# PMNN:Physical Model-driven Neural Network for solving time-fractional differential equations

arXiv: https://arxiv.org/abs/2310.04788 · HTML: https://mlsanity.com/paper/2310.04788/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 45 references that are in the map.
- First posted on arXiv 7 Oct 2023.

## Builds on (most cited first, 7 of 7)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications](https://mlsanity.com/paper/2004.04276/) (2020)

## Built on by (most cited first, 3 of 3)

- [Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Network](https://mlsanity.com/paper/2504.01440/) (2025)
- [MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamics](https://mlsanity.com/paper/2512.17273/) (2025)
- [A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equations](https://mlsanity.com/paper/2605.29688/) (2026)

## Abstract (arXiv)

In this paper, an innovative Physical Model-driven Neural Network (PMNN) method is proposed to solve time-fractional differential equations. It establishes a temporal iteration scheme based on physical model-driven neural networks which effectively combines deep neural networks (DNNs) with interpolation approximation of fractional derivatives. Specifically, once the fractional differential operator is discretized, DNNs are employed as a bridge to integrate interpolation approximation techniques with differential equations. On the basis of this integration, we construct a neural-based iteration scheme. Subsequently, by training DNNs to learn this temporal iteration scheme, approximate solutions to the differential equations can be obtained. The proposed method aims to preserve the intrinsic physical information within the equations as far as possible. It fully utilizes the powerful fitting capability of neural networks while maintaining the efficiency of the difference schemes for fractional differential equations. Moreover, we validate the efficiency and accuracy of PMNN through several numerical experiments.