Nonlinear Model Reduction for Complex Systems using Sparse Optimal Sensor Locations from Learned Nonlinear Libraries

TL;DR

This paper combines sparse sampling and machine learning to develop low-dimensional models for complex nonlinear systems, enabling efficient characterization, stability analysis, and bifurcation classification with nearly optimal sensor placement.

Contribution

It introduces a method to construct nonlinear libraries and optimal sensor locations for local reduced-order models, improving interpretability and robustness in nonlinear system analysis.

Findings

Sparse sampling effectively captures system dynamics.

Nonlinear measurements outperform linear ones under noise.

Local models better represent different physical regimes.

Abstract

We demonstrate the synthesis of sparse sampling and machine learning to characterize and model complex, nonlinear dynamical systems over a range of bifurcation parameters. First, we construct modal libraries using the classical proper orthogonal decomposition to uncover dominant low-rank coherent structures. Here, nonlinear libraries are also constructed in order to take advantage of the discrete empirical interpolation method and projection that allows for the approximation of nonlinear terms in a low-dimensional way. The selected sampling points are shown to be nearly optimal sensing locations for characterizing the underlying dynamics, stability, and bifurcations of complex systems. The use of empirical interpolation points and sparse representation facilitate a family of local reduced-order models for each physical regime, rather than a higher-order global model, which has the benefit of physical interpretability of energy transfer between coherent structures. In particular, the discrete interpolation points and nonlinear modal libraries are used for sparse representation to classify the dynamic bifurcation regime in the complex Ginzburg-Landau equation. It is shown that nonlinear point measurements are more effective than linear measurements when sensor noise is present.