Neural-PDE: a RNN based neural network for solving time dependent PDEs
Neural-PDE: a RNN based neural network for solving time dependent PDEs
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DOI：
10.4310/cis.2022.v22.n2.a3
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发表时间：
2020-09
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通讯作者：
Yihao Hu;Tong Zhao;Shixin Xú;Lizhen Lin;Zhiliang Xu
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作者：
Yihao Hu;Tong Zhao;Shixin Xú;Lizhen Lin;Zhiliang Xu
偏微分方程（PDE）在研究科学和工程中的大量问题中起着至关重要的作用。数值求解非线性和/或高维偏微分方程通常是一项具有挑战性的任务。受传统有限差分和有限元方法以及机器学习新进展的启发，我们提出了一个称为Neural-PDE的序列深度学习框架，它允许通过使用双向LSTM编码器从现有数据中自动学习任何时间相关PDE系统的控制规则，并预测下n个时间步数据。我们提出的框架的一个关键特征是，神经元PDE是能够同时学习和模拟多尺度变量。我们测试了一系列的例子，从一维偏微分方程的高维和非线性复杂流体模型的神经元PDE。实验结果表明，Neural-PDE能够在不知道PDE系统具体形式的情况下学习初始条件、边界条件和微分算子，在20个epoch的训练时间内，Neural-PDE能够有效地提取动力学信息，并产生准确的预测结果。此外，与学习PDE的传统机器学习方法（如CNN和MLP）不同，这些方法需要大量参数来提高模型精度，Neural-PDE在所有时间步长中共享参数，从而大大降低了计算复杂度并导致快速学习算法。
Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional finite difference and finite elements methods and emerging advancements in machine learning, we propose a sequence deep learning framework called Neural-PDE, which allows to automatically learn governing rules of any time-dependent PDE system from existing data by using a bidirectional LSTM encoder, and predict the next n time steps data. One critical feature of our proposed framework is that the Neural-PDE is able to simultaneously learn and simulate the multiscale variables.We test the Neural-PDE by a range of examples from one-dimensional PDEs to a high-dimensional and nonlinear complex fluids model. The results show that the Neural-PDE is capable of learning the initial conditions, boundary conditions and differential operators without the knowledge of the specific form of a PDE system.In our experiments the Neural-PDE can efficiently extract the dynamics within 20 epochs training, and produces accurate predictions. Furthermore, unlike the traditional machine learning approaches in learning PDE such as CNN and MLP which require vast parameters for model precision, Neural-PDE shares parameters across all time steps, thus considerably reduces the computational complexity and leads to a fast learning algorithm.