Physics-informed neural networks have been widely applied to partial differential equations with great success because the physics-informed loss essentially requires no observations or discretization. However, it is difficult to optimize model parameters, and these parameters must be trained for each distinct initial condition. To overcome these challenges in second-order reaction-diffusion type equations, a possible way is to use five-point stencil convolutional neural networks (FCNNs). FCNNs are trained using two consecutive snapshots, where the time step corresponds to the step size of the given snapshots. Thus, the time evolution of FCNNs depends on the time step, and the time step must satisfy its CFL condition to avoid blow-up solutions. In this work, we propose deep FCNNs that have large receptive fields to predict time evolutions with a time step larger than the threshold of the CFL condition. To evaluate our models, we consider the heat, Fisher's, and Allen-Cahn equations with diverse initial conditions. We demonstrate that deep FCNNs retain certain accuracies, in contrast to FDMs that blow up.

翻译：物理信息神经网络因其无需观测数据或离散化的物理信息损失函数，已成功广泛应用于偏微分方程求解。然而，这类模型存在参数优化困难且需针对每个不同初始条件重新训练的局限性。针对二阶反应扩散型方程，五点模板卷积神经网络（FCNNs）提供了一种有效的解决方案。FCNNs通过两个连续快照进行训练，其中时间步长对应于给定快照的采样间隔。因此，FCNNs的时间演化依赖于时间步长，且该步长必须满足CFL条件以避免解发散。本文提出具有大感受野的深度FCNNs，能够使用超过CFL条件阈值的时间步长预测时间演化。我们采用不同初始条件的热方程、Fisher方程和Allen-Cahn方程进行模型评估。结果表明，相较于会发散计算的有限差分法，深度FCNNs能保持特定精度。