Robust Lurie Networks with Controllable Convergent Dynamics

Carl R Richardson, Matthew C. Turner, Steve R. Gunn

Keywords: dynamical systems, convergence, robustness, $k$-contraction analysis, RNNs

TL;DR: A method for learning robust models of convergent dynamical systems.

Abstract: The Lurie Network is proposed as a unifying architecture for modelling time-invariant nonlinear dynamical systems. Many existing continuous-time models including Recurrent Neural Networks and Neural Oscillators are special cases of the Lurie Network when applied to this domain. Motivated by the need for a general inductive bias, shared by many systems, this paper proposes an approach to enable network weights and biases to be trained in such a manner that a generalised concept of stability is guaranteed. This generalised stability measure is that of $k$-contraction which enables global convergence to a point, line or plane in the neural state-space. This result is leveraged to construct a Graph Lurie Network (GLN) satisfying the same convergence properties. Unconstrained parametrisations of these conditions are derived allowing the models to be trained using standard optimisation algorithms, whilst limiting the search space to solutions satisfying the $k$-contraction constraints. Empirical results show significant improvement in terms of prediction accuracy, generalisation and robustness compared to other unconstrained and stability-constrained models. Furthermore, both models consistently learnt representations which respected the convergence behaviour of the dynamics.

Primary Area: learning on time series and dynamical systems

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Submission Number: 9290