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

2026-05-14

AI Summary

This work addresses the problem of constructing black-box dynamic system models from data with guaranteed asymptotic stability. It proposes a novel weight projection method based on the real Schur decomposition, which dynamically projects the quasi-upper-triangular factor of the state matrix onto the nearest stable matrix during training. This approach strictly enforces asymptotic stability in discrete-time state-space neural networks while remaining compatible with backpropagation. By uniquely integrating Schur decomposition with dynamic projection, the method achieves minimal over-parameterization and offers a pre-factorized alternative. Experimental results demonstrate that the model attains accuracy and convergence rates comparable to state-of-the-art stable identification techniques on synthetic linear systems, and achieves efficient training with high accuracy using fewer parameters in stacked architectures on real-world datasets.

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.

Problem

• asymptotic stability
• state-space models
• neural networks
• system identification
• stability constraints

Innovation

• Schur decomposition
• stability projection
• state-space neural networks
• asymptotic stability
• backpropagation-compatible