Model reduction of parametric ordinary differential equations via autoencoders: structure-preserving latent dynamics and convergence analysis

Abstract: We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE is then solved using standard integration in time schemes, and the high-dimensional solution is reconstructed from the low-dimensional one. We investigate the applicability of neural networks for constructing effective autoencoders with the property of reconstructing the input manifold with null representation error. We study the convergence of the reduced-order model to the high-fidelity one. Numerical experiments show the robustness and accuracy of our approach, highlighting its potential to accelerate complex dynamical simulations without sacrificing accuracy. Moreover, we examine how the reduction influences the stability properties of the reconstructed high-dimensional solution.