Sarah Hallerberg

Nahal Sharafi, Christoph Martin, Sarah Hallerberg

Neural networks have become a widely adopted tool for tackling a variety of problems in machine learning and artificial intelligence. In this contribution we use the mathematical framework of local stability analysis to gain a deeper understanding of the learning dynamics of feed forward neural networks. Therefore, we derive equations for the tangent operator of the learning dynamics of three-layer networks learning regression tasks. The results are valid for an arbitrary numbers of nodes and arbitrary choices of activation functions. Applying the results to a network learning a regression task, we investigate numerically, how stability indicators relate to the final training-loss. Although the specific results vary with different choices of initial conditions and activation functions, we demonstrate that it is possible to predict the final training loss, by monitoring finite-time Lyapunov exponents or covariant Lyapunov vectors during the training process.

Christoph Martin, Nahal Sharafi, Sarah Hallerberg

Covariant Lyapunov vectors characterize the directions along which perturbations in dynamical systems grow. They have also been studied as predictors of critical transitions and extreme events. For many applications like, for example, prediction, it is necessary to estimate the vectors from data since model equations are unknown for many interesting phenomena. We propose a novel method for estimating covariant Lyapunov vectors based on data records without knowing the underlying equations of the system. In contrast to previous approaches, our approach can be applied to high-dimensional data-sets. We demonstrate that this purely data-driven approach can accurately estimate covariant Lyapunpov vectors from data records generated by low and high-dimensional dynamical systems. The highest dimension of a time-series from which covariant Lyapunov vectors were estimated in this contribution is 128. Being able to infer covariant Lyapunov vectors from data-records could encourage numerous future applications in data-analysis and data-based predictions.