered during the training procedure, such as overfitting and finding local minima of the objective function and discuss the potential ways of avoiding these issues. The overall aim of the project is to find ways of incorporating the heuristic structure of the bifurcation diagrams, which the current training framework is oblivious to, into the training procedure. As a longer-term goal, we envision a boundary-value-problem-based training algorithm relying on the techniques of numerical bifurcation analysis which could increase the robustness of the training procedure for nonlinear dynamical systems. For more info on the Julia Programming Language, follow us on Twitter: / julialanguage and consider sponsoring us on GitHub: https://github.com/sponsors/JuliaLang 00:00 Welcome! 00:10 Help us add time stamps or captions to this video! See the description for details. Want to help add timestamps to our YouTube videos to help with discoverability? Find out more here: https://github.com/JuliaCommunity/You... Interested in improving the auto generated captions? Get involved here: https://github.com/JuliaCommunity/You...