OS-net: Orbitally Stable Neural Networks
Marieme Ngom, Carlo Graziani
TL;DR
OS-net is a novel neural network architecture tailored for modeling periodic dynamical systems, ensuring stability through ODE theory, and effectively capturing complex behaviors like chaos in systems such as Rossler and Sprott's.
Contribution
The paper introduces OS-net, a stable neural network architecture based on NODEs that leverages ODE stability conditions for modeling periodic and chaotic dynamical data.
Findings
Successfully modeled Rossler and Sprott's systems.
Demonstrated stability and effectiveness in capturing complex dynamics.
Utilized ODE theory to derive stability conditions for neural networks.
Abstract
We introduce OS-net (Orbitally Stable neural NETworks), a new family of neural network architectures specifically designed for periodic dynamical data. OS-net is a special case of Neural Ordinary Differential Equations (NODEs) and takes full advantage of the adjoint method based backpropagation method. Utilizing ODE theory, we derive conditions on the network weights to ensure stability of the resulting dynamics. We demonstrate the efficacy of our approach by applying OS-net to discover the dynamics underlying the R\"{o}ssler and Sprott's systems, two dynamical systems known for their period doubling attractors and chaotic behavior.
Peer Reviews
Decision·Submitted to ICLR 2024
By introducing a regularizer, the paper introduces a novel architecture, OS-net, that bridges the gap between neural networks and periodic dynamical systems, offering a fresh perspective on the design of neural networks for specific types of data.
1. The authors highlight the pivotal role of the activation function, yet the rationale behind the specific activation functions chosen, especially $x+sin(x)$ leading to a periodic system, could be further expounded upon. 2. While the final optimization equation (10) is derived from (9), its original form (5) might not necessarily solve (6). It remains unclear how the authors ensure that the final solution adheres to eq(10). 3. $g$ in equation (9) is not defined. 4. The paper lacks a discussion
The idea of learning Neural ODEs to stabilise periodic orbits seems fairly original. The quality of the theoretical section is generally good and quite clear. I find particularly interesting the links established between known results of the stability theory of dynamical systems and ML applications of Neural ODEs.
The link with related works and literature is poor. I struggled to understand whether the goal of the paper is the same of [1]. In [1] it is clearly stated that the goal is to detect unstable periodic orbits of a dynamical system. From figure 2, I deduce that the training data is the unstable 1-period solution of the Rossler for c=6. Thus, the OS-net is using the unstable 1-period solution itself to learn an ODE that has stable dynamics converging to such 1-period solution. If that is right, the
S1. The authors propose a novel regularization strategy to achieve stability in predicting dynamic systems.
First of all, I need to clarify that my review is based on the assumption that the authors have a correct understanding of their references and that their proofs and derivations are correct. I did not carefully check whether their mathematical proofs have any issues. W1. The introduction of related work is not sufficient. What are the specific works that focus on stability of dynamic systems, and what are the shortcomings of these related works compared to the method proposed in this paper? W
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Taxonomy
TopicsNeural Networks and Applications · Model Reduction and Neural Networks · Neural Networks and Reservoir Computing