Machine-Learning-Based investigations in Complex Systems: Droplet Coalescence, Ising-type models, and Mathematical Models for Cardiac Tissue

Abstract

The importance of artificial intelligence and data driven machine learning is growing exponentially in time as are its applications in investigations of complex phenomena in, e.g., climate-systems science, fluid flows, phase transitions and biological systems, to name but a few. Machine learning (ML) models, such as deep neural networks, are increasingly being used to analyse extensive datasets and to increase accuracy in classification, prediction, dimensionality reduction, modelling, etc. In this thesis, we carry out machine learning-based investigations in a variety of complex systems. We begin with a brief introduction to the mathematical models of liquid-droplet coalescence, phase transitions, and cardiac tissue, as well as the ML models and methods employed in this thesis. Next, we discuss an application of ML to reconstruct flow fields from concentration fields in the context of liquid-droplet coalescence, a problem of significant practical and theoretical interest in fluid dynamics and the statistical mechanics of multiphase flows. We demonstrate that two- dimensional (2D) encoder-decoder convolutional neural networks (CNNs), 2D U-Nets, and three- dimensional (3D) U-Nets can be used to obtain flow fields from concentration fields; here, we conduct investigations using data from 2D and 3D Cahn-Hilliard-Navier-Stokes (CHNS) partial differential equations (PDEs). We then use data from recent experiments on droplet coalescence to illustrate how our method can be applied to obtain the flow field from measurements of the concentration field. We then investigate the phase transitions in Ising spin models with ML. In particular, we combine machine learning (ML) techniques with Monte Carlo (MC) simulations and finite-size scaling (FSS) to study continuous and first-order phase transitions in Ising, Blume-Capel, and Ising-metamagnet spin models. We go beyond earlier studies by demonstrating how to combine neural networks (NNs), trained with data from MC simulations of Ising-type spin models on finite lattices, with FSS to: (a) obtain both thermal and magnetic exponents, respectively, at both critical and tricritical points; (b) derive the NN counterpart of two-scale-factor universality at an Ising-type critical point; and (c) analyze the FSS at a first-order transition. We also obtain the FSS forms for the output of our trained NNs as functions of both temperature and magnetic field. Finally, we explore applications of ML to three different problems in the context of cardiac healthcare. Here, we first investigate the elimination of spiral-wave turbulence in mathematical models for cardiac tissue that utilize partial differential equations, alongside deep learning models. Such numerical models for cardiac tissue admit solutions with spiral- or broken-spiral-wave patterns, which are the mathematical counterparts of ventricular tachycardia (VT) and ventricular fibrillation (VF). These conditions can precipitate sudden cardiac death (SCD), which is the leading cause of mortality in the industrialized world. Secondly, we discuss the prediction of spiral wave tips in the Aliev-Panfilov model for cardiac tissue, using pseudo-ECGs in conjunction with Long Short-Term Memory (LSTM) networks. We demonstrate that our LSTM-based tip-tracking compares favorably with the Iyer-Gray method, which requires the full spatiotemporal evolution of spiral waves to obtain tip trajectories. Here, we also explore predictions with noise and ensemble-based suppression of outliers. Thirdly, we develop a deep-learning-based algorithm to predict the probability of recovery of a comatose patient who has suffered a heart attack by analyzing electroencephalogram (EEG) and electrocardiogram (ECG) data. From hour-long traces for each patient, we extract the associated metrics and use them in combinations with CNNs and LSTM networks to make predictions of the probability of recovery, specifically concerning their Cerebral Performance Category (CPC).