A Hybrid Neural Network -- Polynomial Series Scheme for Learning Invariant Manifolds of Discrete Dynamical Systems
This work addresses the challenge of constructing accurate reduced-order models for dynamical systems, though it appears incremental as it builds on existing polynomial and neural network methods.
The authors tackled the problem of learning invariant manifolds for reduced-order models of discrete dynamical systems by proposing a hybrid scheme combining polynomial series and shallow neural networks, demonstrating that it outperforms pure polynomial and standalone neural network approximations in numerical accuracy.
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.