https://doi.org/10.1140/epjp/s13360-025-07002-8
Regular Article
The gIB-PINN: a new gradient-enhanced physics-informed neural networks for solving forward and inverse problems of partial differential equations
1
College of Science, Minzu University of China, 100081, Beijing, People’s Republic of China
2
College of Science, North China University of Technology, 100144, Beijing, People’s Republic of China
a
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Received:
24
July
2025
Accepted:
27
October
2025
Published online:
4
November
2025
Abstract
Popular physics-informed neural networks (PINNs) for solving partial differential equations (PDEs) integrate the residuals of PDEs and initial and boundary conditions (IBc) as the loss function to approximate the solution of PDEs, but training the IBc part is usually more difficult than training the PDEs part, which consequently leads to the training failure of PINN. In this paper, we incorporate the gradients of IBc into the loss function of PINN to propose a new gradient-enhanced PINN, the gIB-PINN, which has strong abilities to solve the difficulty of training IBc part and also expresses better anti-noise capabilities, and thus improves the predicted accuracy of solution. Moreover, we utilize the neural tangent kernel theory to analyze the efficiency of gIB-PINN. More importantly, the prerequisite of applying gIB-PINN is only the differentiability of IBc; thus, the gIB-PINN is a very simple and effective method and can be performed in a standard feed-forward mode. Consequently, numerical results for solving the forward and inverse problems of PDEs show that the gIB-PINN outperforms the vanilla PINN, the gradient-enhanced PINN, and the state-of-the-art dual-balanced PINN with either fewer training points or fewer neurons in hidden layers, and even solves the problems that the three methods fail to address.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
