On latent dynamics learning in nonlinear reduced order modeling

Abstract

In this work, we present the novel mathematical framework of latent dynamics models (LDMs) for reduced order modeling of parameterized nonlinear time-dependent PDEs. Our framework casts this latter task as a nonlinear dimensionality reduction problem, while constraining the latent state to evolve accordingly to an unknown dynamical system. A time-continuous setting is employed to derive error and stability estimates for the LDM approximation of the full order model (FOM) solution. We analyze the impact of using an explicit Runge–Kutta scheme in the time-discrete setting, resulting in the $\Delta \text{LDM}$ formulation, and further explore the learnable setting, ${\Delta \text{LDM}}_{\theta }$, where deep neural networks approximate the discrete LDM components, while providing a bounded approximation error with respect to the FOM. Moreover, we extend the concept of parameterized Neural ODE – a possible way to build data-driven dynamical systems with varying input parameters – to be a convolutional architecture, where the input parameters information is injected by means of an affine modulation mechanism, while designing a convolutional autoencoder neural network able to retain spatial-coherence, thus enhancing interpretability at the latent level. Numerical experiments, including the Burgers’ and the advection–diffusion–reaction equations, demonstrate the framework’s ability to obtain a time-continuous approximation of the FOM solution, thus being able to query the LDM approximation at any given time instance while retaining a prescribed level of accuracy. Our findings highlight the remarkable potential of the proposed LDMs, representing a mathematically rigorous framework to enhance the accuracy and approximation capabilities of reduced order modeling for time-dependent parameterized PDEs.