Sparsifying priors for Bayesian uncertainty quantification in model discovery (UQ-SINDY)
We propose a probabilistic model discovery method for identifying ordinary differential equations governing the dynamics of observed multivariate data. Our method is based on the sparse identification of nonlinear dynamics (SINDy) framework, where models are expressed as sparse linear combinations of pre-specified candidate functions. Instead of targeting point estimates of the SINDy coefficients, we estimate these coefficients via sparse Bayesian inference. The resulting method, uncertainty quantification SINDy (UQ-SINDy), quantifies not only the uncertainty in the values of the SINDy coefficients due to observation errors and limited data, but also the probability of inclusion of each candidate function in the linear combination. UQ-SINDy is shown to discover accurate models in the presence of noise and with orders-of-magnitude less data than current model discovery methods, thus providing a transformative method for real-world applications which have limited data. A link to this work can be found here.
Structured time-delay models for dynamical systems with connections to Frenet–Serret frame (SHAVOK)
Time-delay embedding and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal. In this paper, we establish a new theoretical connection between HAVOK and the Frenet–Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in the Frenet–Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. A link to this work can be found here.
Centering Improves the Dynamic Mode Decomposition
Originating in the fluids community, DMD has been used in a wide variety of fields with applications to neuroscience, climate science, and finance to name a few. Given spatiotemporal data, DMD decomposes the data into pairs of spatial temporal modes with the assumption that the dynamics are approximately linear, i.e. it is a regression to a best fit linear dynamical model. For many systems of interest, the dynamics we want to model are perturbations about equilibria, in which case subtracting the mean of the data is a natural preprocessing step. In this work, we prove that centering data improves the performance of DMD, and can be used to successfully extract dynamics about equilibria. This is a particularly important and surprising result, since previous work suggested that computing DMD on centered data may be restrictive and have undesirable consequences. The results are illustrated on the Lorenz attractor, in addition to applications to neural recordings and video surveillance. A link to this work can be found here.
Spatiotemporal Intrinsic Mode Decomposition (STIMD)
The analysis of spatiotemporal signals is of critical importance for characterizing emerging large-scale measurements in wide variety of scientific and engineering applications. Many methods have been developed to discover a low dimensional representation of these data. However, many widely used methods such as PCA do not incorporate the temporal structure of the data into the model. In this work we develop a new dimensionality reduction technique, known as the spatiotemporal intrinsic mode decomposition (STIMD) method, which factors spatiotemporal data into a product of spatial modes and temporal modes. Our method allows us to perform instantaneous time frequency analysis by computing a Hilbert transform of the data; in addition, it is possible to make future-state predictions of the spatiotemporal system. A link to this work can be found here.