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.

该研究通过机器学习（特别是傅里叶神经算子和卷积神经网络）来学习参数化偏微分方程的时间推进算子。研究的目标是扩展现有的算子学习方法以处理表示PDE参数的额外输入，以便准确预测短期解并在不同参数条件下提供稳健的长期统计数据，从而节省计算成本并加速工程模拟的发展。通过对一维PDE和来自Navier-Stokes方程的真实火焰前沿演化数据的直接数值模拟进行评估，我们开发和比较基于FNO和CNN的参数化学习方法的有效性。

非稳定火焰演化的时间前进算子的参数化学习