Improving Predictive Performance of Low-Order Models Using Bounded Trigonometric Closures

Low-order models of high-dimensional systems are useful in many settings requiring prediction, analysis, and control. However, predictive fidelity in low-order models can be compromised due to the inherent truncation of higher-order modes required to construct such models, either directly from the governing equations or using data-driven techniques. Previous studies have proposed the use of quadratic and higher-order polynomial closures to account for these truncated modes without increasing the model order. While these closures have been shown to improve predictive accuracy over short time periods, the underlying models are not guaranteed to be bounded and can result in a deterioration in predictive capability over longer time horizons. In this work, we propose a trigonometric closure model that improves predictive performance over long time periods. Trigonometric closures are formed as pairs of sine and cosine functions acting on appropriately weighted state variables, enabling prediction over a wide range of frequencies as needed to model the truncated modes. We mathematically prove that introducing the proposed trigonometric closure to a stable linear model of the system dynamics guarantees long-time boundedness of the resulting model. We demonstrate the utility of the trigonometric closures for improving a low-order linear model on two test cases: (i) a transient cylinder wake, and (ii) the periodic vortex shedding of a cylinder wake. The resulting models are found to improve predictive accuracy and maintain boundedness over hundreds of convective time units.