Variationally mimetic operator network approach to transient viscous flows

TL;DR

The paper extends the Variationally Mimetic Operator Network (VarMiON) to transient viscous flows, demonstrating its effectiveness in predicting solutions for low-to-moderate Reynolds number flows with good accuracy.

Contribution

It introduces a formulation of VarMiON for vector fields and transient problems, applying it to Navier-Stokes equations in simplified regimes.

Findings

Accurately predicts solutions in three flow geometries.

Shows good agreement with finite-element solutions.

Extends VarMiON to time-dependent viscous flows.

Abstract

The Variationally Mimetic Operator Network (VarMiON) approach is a machine learning technique, originally developed to predict the solution of elliptic differential problems, that combines operator networks with a structure inherited from the variational formulation of the equations. We investigate the capabilities of this method in the context of viscous flows, by extending its formulation to vector-valued unknown fields and with a particular emphasis on the space-time approximation context necessary to deal with transient flows. As a first step, we restrict attention to the regime of low-to-moderate Reynolds numbers, in which the Navier--Stokes equations can be linearized to give the time-dependent Stokes problem for incompressible fluids. The details of the method as well as its performance are illustrated in three paradigmatic flow geometries where we obtain a very good agreement between the VarMiON predictions and reference finite-element solutions.