Deep learning of normal form autoencoders for universal, parameter-dependent dynamics

A long-standing goal in dynamical systems is to construct reduced-order models for high-dimensional spatiotemporal data that capture the underlying parametric dependence of the data. In this work, we introduce a deep learning technique to discover a single low-dimensional model for such data that captures the underlying parametric dependence in terms of a normal form. A normal form is a symbolic expression, or universal unfolding, that describes how a the reduced-order differential equation model varies with respect to a bifurcation parameter. Our approach introduces coupled autoencoders for the state and parameter, with the latent variables constrained to adhere to a given normal form. We demonstrate our method on one-parameter bifurcations that occur in the canonical Lorenz96 equations and a neural field equation. This method demonstrates how normal forms can be leveraged as canonical and universal building blocks in deep learning approaches for model discovery and reduced-order modeling.