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

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