Chaotic dynamics are ubiquitous in many real-world systems, ranging from biological and industrial processes to climate dynamics and the spread of viruses. These systems are characterized by high sensitivity to initial conditions, making it challenging to predict their future behavior confidently. In this study, we propose a novel deep-learning framework that addresses this challenge by directly exploiting the long-term compounding of local prediction errors during model training, aiming to extend the time horizon for reliable predictions of chaotic systems. Our approach observes the future trajectories of initial errors at a time horizon, modeling the evolution of the loss to that point through the use of two major components: (1) a recurrent architecture (Error Trajectory Tracing) designed to trace the trajectories of predictive errors through phase space, and (2) a training regime, Horizon Forcing, that pushes the model’s focus out to a predetermined time horizon. We validate our method on three classic chaotic systems and six real-world time series prediction tasks with chaotic characteristics. The results show that our approach outperforms the state-of-the-art methods.

The understanding of chaotic systems is challenging not only for theoretical research but also for many important applications. Chaotic behavior is found in many nonlinear dynamical systems, such as those found in climate dynamics, weather, the stock market, and the space-time dynamics of virus spread. A reliable solution for these systems must handle their complex space-time dynamics and sensitive dependence on initial conditions. We develop a deep learning framework to push the time horizon at which reliable predictions can be made further into the future by better evaluating the consequences of local errors when modeling nonlinear systems. Our approach observes the future trajectories of initial errors at a time horizon to model the evolution of the loss to that point with two major components: 1) a recurrent architecture, Error Trajectory Tracing, that is designed to trace the trajectories of predictive errors through phase space, and 2) a training regime, Horizon Forcing, that pushes the model’s focus out to a predetermined time horizon. We validate our method on classic chaotic systems and real-world time series prediction tasks with chaotic characteristics, and show that our approach outperforms the current state-of-the-art methods.

The study of model bias and variance with respect to decision boundaries is critically important in supervised learning and artificial intelligence. There is generally a tradeoff between the two, as fine-tuning of the decision boundary of a classification model to accommodate more boundary training samples (i.e., higher model complexity) may improve training accuracy (i.e., lower bias) but hurt generalization against unseen data (i.e., higher variance). By focusing on just classification boundary fine-tuning and model complexity, it is difficult to reduce both bias and variance. To overcome this dilemma, we take a different perspective and investigate a new approach to handle inaccuracy and uncertainty in the training data labels, which are inevitable in many applications where labels are conceptual entities and labeling is performed by human annotators. The process of classification can be undermined by uncertainty in the labels of the training data; extending a boundary to accommodate an inaccurately labeled point will increase both bias and variance. Our novel method can reduce both bias and variance by estimating the pointwise label uncertainty of the training set and accordingly adjusting the training sample weights such that those samples with high uncertainty are weighted down and those with low uncertainty are weighted up. In this way, uncertain samples have a smaller contribution to the objective function of the model’s learning algorithm and exert less pull on the decision boundary. In a real-world physical activity recognition case study, the data present many labeling challenges, and we show that this new approach improves model performance and reduces model variance.