Dynamic mode decomposition for multiscale nonlinear physics

We present a data-driven method for separating complex, multiscale systems into their constituent timescale components using a recursive implementation of dynamic mode decomposition (DMD). Local linear models are built from windowed subsets of the data, and dominant timescales are discovered using spectral clustering on their eigenvalues. This approach produces time series data for each identified component, which sum to a faithful reconstruction of the input signal. It differs from most other methods in the field of multiresolution analysis (MRA) in that it (1) accounts for spatial and temporal coherencies simultaneously, making it more robust to scale overlap between components, and (2) yields a closed-form expression for local dynamics at each scale, which can be used for short-term prediction of any or all components. Our technique is an extension of multi-resolution dynamic mode decomposition (mrDMD), generalized to treat a broader variety of multiscale systems and more faithfully reconstruct their isolated components. In this paper we present an overview of our algorithm and its results on two example physical systems, and briefly discuss some advantages and potential forecasting applications for the technique.

Figure 1

Sliding a window filter across a longer time-series data set. For each step of the window's movement, a new DMD result is obtained. Note that the width of the window is such that it contains multiple complete oscillations of the fast scale but only a fraction of a period of the slow scale.

Figure 2

Spectra of the (modulus squared) frequencies obtained by the sliding-window DMD procedure. Frequencies are plotted at the midpoints of the windows from which they were computed. Colors denote the cluster labels assigned to each point retroactively.

Figure 3

Histogram of all $|{\omega }_j^k{|}^2$ with the $k$-medians cluster centroids overlaid in color. Note that the outliers visible in Fig. 2 are vastly outnumbered by in-band data points.

Figure 4

DMD reconstructions ${\overline{\mathbf{x}}}^k\left(t\right)$ plotted over input data $x\left(t\right)$ for four nonoverlapping windows (delineated by dotted lines).

Figure 5

Global DMD reconstruction ${\overline{\mathbf{x}}}^{\text{global}}\left(t\right)$ plotted over the input data $x\left(t\right).$

Figure 6

Scale separated reconstructions ${\overline{\mathbf{x}}}_{\text{slow}}^{\text{global}}\left(t\right)$ (blue) and ${\overline{\mathbf{x}}}_{\text{fast}}^{\text{global}}\left(t\right)$ (red) plotted over the input data $x\left(t\right)$ (black).

Figure 7

Power spectra of the input signal (black) and fast- and slow-scale reconstructions (red and blue, respectively). The four variables of each signal are summed to compute frequency content.

Figure 8

Comparison of two scale separation techniques. The measurement signal (top) is constructed from two components, generated by the FitzHugh-Nagumo (center) and Duffing (bottom) models. The ground-truth signal separation is plotted in black, with the results of the two data-driven methods overlaid in color. The sliding-window DMD approach is much more successful in recovering the true components.

Figure 9

Histogram of frequency results of sliding window DMD on the three-body planetary system. $k$-medians cluster centroids ($k=3$) are overlaid in color.

Figure 10

Scale-separated reconstructions of three components (color) overlaid on the Cartesian input data (black). Note that for the short domain plotted (150 years), the slow-scale component in blue looks like a constant.

Figure 11

The isolated slow-scale component obtained by the first pass of sliding-window DMD on the three-body planetary system. Plotted in blue as Component 1 in Fig. 10, the full scope of its dynamics is revealed over this longer time domain. The multiscale behavior evident within this single component motivates our recursive approach to scale separation.

Figure 12

Scale-separated reconstructions obtained from four recursive applications of the sliding-window DMD procedure to the three-body planetary system. Dotted lines are used to indicate the relative timescales between each successive recursion. Only the $x$ coordinates are plotted. Solid lines denote $x_{\text{Jupiter}}$ and dot-dashed lines denote $x_{\text{Saturn}}$.

Figure 13

The full reconstruction to the three-body planetary system obtained by summing the scale-separated components ($x_{\text{Jupiter}}$ in solid green, $x_{\text{Saturn}}$ in dot-dashed green) plotted over the original simulation data (black). Each plot here contains the same result, visualized over progressively longer timespans. In the last two plots, only the upper-bound envelope has been plotted for ease of visualization. The final plot shows moving averages to show the conformity even over the longest timescales.

Figure 14

Keplerian orbital elements for the orbit of Jupiter, computed from the Cartesian simulation data of the three-body planetary system. The three plots show the same data over three distinct timescales.

Figure 15

Ensemble componentwise forecasts from the sliding-window DMD model on the three-body planetary system. The dotted vertical lines represent the “current time”; everything to their right is the forecasted trajectory. At each scale, the unperturbed DMD forecast (black) is overlaid by an ensemble of predictions generated by sampling $b_j^k$, ${\mathit{\varphi }}_j^k$, and ${\omega }_j^k$ according to their statistics in the preceding windowed iterations (constraining all ${\mathit{\varphi }}_j^k$ to maintain unit length). Note that these trajectories are all nearly identical in the fast-scale components, where the variance of DMD parameters is minimal.