On latent dynamics learning in nonlinear reduced order modeling

TL;DR

This paper introduces latent dynamics models (LDMs) as a new mathematical framework for reduced order modeling of nonlinear, parameterized, time-dependent PDEs, combining nonlinear dimensionality reduction with dynamical system constraints.

Contribution

The work develops a novel LDM framework, analyzes its stability and error, and extends neural ODE concepts with convolutional autoencoders for improved interpretability and accuracy.

Findings

LDMs provide accurate, multi-query approximations of PDE solutions.

The $ ext{LDM}_ heta$ approach bounds approximation errors.

Numerical experiments demonstrate the framework's effectiveness.

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 - recently proposed as 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-reaction-diffusion equations, demonstrate the framework's ability to obtain, in a multi-query context, 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.