Tractable Dendritic RNNs for Reconstructing Nonlinear Dynamical Systems

TL;DR

This paper introduces a dendritically expanded piecewise-linear RNN that effectively reconstructs nonlinear dynamical systems, combining interpretability, tractability, and improved approximation capacity over existing methods.

Contribution

It proposes a novel dendritic computation-inspired expansion of PLRNNs, enhancing their ability to model complex nonlinear dynamics with fewer parameters and maintaining interpretability.

Findings

Achieves better reconstruction accuracy on benchmarks

Uses fewer parameters and lower dimensions

Outperforms comparable methods in fidelity and efficiency

Abstract

In many scientific disciplines, we are interested in inferring the nonlinear dynamical system underlying a set of observed time series, a challenging task in the face of chaotic behavior and noise. Previous deep learning approaches toward this goal often suffered from a lack of interpretability and tractability. In particular, the high-dimensional latent spaces often required for a faithful embedding, even when the underlying dynamics lives on a lower-dimensional manifold, can hamper theoretical analysis. Motivated by the emerging principles of dendritic computation, we augment a dynamically interpretable and mathematically tractable piecewise-linear (PL) recurrent neural network (RNN) by a linear spline basis expansion. We show that this approach retains all the theoretically appealing properties of the simple PLRNN, yet boosts its capacity for approximating arbitrary nonlinear dynamical systems in comparatively low dimensions. We employ two frameworks for training the system, one combining back-propagation-through-time (BPTT) with teacher forcing, and another based on fast and scalable variational inference. We show that the dendritically expanded PLRNN achieves better reconstructions with fewer parameters and dimensions on various dynamical systems benchmarks and compares favorably to other methods, while retaining a tractable and interpretable structure.