论文标题
基于稀疏和嘈杂数据的积分形式的基于偏微分方程的深度学习发现
Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data
论文作者
论文摘要
近年来，数据驱动的部分微分方程（PDE）引起了人们越来越多的关注。尽管已经取得了重大进展，但仍然存在某些未解决的问题。例如，对于具有高级导数的PDE，现有方法的性能不令人满意，尤其是当数据稀疏和嘈杂时。也很难发现将异质参数嵌入到部分差分运算符中的异质参数PDE。在这项工作中，提出了一个新的框架，结合了深度学习和整体形式，以同时处理上述问题，并提高PDE发现的准确性和稳定性。在框架中，首先对深度神经网络进行了观察数据，以生成元数据并计算衍生物。然后，定义了统一的积分形式，并采用遗传算法来发现最佳结构。最后，计算参数的值，以及参数是常数还是变量。数值实验证明，与积分形式的利用相比，与现有方法相比，我们提出的算法对噪声和更准确。我们提出的算法还能够通过稀疏和嘈杂的数据准确地发现具有高阶导数或异质参数的PDE。
Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order derivatives, the performance of existing methods is unsatisfactory, especially when the data are sparse and noisy. It is also difficult to discover heterogeneous parametric PDEs where heterogeneous parameters are embedded in the partial differential operators. In this work, a new framework combining deep-learning and integral form is proposed to handle the above-mentioned problems simultaneously, and improve the accuracy and stability of PDE discovery. In the framework, a deep neural network is firstly trained with observation data to generate meta-data and calculate derivatives. Then, a unified integral form is defined, and the genetic algorithm is employed to discover the best structure. Finally, the value of parameters is calculated, and whether the parameters are constants or variables is identified. Numerical experiments proved that our proposed algorithm is more robust to noise and more accurate compared with existing methods due to the utilization of integral form. Our proposed algorithm is also able to discover PDEs with high-order derivatives or heterogeneous parameters accurately with sparse and noisy data.
