Applications and Machine-Learning Prediction of Nonlinear Dynamical Systems

Predicting nonlinear dynamical systems has been a long-standing challenge in science. This field is currently witnessing a revolution with the advent of machine learning methods. Concurrently, the analysis of dynamics in various nonlinear complex systems continues to be crucial. Guided by these directions, I conduct the following studies. Predicting critical transitions and transient states in nonlinear dynamics is a complex problem. I developed a solution called parameter-aware reservoir computing, which uses machine learning to track how system dynamics change with a driving parameter. I show that the transition point can be accurately predicted while trained in a sustained functioning regime before the transition. Notably, it can also predict if the system will enter a transient state, the distribution of transient lifetimes, and their average before a final collapse, which are crucial for management. I introduce a machine-learning-based digital twin for monitoring and predicting the evolution of externally driven nonlinear dynamical systems, where reservoir computing is exploited. Extensive tests on various models, encompassing optics, ecology, and climate, verify the approach’s effectiveness. The digital twins can extrapolate unknown system dynamics, continually forecast and monitor under non-stationary external driving, infer hidden variables, adapt to different driving waveforms, and extrapolate bifurcation behaviors across varying system sizes. Integrating engineered gene circuits into host cells poses a significant challenge in synthetic biology due to circuit-host interactions, such as growth feedback. I conducted systematic studies on hundreds of circuit structures exhibiting various functionalities, and identified a comprehensive categorization of growth-induced failures. I discerned three dynamical mechanisms behind these circuit failures. Moreover, my comprehensive computations reveal a scaling law between the circuit robustness and the intensity of growth feedback. A class of circuits with optimal robustness is also identified. Chimera states, a phenomenon of symmetry-breaking in oscillator networks, traditionally have transient lifetimes that grow exponentially with system size. However, my research on high-dimensional oscillators leads to the discovery of ’short-lived’ chimera states. Their lifetime increases logarithmically with system size and decreases logarithmically with random perturbations, indicating a unique fragility. To understand these states, I use a transverse stability analysis supported by simulations.