Solving nonlinear evolution partial differential equations has been a longstanding computational challenge. In this paper, we present a universal paradigm of learning the system and extracting patterns from data generated from experiments. Specifically, this framework approximates the latent solution with a deep neural network, which is trained with the constraint of underlying physical laws usually expressed by some equations. In particular, we test the effectiveness of the approach for the Burgers’ equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions. The results also indicate that for soliton solutions, the model training costs significantly less time than other initial conditions.

中文翻译：

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使用深度学习求解二阶非线性演化偏微分方程

求解非线性演化偏微分方程一直是一个长期的计算挑战。在本文中，我们提出了一种学习系统并从实验生成的数据中提取模式的通用范式。具体来说，这个框架用一个深度神经网络来逼近潜在解，该网络是在底层物理定律的约束下训练的，通常由一些方程表示。特别是，我们测试了 Burgers 方程的方法的有效性，该方程用作不同初始和边界条件下的二阶非线性演化方程的示例。结果还表明，对于孤子解决方案，模型训练的时间比其他初始条件要少得多。