ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients
Abstract
Physics-informed neural networks (PINNs) have emerged as a meshless alternative to conventional numerical methods for solving partial differential equations (PDEs). However, their limited ability to capture sharp gradients can lead to substantial errors when resolving boundary and interior layers. Here, we introduce an energy-norm-enhanced PINN (ENPINN) that incorporates gradient information and variational structure into the loss function to improve the resolution of layer-dominated solutions. We first examine two related formulations: weak-loss PINNs (WLPINNs), which incorporate test functions into the conventional PINN residual, and gradient-enhanced PINNs (gPINNs), which augment the loss with spatial derivatives of the PDE residual. By analyzing these formulations, we identify their limitations in resolving steep solution gradients and motivate the systematic construction of ENPINN. We establish theoretically how the energy-norm error depends on the ENPINN loss and show that a suitably modified residual-derivative term is essential for accurately capturing boundary layers. We further establish the existence of neural-network approximations with arbitrarily small energy error and derive corresponding derivative bounds, providing a theoretical foundation for the proposed framework. The performance of ENPINN is assessed through systematic comparisons with existing PINN variants for convection-diffusion-reaction problems exhibiting steep gradients. Numerical experiments include a combustion model, a coupled multi-scale system, a two-dimensional Burgers equation with an interior layer, and a three-dimensional time-dependent problem.
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BibTeX
@misc{maity2026enpinn,
title = {{ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients}},
author = {Subhendu Maity and Pratibhamoy Das and Arihant Patawari and Ameya D. Jagtap},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29499},
eprint = {2608.29499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29499},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients
AU - Subhendu Maity
AU - Pratibhamoy Das
AU - Arihant Patawari
AU - Ameya D. Jagtap
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29499
UR - https://arxiv.org/abs/2608.29499
AB - Physics-informed neural networks (PINNs) have emerged as a meshless alternative to conventional numerical methods for solving partial differential equations (PDEs). However, their limited ability to capture sharp gradients can lead to substantial errors when resolving boundary and interior layers. Here, we introduce an energy-norm-enhanced PINN (ENPINN) that incorporates gradient information and variational structure into the loss function to improve the resolution of layer-dominated solutions. We first examine two related formulations: weak-loss PINNs (WLPINNs), which incorporate test functions into the conventional PINN residual, and gradient-enhanced PINNs (gPINNs), which augment the loss with spatial derivatives of the PDE residual. By analyzing these formulations, we identify their limitations in resolving steep solution gradients and motivate the systematic construction of ENPINN. We establish theoretically how the energy-norm error depends on the ENPINN loss and show that a suitably modified residual-derivative term is essential for accurately capturing boundary layers. We further establish the existence of neural-network approximations with arbitrarily small energy error and derive corresponding derivative bounds, providing a theoretical foundation for the proposed framework. The performance of ENPINN is assessed through systematic comparisons with existing PINN variants for convection-diffusion-reaction problems exhibiting steep gradients. Numerical experiments include a combustion model, a coupled multi-scale system, a two-dimensional Burgers equation with an interior layer, and a three-dimensional time-dependent problem.
ER -