The Neural Network shifted-Proper Orthogonal Decomposition: a Machine Learning Approach for Non-linear Reduction of Hyperbolic Equations

TL;DR

This paper introduces a deep learning-based method to automatically identify optimal non-linear transformations for reduced order modeling of hyperbolic equations with unknown advection fields, improving accuracy and applicability.

Contribution

It presents a novel data-driven deep learning approach that generalizes linear subspace methods to non-linear hyperbolic problems without prior knowledge of advection speeds.

Findings

Successfully applied to multiphase simulation

Validated against simple test cases

Improves reduction accuracy for hyperbolic equations

Abstract

Models with dominant advection always posed a difficult challenge for projection-based reduced order modelling. Many methodologies that have recently been proposed are based on the pre-processing of the full-order solutions to accelerate the Kolmogorov N-width decay thereby obtaining smaller linear subspaces with improved accuracy. These methods however must rely on the knowledge of the characteristic speeds in phase space of the solution, limiting their range of applicability to problems with explicit functional form for the advection field. In this work we approach the problem of automatically detecting the correct pre-processing transformation in a statistical learning framework by implementing a deep-learning architecture. The purely data-driven method allowed us to generalise the existing approaches of linear subspace manipulation to non-linear hyperbolic problems with unknown advection fields. The proposed algorithm has been validated against simple test cases to benchmark its performances and later successfully applied to a multiphase simulation.