We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our approach addresses the time evolution by applying classical ODEs techniques and uses ELM-based collocation for solving the resulting stationary elliptic problems. In this framework, the $\theta$-method and Backward Difference Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challeging problems for the stability and accuracy properties of the methods. The results of numerical experiments confirm that ELM-based solution techniques combined with BDF methods can provide high-accuracy solutions of parabolic PDEs.
翻译:我们通过极端学习机器神经网络(ELMs)来调查抛物体PDE的分辨率,它们有一个单一的隐藏层,可以与深学习神经网络相比以较低的计算成本进行培训。我们的方法通过应用古典的 ODE 技术来应对时间演变,并使用基于ELM 的合用点来解决由此产生的固定的椭圆问题。在这个框架内,对一些线性抛物体PDE(BDF)技术进行了调查,这些技术对方法的稳定性和准确性产生分歧。数字实验的结果证实,基于ELM 的解决方案技术与BDF方法相结合,可以提供高精度的参数PDE解决方案。