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In this paper, we develop a physics-informed neural network (PINN) model forparabolic problems with a sharply perturbed initial condition. As an example ofa parabolic problem, we consider the advection-dispersion equation (ADE) with apoint (Gaussian) source initial condition. In the $d$-dimensional ADE,perturbations in the initial condition decay with time $t$ as $t^{-d/2}$, whichcan cause a large approximation error in the PINN solution. Localized largegradients in the ADE solution make the (common in PINN) Latin hypercubesampling of the equation's residual highly inefficient. Finally, the PINNsolution of parabolic equations is sensitive to the choice of weights in theloss function. We propose a normalized form of ADE where the initialperturbation of the solution does not decrease in amplitude and demonstratethat this normalization significantly reduces the PINN approximation error. Wepropose criteria for weights in the loss function that produce a more accuratePINN solution than those obtained with the weights selected via other methods.Finally, we proposed an adaptive sampling scheme that significantly reduces thePINN solution error for the same number of the sampling (residual) points. Wedemonstrate the accuracy of the proposed PINN model for forward, inverse, andbackward ADEs.

Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions