Gradient Enhanced Self-Training Physics-Informed Neural Network (gST-PINN) for Solving Nonlinear Partial Differential Equations

• URL: http://arxiv.org/abs/2510.10483v1
• Date: Sun, 12 Oct 2025 07:29:02 GMT
• Title: Gradient Enhanced Self-Training Physics-Informed Neural Network (gST-PINN) for Solving Nonlinear Partial Differential Equations
• Authors: Narayan S Iyer, Bivas Bhaumik, Ram S Iyer, Satyasaran Changdar,
• Abstract summary: Partial differential equations (PDEs) provide a mathematical foundation for simulating and understanding intricate behaviors.<n>Data$-$driven approaches like Physics$-$Informed Neural Networks (PINNs) have been developed.<n> PINNs often struggle with challenges such as limited precision, slow training dynamics, lack of labeled data availability, and inadequate handling of multi$-$physics interactions.<n>We propose a Gradient Enhanced Self$-$Training PINN (gST$-$PINN) method that specifically introduces a gradient based pseudo point self$-$learning algorithm for solving PDEs.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Partial differential equations (PDEs) provide a mathematical foundation for simulating and understanding intricate behaviors in both physical sciences and engineering. With the growing capabilities of deep learning, data$-$driven approaches like Physics$-$Informed Neural Networks (PINNs) have been developed, offering a mesh$-$free, analytic type framework for efficiently solving PDEs across a wide range of applications. However, traditional PINNs often struggle with challenges such as limited precision, slow training dynamics, lack of labeled data availability, and inadequate handling of multi$-$physics interactions. To overcome these challenging issues of PINNs, we proposed a Gradient Enhanced Self$-$Training PINN (gST$-$PINN) method that specifically introduces a gradient based pseudo point self$-$learning algorithm for solving PDEs. We tested the proposed method on three different types of PDE problems from various fields, each representing distinct scenarios. The effectiveness of the proposed method is evident, as the PINN approach for solving the Burgers$'$ equation attains a mean square error (MSE) on the order of $10^{-3}$, while the diffusion$-$sorption equation achieves an MSE on the order of $10^{-4}$ after 12,500 iterations, with no further improvement as the iterations increase. In contrast, the MSE for both PDEs in the gST$-$PINN model continues to decrease, demonstrating better generalization and reaching an MSE on the order of $10^{-5}$ after 18,500 iterations. Furthermore, the results show that the proposed purely semi$-$supervised gST$-$PINN consistently outperforms the standard PINN method in all cases, even when solution of the PDEs are unavailable. It generalizes both PINN and Gradient$-$enhanced PINN (gPINN), and can be effectively applied in scenarios prone to low accuracy and convergence issues, particularly in the absence of labeled data.

Related papers

• Towards Fast Option Pricing PDE Solvers Powered by PIELM [0.0]
  Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving the forward and inverse problems of partial differential equations (PDEs) using deep learning.<n>This paper introduces Physics-Informed Extreme Learning Machines (PIELMs) as fast alternative to PINNs for solving both forward and inverse problems in financial PDEs.
  arXiv  Detail & Related papers  (2025-10-05T18:50:49Z)
• HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions [16.904297509040777]
  We present HyPINO, a multi-physics neural operator designed for zeroshot generalization across a broad class of parametric PDEs.<n>The model maps PDE parametrizations to target Physics-Informed Neural Networks (PINNs) and can handle linear elliptic, hyperbolic, and parabolic equations in two dimensions.<n>HyPINO achieves strong zero-shot accuracy on seven benchmark problems from PINN, outperforming U-Nets, literature, and Physics-Informed Neural Operators (PINO)
  arXiv  Detail & Related papers  (2025-09-05T13:59:25Z)
• High precision PINNs in unbounded domains: application to singularity formulation in PDEs [83.50980325611066]
  We study the choices of neural network ansatz, sampling strategy, and optimization algorithm.<n>For 1D Burgers equation, our framework can lead to a solution with very high precision.<n>For the 2D Boussinesq equation, we obtain a solution whose loss is $4$ digits smaller than that obtained in citewang2023asymptotic with fewer training steps.
  arXiv  Detail & Related papers  (2025-06-24T02:01:44Z)
• ANaGRAM: A Natural Gradient Relative to Adapted Model for efficient PINNs learning [2.3129637440028197]
  Physics Informed Neural Networks (PINNs) have received strong interest as a method to solve PDE driven systems.<n>We propose a natural gradient approach to PINNs which contributes to speed-up and improve the accuracy of the training.
  arXiv  Detail & Related papers  (2024-12-14T10:38:09Z)
• PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations [5.4087282763977855]
  We propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network.<n>Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training.<n> Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs.
  arXiv  Detail & Related papers  (2024-12-08T16:58:29Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• iPINNs: Incremental learning for Physics-informed neural networks [66.4795381419701]
  Physics-informed neural networks (PINNs) have recently become a powerful tool for solving partial differential equations (PDEs) We propose incremental PINNs that can learn multiple tasks sequentially without additional parameters for new tasks and improve performance for every equation in the sequence. Our approach learns multiple PDEs starting from the simplest one by creating its own subnetwork for each PDE and allowing each subnetwork to overlap with previously learnedworks.
  arXiv  Detail & Related papers  (2023-04-10T20:19:20Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Method [70.19159220248805]
  Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs) We propose the generative adversarial neural network (GA-PINN), which integrates the generative adversarial (GA) mechanism with the structure of PINNs. Inspired from the weighting strategy of the Adaboost method, we then introduce a point-weighting (PW) method to improve the training efficiency of PINNs.
  arXiv  Detail & Related papers  (2022-05-18T06:50:44Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• dNNsolve: an efficient NN-based PDE solver [62.997667081978825]
  We introduce dNNsolve, that makes use of dual Neural Networks to solve ODEs/PDEs. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions.
  arXiv  Detail & Related papers  (2021-03-15T19:14:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.