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

arXiv: https://arxiv.org/abs/2208.08635 · HTML: https://mlsanity.com/paper/2208.08635/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 38 references that are in the map.
- First posted on arXiv 18 Aug 2022.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [Exponential Convergence of the Deep Neural Network Approximation for Analytic Functions](https://mlsanity.com/paper/1807.00297/) (2018)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [Physics-Informed CoKriging: A Gaussian-Process-Regression-Based Multifidelity Method for Data-Model Convergence](https://mlsanity.com/paper/1811.09757/) (2018)
- [Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems](https://mlsanity.com/paper/2108.00037/) (2021)
- [Weight Initialization of Deep Neural Networks(DNNs) using Data Statistics](https://mlsanity.com/paper/1710.10570/) (2017)

## Built on by (most cited first, 7 of 7)

- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [BiLO: Bilevel Local Operator Learning for PDE Inverse Problems](https://mlsanity.com/paper/2404.17789/) (2024)
- [Randomized Physics-Informed Neural Networks for Bayesian Data Assimilation](https://mlsanity.com/paper/2407.04617/) (2024)
- [Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective](https://mlsanity.com/paper/2506.19805/) (2025)
- [A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/2509.13717/) (2025)
- [Randomized Physics-Informed Machine Learning for Uncertainty Quantification in High-Dimensional Inverse Problems](https://mlsanity.com/paper/2312.06177/) (2023)
- [Physics-informed Convolutional Neural Network for Microgrid Economic Dispatch](https://mlsanity.com/paper/2404.18362/) (2024)

## Abstract (arXiv)

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.