论文标题
稀疏和嘈杂数据的参数部分微分方程的深度学习
Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Data
论文作者
论文摘要
数据驱动的方法最近在从时空数据中发现部分微分方程（PDE）方面取得了巨大进展。但是，仍有一些挑战待解决，包括稀疏的嘈杂数据，不完整的候选库以及空间或时间变化的系数。在这项工作中，提出了一个新的框架，该框架结合了神经网络，遗传算法和自适应方法，以同时解决所有这些挑战。在框架中，训练有素的神经网络用于计算衍生物并生成大量的元数据，从而解决了稀疏嘈杂数据的问题。接下来，使用遗传算法来发现PDE的形式和具有不完整候选库的相应系数。最后，引入了两步自适应方法，以发现具有空间或时间变化系数的参数PDE。在此方法中，首先发现了参数PDE的结构，然后确定了变化系数的一般形式。提出的算法在汉堡方程，对流扩散方程，波方程和KDV方程上进行了测试。结果表明，此方法对于稀疏和嘈杂的数据是可靠的，并且能够使用不完整的候选库来发现参数PDE。
Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete candidate library, and spatially- or temporally-varying coefficients. In this work, a new framework, which combines neural network, genetic algorithm and adaptive methods, is put forward to address all of these challenges simultaneously. In the framework, a trained neural network is utilized to calculate derivatives and generate a large amount of meta-data, which solves the problem of sparse noisy data. Next, genetic algorithm is utilized to discover the form of PDEs and corresponding coefficients with an incomplete candidate library. Finally, a two-step adaptive method is introduced to discover parametric PDEs with spatially- or temporally-varying coefficients. In this method, the structure of a parametric PDE is first discovered, and then the general form of varying coefficients is identified. The proposed algorithm is tested on the Burgers equation, the convection-diffusion equation, the wave equation, and the KdV equation. The results demonstrate that this method is robust to sparse and noisy data, and is able to discover parametric PDEs with an incomplete candidate library.
