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.

翻译：在本文中,我们开发了一个物理知情神经网络模型(PINN),用于处理初步条件受到剧烈扰动的抛物线问题。作为抛物线问题的一个实例,我们考虑的是具有点(Gausian)源初始条件的平面分散方程式(ADE)。在以美元为维值的ADE中,最初状态衰减的扰动以美元为美元=d/2}美元计时,这可能导致PINN解决方案出现很大的近似错误。亚非方案本地化的大梯度使(在PINN中常见的)该方程式残余的拉丁超立方体取样效率非常低。最后,抛物方方方方方程式的PINN解决方案对损失功能中加权选择的敏感度。我们提出了一种正常的ADE模式,即最初扰动不会减少振动,并表明这种正常化模式会大大降低PINN的近似误差。我们提出了损失函数中的权重标准,这比以加权方式获得的PINN解决方案更为精确。最后,我们提议了通过其他取样法的平面方法来大幅降低PIN的平面方案。