有限差分神经网络：偏微分方程的快速预测
Finite Difference Neural Networks: Fast Prediction of Partial Differential Equations
摘要：
发现复杂系统的潜在行为是许多科学和工程学科的重要课题。在本文中，我们提出了一种新的神经网络框架，有限差分神经网络（FDNet），学习偏微分方程的数据。具体来说，我们提出的有限差分启发网络的目的是学习基本的偏微分方程从轨迹数据，并迭代估计未来的动力学行为，只使用几个可训练的参数。我们说明了我们的框架上的热方程的性能（预测能力），有和没有噪音和/或强迫，并比较我们的结果向前欧拉方法。此外，我们使用Hessian-Free信赖域方法训练网络的优点。
英文摘要：
Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters. We illustrate the performance (predictive power) of our framework on the heat equation, with and without noise and/or forcing, and compare our results to the Forward Euler method. Moreover, we show the advantages of using a Hessian-Free Trust Region method to train the network.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
