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

arXiv: https://arxiv.org/abs/2510.10483 · HTML: https://mlsanity.com/paper/2510.10483/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 29 references that are in the map.
- First posted on arXiv 12 Oct 2025.

## Builds on (most cited first, 10 of 10)

- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Confidence Regularized Self-Training](https://mlsanity.com/paper/1908.09822/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Self-Training: A Survey](https://mlsanity.com/paper/2202.12040/) (2022)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [PINNSim: A Simulator for Power System Dynamics based on Physics-Informed Neural Networks](https://mlsanity.com/paper/2303.10256/) (2023)
- [ST-PINN: A Self-Training Physics-Informed Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2306.09389/) (2023)

## Built on by (most cited first, 1 of 1)

- [Pseudo-differential-enhanced physics-informed neural networks](https://mlsanity.com/paper/2602.14663/) (2026)

## Abstract (arXiv)

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.