A Novel Schur-Decomposition-Based Weight Projection Method for Stable State-Space Neural-Network Architectures

TL;DR

This paper presents a Schur-decomposition-based projection method to ensure stability in state-space neural networks, improving robustness and convergence for modeling dynamical systems.

Contribution

It introduces a novel stability-ensuring projection scheme compatible with backpropagation, enhancing neural network stability without significant complexity increase.

Findings

Achieves accuracy comparable to state-of-the-art methods

Ensures stable dynamics with minimal overparameterization

Facilitates convergence in training neural architectures

Abstract

Building black-box models for dynamical systems from data is a challenging problem in machine learning, especially when asymptotic stability guarantees are required. In this paper, we introduce a novel stability-ensuring and backpropagation-compatible projection scheme based on the Schur decomposition for the state matrix of linear discrete-time state-space layers, as well as an alternative pre-factorized formulation of the methodology. The proposed methods dynamically project the quasi-triangular factor of the state matrix's real Schur decomposition onto its nearest stable peer, ensuring stable dynamics with minimal overparameterization. Experiments on synthetic linear systems demonstrate that the method achieves accuracy and convergence rates comparable to those of state-of-the-art stable-system identification techniques, despite a marginal increase in computational complexity. Furthermore, the lower weight count facilitates convergence during training without sacrificing accuracy in stacked neural-network architectures with static nonlinearities targeting real-world datasets. These results suggest that the Schur-based projection provides a numerically robust framework for identifying complex dynamics on par with the State of the Art while satisfying strict asymptotic-stability requirements.