A Hybrid Neural Network -- Polynomial Series Scheme for Learning Invariant Manifolds of Discrete Dynamical Systems

• URL: http://arxiv.org/abs/2506.13950v1
• Date: Mon, 16 Jun 2025 19:42:34 GMT
• Title: A Hybrid Neural Network -- Polynomial Series Scheme for Learning Invariant Manifolds of Discrete Dynamical Systems
• Authors: Dimitrios G. Patsatzis, Nikolaos Kazantzis, Ioannis G. Kevrekidis, Constantinos Siettos,
• Abstract summary: We propose a hybrid machine learning scheme to learn, in physics-informed and numerical analysis-informed fashion.<n>The proposed scheme combines series with shallow neural networks, exploiting the complementary strengths of both approaches.<n>We demonstrate that the proposed hybrid scheme outperforms both pure approximations and standalone neural networks.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose a hybrid machine learning scheme to learn -- in physics-informed and numerical analysis-informed fashion -- invariant manifolds (IM) of discrete maps for constructing reduced-order models (ROMs) for dynamical systems. The proposed scheme combines polynomial series with shallow neural networks, exploiting the complementary strengths of both approaches. Polynomials enable an efficient and accurate modeling of ROMs with guaranteed local exponential convergence rate around the fixed point, where, under certain assumptions, the IM is demonstrated to be analytic. Neural networks provide approximations to more complex structures beyond the reach of the polynomials' convergence. We evaluate the efficiency of the proposed scheme using three benchmark examples, examining convergence behavior, numerical approximation accuracy, and computational training cost. Additionally, we compare the IM approximations obtained solely with neural networks and with polynomial expansions. We demonstrate that the proposed hybrid scheme outperforms both pure polynomial approximations (power series, Legendre and Chebyshev polynomials) and standalone shallow neural network approximations in terms of numerical approximation accuracy.

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