Learning Low-Dimensional Quadratic-Embeddings of High-Fidelity Nonlinear Dynamics using Deep Learning

TL;DR

This paper introduces a deep learning approach to find low-dimensional quadratic embeddings of high-dimensional nonlinear dynamical systems, enabling simpler models for complex processes like flow dynamics and reactors.

Contribution

It proposes a novel autoencoder-based method combined with Runge-Kutta integration to identify low-dimensional quadratic models from high-fidelity data.

Findings

Successfully applied to flow dynamics example

Effectively captures oscillatory reactor behavior

Reduces complexity of high-dimensional models

Abstract

Learning dynamical models from data plays a vital role in engineering design, optimization, and predictions. Building models describing dynamics of complex processes (e.g., weather dynamics, or reactive flows) using empirical knowledge or first principles are onerous or infeasible. Moreover, these models are high-dimensional but spatially correlated. It is, however, observed that the dynamics of high-fidelity models often evolve in low-dimensional manifolds. Furthermore, it is also known that for sufficiently smooth vector fields defining the nonlinear dynamics, a quadratic model can describe it accurately in an appropriate coordinate system, conferring to the McCormick relaxation idea in nonconvex optimization. Here, we aim at finding a low-dimensional embedding of high-fidelity dynamical data, ensuring a simple quadratic model to explain its dynamics. To that aim, this work leverages deep learning to identify low-dimensional quadratic embeddings for high-fidelity dynamical systems. Precisely, we identify the embedding of data using an autoencoder to have the desired property of the embedding. We also embed a Runge-Kutta method to avoid the time-derivative computations, which is often a challenge. We illustrate the ability of the approach by a couple of examples, arising in describing flow dynamics and the oscillatory tubular reactor model.