Koopman Reduced Order Modeling with Confidence Bounds

TL;DR

Koopman Reduced Order Modeling with Confidence Bounds introduces a data-driven ROM that splits the Koopman evolution into a deterministic modal reconstruction and a stochastic residual, enabling distributional forecasts with confidence bounds. By modeling the residual as a Gaussian modal component within the retained subspace and an orthogonal innovation, the method constructs prediction intervals around the deterministic prediction and provides a criterion to select the minimum number of modes via a normality test. The approach is demonstrated on a linear modal system, a network of switched anharmonic oscillators, and a noisy Kuramoto model, showing that the true trajectories largely lie within the predicted bounds and that the modal count governs the bias-variance trade-off. This work offers a practical uncertainty quantification framework for data-driven Koopman ROMs, with potential applications to systems experiencing changing topology and stochastic perturbations.

Abstract

This paper introduces a reduced order modeling technique based on Koopman operator theory that gives confidence bounds on the model's predictions. It is based on a data-driven spectral decomposition of the Koopman operator. The reduced order model is constructed using a finite number of Koopman eigenvalues and modes, while the rest of spectrum is treated as a noise process. This noise process is used to extract the confidence bounds. Additionally, we propose a heuristic algorithm to choose the number of deterministic modes to keep in the model.

Figures (19)

• Figure 1: Linear Modal Model Eigenvalues: Eigenvalues and relative mode weights for 50-mode ROM with various levels of noise. Modes and eigenvalues are ordered by their weights computed via \ref{['eq:mode-weights']}. The relative strength of a mode in the ROM is determined by \ref{['eq:model-reconstruction-coeff']} and is shown in the plots. ROM eigenvalues' (red) opacity is determined by the relative strength of the corresponding mode. The observable was a delay-embedding of the state vector with a delay of 300. When the noise is zero (a), the true eigenvalues are captured and the extra eigenvalues have (numerically) zero weight. With increasing the levels of noise, the true eigenvalues are still captured, but some of the extra eigenvalues have corresponding modes with non-trivial weights. As the noise increases, (b), (c), (d), the number of modes with non-trivial weight increases to capture the stochasticity in the underlying model.
• Figure 2: Linear Modal Model Noise Distributions: Noise distributions in the first coordinate for a 50-mode ROM. Standard deviation of $\xi$ in \ref{['eq:linear-modal-model']} is 0.25. The observable was a delay-embedding of the state vector with a delay of 300. (a) Modal noise. Green is the histogram, orange is a fitted Gaussian distribution, and pink is a kernel density approximation. (b) Innovation noise. Blue is the histogram, orange is a fitted Gaussian distribution, and pink is a kernel density approximation.
• Figure 3: Linear Modal Model Reconstruction: Comparison of the 50-mode projected ROM and confidence bounds (eq.'s \ref{['eq:projected-ROM-woth-modal-noise']} and \ref{['eq:ROM-prediction-interval']}) versus the true signal for the first coordinate. Standard deviation of $\xi$ in \ref{['eq:linear-modal-model']} is 0.25. The true signal is given by the blue trace, the ROM reconstruction is the orange trace, and the green band is $\pm$2 standard deviations of the modal noise. The green band represent the 95% confidence interval for the ROM reconstruction. The top trace is the real part of the signal. The bottom trace is the imaginary part.
• Figure 4: Linear Modal Model Error and Residence Times: Standard deviation of $\xi$ in \ref{['eq:linear-modal-model']} is 0.25. For both the error and the residence times, the metrics were computed for each coordinate and then averaged over coordinates. (a) Error vs. number of modes in the ROM. (b) Fraction of time the true signal is in the confidence bounds vs. number of modes in the ROM. The error never drops too low (minimum at 40 modes) suggesting that the point prediction of the ROM is never that accurate. However, the residence times show a sharp transition and plateau around 50 modes suggesting that 50 modes capture the "coherent" part of the dynamics well and the rest of the dynamics are well-modeled by a stochastic term.
• Figure 5: Anharmonic Oscillators. Computed Koopman eigenvalues for the switched anharmonic oscillator model with added added Gaussian noise $(\sim N(0, 0.05^2)$. The eigenvalues of lower-order models are always a subset of those of higher-order models.