Physics-Informed Neural networks (PINNs) are mesh-free Deep Learning (DL) framework to solve Partial Differential equations (PDEs). This technique embeds physical laws directly into the training process, enabling the solution of forward and inverse problems governed by PDEs. Unlike traditional neural networks, PINNs incorporate the governing equations, initial conditions, and boundary conditions directly into the loss function. Automatic differentiation in PINNs avoids truncation errors and ensures high precision enforcing the governing equations. Despite their advantages, PINNs face several challenges. PINNs struggle to solve Convection-Diffusion Equations (CDEs), particularly at the region where the convection term dominated. To overcome this problem, an extended form of PINNs is discussed here. Adaptive Gradient-enhanced PINNs (AG-PINNs) are extensions of PINNs, where Residual-based Adaptive Refinement (RAR) and the derivatives of the governing equations are also enforced during training. However, adding gradient constraints leads to over-constraining the network, increased computational cost, and inefficient learning in smooth regions. This motivates RAR, which improves the solution accuracy while avoiding over-constraining the neural network in smooth regions. In this paper we discuss convection-diffusion equation with high Péclet number. As Pé increased the convection terms dominated so it become challenging for standard PINNs, to mitigate these challenges AG-PINNs is used. AG-PINNs is better than standard PINNs which is shown in this paper by comparing results of AG-PINNs with standard PINNs technique. This work is carried out through Python Jupiter Notebook in a deepXDE library.
Javeed Ahmad “Adaptive Gradient-Enhanced PINNs for Numerical Solution of Convection-Diffusion Equa Vol. 13 Issue 02 PP. 13-21 February 2026. https://doi.org/10.5281/zenodo.18753514.
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