In recent years, Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations alongside numerical methods because PINNs can be trained without observations and deal with continuous-time problems directly. In contrast, optimizing the parameters of such models is difficult, and individual training sessions must be performed to predict the evolutions of each different initial condition. To alleviate the first problem, observed data can be injected directly into the loss function part. To solve the second problem, a network architecture can be built as a framework to learn a finite difference method. In view of the two motivations, we propose Five-point stencil CNNs (FCNNs) containing a five-point stencil kernel and a trainable approximation function for reaction-diffusion type equations including the heat, Fisher's, Allen-Cahn, and other reaction-diffusion equations with trigonometric function terms. We show that FCNNs can learn finite difference schemes using few data and achieve the low relative errors of diverse reaction-diffusion evolutions with unseen initial conditions. Furthermore, we demonstrate that FCNNs can still be trained well even with using noisy data.

翻译：近年来，物理信息神经网络（PINNs）与数值方法一起被广泛用于求解偏微分方程，因为PINNs无需观测数据即可训练，并能直接处理连续时间问题。然而，此类模型的参数优化十分困难，且需要对每个不同初始条件的演化进行单独训练。为缓解第一个问题，可将观测数据直接注入损失函数部分；为解决第二个问题，可构建一个网络架构作为学习有限差分方法的框架。基于这两个动机，我们提出了五点模板卷积神经网络（FCNNs），该网络包含五点模板卷积核和一个可训练逼近函数，适用于反应-扩散型方程，包括热传导方程、Fisher方程、Allen-Cahn方程及其他含三角函数项的反应-扩散方程。我们证明，FCNNs能利用少量数据学习有限差分格式，并对未见过的初始条件的多种反应-扩散演化实现低相对误差。此外，我们证明即使使用含噪声数据，FCNNs仍能得到良好训练。