论文标题
抛物线分化方程的物理信息神经网络方法，初始条件急剧干扰
Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions
论文作者
论文摘要
在本文中，我们开发了一个具有剧烈干扰初始条件的抛物线问题的物理信息神经网络（PINN）模型。作为抛物线问题的一个示例，我们考虑具有点（高斯）源初始条件的对流 - 分散方程（ADE）。在$ d $维的ade中，在初始条件下的扰动，随时间$ t $ as $ t^{ - d/2} $，这可能会在Pinn解决方案中造成较大的近似错误。 ADE溶液中的局部大梯度使方程式残留效率低下的（PINN）拉丁超立方体采样（常见）。最后，抛物线方程的PINN解对损耗函数中重量的选择敏感。我们提出了一种归一化的ADE形式，其中溶液的初始扰动不会降低幅度，并证明该归一化显着降低了PINN近似误差。我们提出了损耗函数中的权重标准，该标准比通过其他方法选择的权重获得的PINN解决方案更准确。最后，我们提出了一种自适应采样方案，该方案可显着减少相同数量的采样（残留）点的PINN溶液误差。我们证明了提出的PINN模型的前进，反向和向后ADE的准确性。
In this paper, we develop a physics-informed neural network (PINN) model for parabolic problems with a sharply perturbed initial condition. As an example of a parabolic problem, we consider the advection-dispersion equation (ADE) with a point (Gaussian) source initial condition. In the $d$-dimensional ADE, perturbations in the initial condition decay with time $t$ as $t^{-d/2}$, which can cause a large approximation error in the PINN solution. Localized large gradients in the ADE solution make the (common in PINN) Latin hypercube sampling of the equation's residual highly inefficient. Finally, the PINN solution of parabolic equations is sensitive to the choice of weights in the loss function. We propose a normalized form of ADE where the initial perturbation of the solution does not decrease in amplitude and demonstrate that this normalization significantly reduces the PINN approximation error. We propose criteria for weights in the loss function that produce a more accurate PINN solution than those obtained with the weights selected via other methods. Finally, we proposed an adaptive sampling scheme that significantly reduces the PINN solution error for the same number of the sampling (residual) points. We demonstrate the accuracy of the proposed PINN model for forward, inverse, and backward ADEs.
