Parametric Learning of Time-Advancement Operators for Unstable Flame
  Evolution
Rixin Yu Erdzan Hodzic
This study investigates the application of machine learning, specifically
Fourier Neural Operator (FNO) and Convolutional Neural Network (CNN), to learn
time-advancement operators for parametric partial differential equations
(PDEs). Our focus is on extending existing operator learning methods to handle
additional inputs representing PDE parameters. The goal is to create a unified
learning approach that accurately predicts short-term solutions and provides
robust long-term statistics under diverse parameter conditions, facilitating
computational cost savings and accelerating development in engineering
simulations. We develop and compare parametric learning methods based on FNO
and CNN, evaluating their effectiveness in learning parametric-dependent
solution time-advancement operators for one-dimensional PDEs and realistic
flame front evolution data obtained from direct numerical simulations of the
Navier-Stokes equations.