Generalized Quadratic-Embeddings for Nonlinear Dynamics using Deep Learning

The engineering design process (e.g., control and forecasting) relies on mathematical modeling, describing the underlying dynamic behavior. For complex dynamics behavior, modeling procedures, as well as models, can be intricated, which can make the design process cumbersome. Therefore, it is desirable to have a common model structure, which is also simple enough, for all nonlinear dynamics to enhance design processes. The simplest dynamical model -- one can think of -- is linear, but linear models are often not expressive enough to apprehend complex dynamics. In this work, we propose a modeling approach for nonlinear dynamics and discuss a common framework to model nonlinear dynamic processes, which is built upon a \emph{lifting-principle}. The preeminent idea of the principle is that smooth nonlinear systems can be written as quadratic systems in an appropriate lifted coordinate system without any approximation error. Hand-designing these coordinates is not straightforward. In this work, we utilize deep learning capabilities and discuss suitable neural network architectures to find such a coordinate system using data. We present innovative neural architectures and the corresponding objective criterion to achieve our goal. We illustrate the approach using data coming from applications in engineering and biology.