Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equations
This work addresses a long-standing problem in dynamical systems for researchers, offering a data-driven approach to model discovery and reduced-order modeling, though it appears incremental as it builds on existing normal form theory with deep learning.
The authors tackled the challenge of discovering universal, parameter-dependent governing equations for complex systems by introducing deep learning autoencoders that learn coordinate transformations to capture canonical normal forms, demonstrating the method on example problems with various bifurcations.
Complex systems manifest a small number of instabilities and bifurcations that are canonical in nature, resulting in universal pattern forming characteristics as a function of some parametric dependence. Such parametric instabilities are mathematically characterized by their universal un-foldings, or normal form dynamics, whereby a parsimonious model can be used to represent the dynamics. Although center manifold theory guarantees the existence of such low-dimensional normal forms, finding them has remained a long standing challenge. In this work, we introduce deep learning autoencoders to discover coordinate transformations that capture the underlying parametric dependence of a dynamical system in terms of its canonical normal form, allowing for a simple representation of the parametric dependence and bifurcation structure. The autoencoder constrains the latent variable to adhere to a given normal form, thus allowing it to learn the appropriate coordinate transformation. We demonstrate the method on a number of example problems, showing that it can capture a diverse set of normal forms associated with Hopf, pitchfork, transcritical and/or saddle node bifurcations. This method shows how normal forms can be leveraged as canonical and universal building blocks in deep learning approaches for model discovery and reduced-order modeling.