Intelligent data-informed study of ionospheric TEC dynamics: learning partial differential equations via PINN, PDE-Net2, and SINDy

Abstract

The ionosphere is a critical region of near-Earth space, directly influencing satellite navigation and shortwave communication quality. The total electron content (TEC) is a key parameter for ionospheric physics, and AI-based research on TEC has become a major focus in space weather studies. However, current AI models often function as “black boxes” with limited physical interpretability, hindering our understanding of ionospheric dynamics. We employed two mainstream neural networks combined with partial differential equations (PDEs): PDE-Net2 (a deep learning technique capable of automatically extracting PDEs from data), physics-informed neural networks, and SINDy (a traditional method for sparse identification of PDEs), to compare the performance of these methods in reconstructing ionospheric TEC data. The comparison shows that PDE-Net2 significantly outperforms the other methods in reconstructing TEC data. Its performance metrics indicate superior effectiveness in TEC reconstruction. By directly extracting PDEs from PDE-Net2, we analyzed the expressions and found that the longitudinal convection term (e.g., $$ \frac{\partial u}{\partial x} $$) and the latitudinal diffusion term (e.g., $$ \frac{\partial^2 u}{\partial y^2} $$) have the largest coefficients. This suggests that the longitudinal electron transport process in the ionosphere is the most dominant, potentially linked to the effects of longitudinal winds and diurnal solar radiation variations. Additionally, the latitudinal diffusion process plays an important role, which may involve nonlinear coupling between the Earth’s magnetosphere and ionosphere.

Keywords

Partial differential equation / PDE-Net2.0 / PINN / SINDy / TEC

References

[1] Jiang H,Wang Z.Assessment of spatial and temporal TEC variations derived from ionospheric models over the polar regions.J Geod2019;93:455-71

[2] Liu L,Yao Y.Forecasting global ionospheric TEC using deep learning approach.Space Weather2020;18:e2020SW002501

[3] Tang J,Yang D.An approach for predicting global ionospheric TEC using machine learning.Remote Sens2022;14:1585

[4] Cho K,Gulcehre C. Learning phrase representations using RNN encoder-decoder for statistical machine translation. arXiv 2014; arXiv:1406.1078. https://doi.org/10.48550/arXiv.1406.1078. (accessed 20 May 2025)

[5] Hochreiter S.Long short-term memory.Neural Comput1997;9:1735-80

[6] Yu Y,Hu C.A review of recurrent neural networks: LSTM cells and network architectures.Neural Comput2019;31:1235-70

[7] Sun W,Huang X.Forecasting of ionospheric vertical total electron content (TEC) using LSTM networks. In: 2017 International Conference on Machine Learning and Cybernetics (ICMLC), Ningbo, China. Jul 09-12, 2017. IEEE; 2017. pp. 340-4.

[8] Chen Z,Li H,Deng X.Prediction of global ionosphere TEC base on deep learning. ESS. Open. Archive. 2021.

[9] Chen Z,Li H.Storm-time characteristics of ionospheric model (MSAP) based on multi-algorithm fusion.Space Weather2024;22:e2022SW003360

[10] Liu L,Liu Y.ML prediction of global ionospheric TEC maps.Space Weather2022;20:e2022SW003135

[11] Xia G,Wang C.ED-ConvLSTM: a novel global ionospheric total electron content medium-term forecast model.Space Weather2022;20:e2021SW002959

[12] Ren X,Ren Z,Xiong B.Deep learning-based prediction of global ionospheric TEC during storm periods: mixed CNN-BiLSTM method.Space Weather2024;22:e2024SW003877

[13] Raissi M,Karniadakis GE.Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.J Comput Phys2019;378:686-707

[14] Long Z,Dong B.PDE-Net 2.0: learning PDEs from data with a numeric-symbolic hybrid deep network.J Comput Phys2019;399:108925

[15] Champion K,Kutz JN.Data-driven discovery of coordinates and governing equations.Proc Natl Acad Sci U S A2019;116:22445-51

[16] Chen Y,Liu Q,Zhang D.Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE).Phys Rev Res2022;4:023174

[17] Yazdani S.Data-driven discovery of turbulent flow equations using physics-informed neural networks.Phys Fluids2024;36:035107