  This work explores the physics-driven machine learning technique Operator
Inference (OpInf) for predicting the state of chaotic dynamical systems. OpInf
provides a non-intrusive approach to infer approximations of polynomial
operators in reduced space without having access to the full order operators
appearing in discretized models. Datasets for the physics systems are generated
using conventional numerical solvers and then projected to a low-dimensional
space via Principal Component Analysis (PCA). In latent space, a least-squares
problem is set to fit a quadratic polynomial operator, which is subsequently
employed in a time-integration scheme in order to produce extrapolations in the
same space. Once solved, the inverse PCA operation is applied to reconstruct
the extrapolations in the original space. The quality of the OpInf predictions
is assessed via the Normalized Root Mean Squared Error (NRMSE) metric from
which the Valid Prediction Time (VPT) is computed. Numerical experiments
considering the chaotic systems Lorenz 96 and the Kuramoto-Sivashinsky equation
show promising forecasting capabilities of the OpInf reduced order models with
VPT ranges that outperform state-of-the-art machine learning methods such as
backpropagation and reservoir computing recurrent neural networks [1], as well
as Markov neural operators [2].
