面向反应扩散方程的五点模板卷积的深层化研究
Going Deeper with Five-point Stencil Convolutions for Reaction-Diffusion Equations
AI总结：
本文提出具有大感受野的深层五点模板卷积神经网络，用于在超过 CFL 条件阈值的时间步长下稳定预测反应扩散方程的时间演化，并在热方程、Fisher 方程和 Allen-Cahn 方程上验证其精度优于会爆炸的有限差分法。
摘要：
物理信息神经网络已广泛应用于偏微分方程并取得巨大成功，因为物理信息损失本质上不需要观测数据或离散化。然而，模型参数难以优化，并且这些参数必须针对每个不同的初始条件分别训练。为了克服二阶反应扩散型方程中的这些挑战，一种可行的方法是使用五点模板卷积神经网络（FCNNs）。FCNNs 使用两个连续快照进行训练，其中时间步长对应于给定快照的步长。因此，FCNNs 的时间演化依赖于时间步长，并且时间步长必须满足其 CFL 条件以避免解爆炸。在这项工作中，我们提出了具有大感受野的深层 FCNNs，以在大于 CFL 条件阈值的时间步长下预测时间演化。为了评估我们的模型，我们考虑了具有多种初始条件的热方程、Fisher 方程和 Allen-Cahn 方程。我们证明，与发生爆炸的有限差分法（FDMs）相比，深层 FCNNs 保持了一定的精度。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
