NeuroKoopman Dynamic Causal Discovery

In many real-world applications where the system dynamics has an underlying interdependency among its variables (such as power grid, economics, neuroscience, omics networks, environmental ecosystems, and others), one is often interested in knowing whether the past values of one time series influences the future of another, known as Granger causality, and the associated underlying dynamics. This paper introduces a Koopman-inspired framework that leverages neural networks for data-driven learning of the Koopman bases, termed NeuroKoopman Dynamic Causal Discovery (NKDCD), for reliably inferring the Granger causality along with the underlying nonlinear dynamics. NKDCD employs an autoencoder architecture that lifts the nonlinear dynamics to a higher dimension using data-learned bases, where the lifted time series can be reliably modeled linearly. The lifting function, the linear Granger causality lag matrices, and the projection function (from lifted space to base space) are all represented as multilayer perceptrons and are all learned simultaneously in one go. NKDCD also utilizes sparsity-inducing penalties on the weights of the lag matrices, encouraging the model to select only the needed causal dependencies within the data. Through extensive testing on practically applicable datasets, it is shown that the NKDCD outperforms the existing nonlinear Granger causality discovery approaches.

Data-Driven Linear Koopman Embedding for Model-Predictive Power System Control

This paper presents a linear Koopman embedding for model predictive emergency voltage regulation in power systems, by way of a data-driven lifting of the system dynamics into a higher dimensional linear space over which the MPC (model predictive control) is exercised, thereby scaling as well as expediting the MPC computation for its real-time implementation for practical systems. We develop a {\em Koopman-inspired deep neural network} (KDNN) architecture for the linear embedding of the voltage dynamics subjected to reactive controls. The training of the KDNN for the purposes of linear embedding is done using the simulated voltage trajectories under a variety of applied control inputs and load conditions. The proposed framework learns the underlying system dynamics from the input/output data in the form of a triple of transforms: A Neural Network (NN)-based lifting to a higher dimension, a linear dynamics within that higher dynamics, and an NN-based projection to original space. This approach alleviates the burden of an ad-hoc selection of the basis functions for the purposes of lifting to higher dimensional linear space. The MPC is computed over the linear dynamics, making the control computation scalable and also real-time.

Noise-robust Clustering

This paper presents noise-robust clustering techniques in unsupervised machine learning. The uncertainty about the noise, consistency, and other ambiguities can become severe obstacles in data analytics. As a result, data quality, cleansing, management, and governance remain critical disciplines when working with Big Data. With this complexity, it is no longer sufficient to treat data deterministically as in a classical setting, and it becomes meaningful to account for noise distribution and its impact on data sample values. Classical clustering methods group data into "similarity classes" depending on their relative distances or similarities in the underlying space. This paper addressed this problem via the extension of classical $K$-means and $K$-medoids clustering over data distributions (rather than the raw data). This involves measuring distances among distributions using two types of measures: the optimal mass transport (also called Wasserstein distance, denoted $W_2$) and a novel distance measure proposed in this paper, the expected value of random variable distance (denoted ED). The presented distribution-based $K$-means and $K$-medoids algorithms cluster the data distributions first and then assign each raw data to the cluster of data's distribution.