论文标题
使用FCNN学习反应扩散类型方程的有限差异方法
Learning finite difference methods for reaction-diffusion type equations with FCNN
论文作者
论文摘要
近年来，物理知识的神经网络（PINN）已被广泛用于解决部分微分方程与数值方法，因为可以在没有观察的情况下训练PINN并直接处理连续的时间问题。相反，优化此类模型的参数很困难，并且必须进行单个培训课程以预测每个不同初始条件的演变。为了减轻第一个问题，可以将观察到的数据直接注入损耗函数部分。为了解决第二个问题，可以构建网络体系结构作为学习有限差异方法的框架。鉴于这两种动机，我们提出了五点模板CNN（FCNN），其中包含五点模板内核和可训练的近似函数，用于反应扩散类型方程，包括热，Fisher's，allen-cahn和其他带有三角函数函数项的反应 - 反应方程。我们表明，FCNN可以使用很少的数据学习有限差异方案，并在没有看到的初始条件下实现不同反应扩散发展的低相对误差。此外，我们证明即使使用嘈杂的数据，FCNN仍然可以接受良好的训练。
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.
