具有FCNN的反应扩散型方程的学习有限差分方法
Learning finite difference methods for reaction-diffusion type equations with FCNN
摘要：
近年来，物理信息神经网络（PINNs）被广泛应用于求解偏微分方程和数值方法，因为PINNs可以在没有观测值的情况下进行训练并直接处理连续时间问题。相比之下，优化这些模型的参数是困难的，并且必须执行单独的训练会话以预测每个不同初始条件的演变。为了缓解第一个问题，可以将观察到的数据直接注入到损失函数部分。为了解决第二个问题，可以建立一个网络架构作为学习有限差分方法的框架。鉴于这两个动机，我们提出了五点模板CNN（FCNN）包含一个五点模板内核和一个可训练的近似函数的反应扩散型方程，包括热，费舍尔，艾伦-卡恩，和其他反应扩散方程与三角函数项。我们证明了FCNN可以使用很少的数据学习有限差分格式，并在初始条件未知的情况下实现各种反应扩散演化的低相对误差。此外，我们证明了即使使用噪声数据，FCNN仍然可以很好地训练。
英文摘要：
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.
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