Nonlinear model order reduction for problems with microstructure using mesh informed neural networks

TL;DR

This paper introduces a mesh-informed neural network approach combining POD to efficiently model complex microstructured problems in computational physics, outperforming traditional methods especially in multi-scale scenarios.

Contribution

It presents a novel nonintrusive reduced order modeling strategy using sparse MINNs integrated with POD for microstructure problems, addressing limitations of classical ROMs.

Findings

Effective approximation of microstructure problems demonstrated on benchmarks

Outperforms traditional projection-based ROMs in multi-scale scenarios

Successfully applied to real-life microcirculation impact analysis

Abstract

Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to capture fine details at small scales. As a result, simulating such phenomena becomes unaffordable for many-query applications, such as parametrized systems with multiple scale-dependent features. Traditional projection-based reduced order models (ROMs) fail to resolve these issues, even for second-order elliptic PDEs commonly found in engineering applications. To address this, we propose an alternative nonintrusive strategy to build a ROM, that combines classical proper orthogonal decomposition (POD) with a suitable neural network (NN) model to account for the small scales. Specifically, we employ sparse mesh-informed neural networks (MINNs), which handle both spatial dependencies in the solutions and model parameters simultaneously. We evaluate the performance of this strategy on benchmark problems and then apply it to approximate a real-life problem involving the impact of microcirculation in transport phenomena through the tissue microenvironment.