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.
翻译:局部差异方程式(PDEs)在研究大量科学和工程问题方面发挥着关键作用。 量化解决非线性和/或高维PDEs通常是一项具有挑战性的任务。 在传统的有限差异和有限元素方法以及机械学习新进步的启发下,我们提议了一个叫Neal-PDE的顺序深层次学习框架,它允许使用双向 LSTM 编码器从现有数据中自动学习任何时间依赖PDE系统的规则,并预测下个时间步骤数据。我们拟议框架的一个关键特征是,Neal-PDE能够同时学习和模拟多尺度变量。我们通过从一维PDEs到高维和非线性复杂流体模型等一系列实例测试神经-PDE。结果显示,Neal-PE能够在不了解PDE系统具体形式的情况下,从初始条件、边界条件和不同操作者之间学习初始动态动态,并因此在快速计算中学习最精确的 RMS-DPL 模型, 并由此在快速计算中进行大量学习。