Phase space learning with neural networks

TL;DR

This paper introduces a neural network-based autoencoder that learns the phase space dynamics of PDEs directly in a reduced latent space, enabling efficient prediction and analysis of complex dynamical behaviors.

Contribution

It presents a novel deep learning architecture that captures PDE dynamics in a latent space without intermediate reconstructions, improving efficiency and predictive capabilities.

Findings

Successfully models PDE dynamics in latent space

Predicts unseen bifurcations accurately

Maintains physical plausibility of learned trajectories

Abstract

This work proposes an autoencoder neural network as a non-linear generalization of projection-based methods for solving Partial Differential Equations (PDEs). The proposed deep learning architecture presented is capable of generating the dynamics of PDEs by integrating them completely in a very reduced latent space without intermediate reconstructions, to then decode the latent solution back to the original space. The learned latent trajectories are represented and their physical plausibility is analyzed. It is shown the reliability of properly regularized neural networks to learn the global characteristics of a dynamical system's phase space from the sample data of a single path, as well as its ability to predict unseen bifurcations.