A Novel Schur-Decomposition-Based Weight Projection Method for Stable State-Space Neural-Network 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, despite a marginal increase in computational complexity, the method achieves accuracy and convergence rates comparable to those of state-of-the-art stable-system identification techniques. 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.

32 pages, 13 figures. Source code at this https URL

Machine Learning (cs.LG); Systems and Control (eess.SY)

I.5.1; I.5.4; I.6.5

arXiv:2605.14489 [cs.LG]

(or arXiv:2605.14489v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2605.14489 Focus to learn more arXiv-issued DOI via DataCite

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