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Application of Graph Theory to Analyze Gaps in Water and Utility Networks to Enhance the Resilience of Urban Infrastructure

Authors

  • zeyad abed Al-Nahrain University Author

DOI:

https://doi.org/10.65204/djes.v3i1.639

Keywords:

Temporal graphs, Inverse modeling, Differential equations, Graph neural networks, Physics-informed machine learning

Abstract

The temporal graph data offers an effective model of complex time-dependent systems across various systems such as traffic networks, brain connectivity, weather forecasting, epidemiology and physical systems. But there is still a real challenge of closing the divide between the discrete-time observations of networks and continuous-time dynamical models. It is a systematic review of the approaches to extracting effective differential equations to the data of temporal graphs by means of inverse modeling, focusing on rigorous mathematical backgrounds. We divide the existing methods into such families as graph neural differential equations, continuous-time representation learning, PDE discovery frameworks, operator learning methods, and physics-informed universal differential systems. The theoretical matters covered in the review are identifiability, stability, scalability, and robustness, with practical issues being model validation and cross-domain applicability. Inverse methods have been applied to solve problems in traffic, neuroscience, weather, epidemiology, and groundwater systems to demonstrate how the inventive methods can be used to reconstruct interpretable and physically significant governing equations. The synthesis gives a full conceptual and mathematical view of the temporal graph based inverse modeling and sets the basis of future studies in data discovery of complex dynamical systems.

References

References:

Gravina, A., Zambon, D., Bacciu, D., & Alippi, C. (2024). Temporal graph odes for irregularly-sampled time series. arXiv preprint arXiv:2404.19508.

Liu, Z., Wang, X., Wang, B., Huang, Z., Yang, C., & Jin, W. (2025, August). Graph odes and beyond: A comprehensive survey on integrating differential equations with graph neural networks. In Proceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining V. 2 (pp. 6118-6128).

Eliasof, M., Haber, E., Treister, E., & Schönlieb, C. B. B. (2024, April). On the temporal domain of differential equation inspired graph neural networks. In International Conference on Artificial Intelligence and Statistics (pp. 1792-1800). PMLR.

Wang, Q., Feng, S., & Han, M. (2025). Causal graph convolution neural differential equation for spatio-temporal time series prediction. Applied Intelligence, 55(7), 671.

Fang, Z., Long, Q., Song, G., & Xie, K. (2021, August). Spatial-temporal graph ode networks for traffic flow forecasting. In Proceedings of the 27th ACM SIGKDD conference on knowledge discovery & data mining (pp. 364-373).

Tang, H., Liu, G., Dai, S., Ye, K., Zhao, K., Wang, W., ... & Zhan, L. (2024, October). Interpretable spatio-temporal embedding for brain structural-effective network with ordinary differential equation. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 227-237). Cham: Springer Nature Switzerland.

Shi, H., & Zhang, W. (2025). Learning continuous spatial-temporal evolution via dynamic graph neural differential equations for traffic flow forecasting. Information Sciences, 700, 121845.

Han, M., & Wang, Q. (2024). Adaptive graph convolution neural differential equation for spatio-temporal time series prediction. IEEE Transactions on Knowledge and Data Engineering.

Zhou, J., Qin, X., Ding, Y., & Ma, H. (2023). Spatial–Temporal Dynamic Graph Differential Equation Network for Traffic Flow Forecasting. Mathematics, 11(13), 2867.

Liang, Y., Ouyang, K., Yan, H., Wang, Y., Tong, Z., & Zimmermann, R. (2021, August). Modeling Trajectories with Neural Ordinary Differential Equations. In IJCAI (pp. 1498-1504).

Iakovlev, V., Heinonen, M., & Lähdesmäki, H. (2020). Learning continuous-time pdes from sparse data with graph neural networks. arXiv preprint arXiv:2006.08956.

Brunton, S. L., & Kutz, J. N. (2024). Promising directions of machine learning for partial differential equations. Nature Computational Science, 4(7), 483-494.

Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., & Anandkumar, A. (2020). Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895.

Tanyu, D. N., Ning, J., Freudenberg, T., Heilenkötter, N., Rademacher, A., Iben, U., & Maass, P. (2023). Deep learning methods for partial differential equations and related parameter identification problems. Inverse Problems, 39(10), 103001.

Zhang, D., Guo, L., & Karniadakis, G. E. (2020). Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks. SIAM Journal on Scientific Computing, 42(2), A639-A665.

Maslyaev, M., Hvatov, A., & Kalyuzhnaya, A. V. (2021). Partial differential equations discovery with EPDE framework: Application for real and synthetic data. Journal of Computational Science, 53, 101345.

Li, X., Wong, T. K. L., Chen, R. T., & Duvenaud, D. (2020, June). Scalable gradients for stochastic differential equations. In International Conference on Artificial Intelligence and Statistics (pp. 3870-3882). PMLR.

Xia, Y., Liang, Y., Wen, H., Liu, X., Wang, K., Zhou, Z., & Zimmermann, R. (2023). Deciphering spatio-temporal graph forecasting: A causal lens and treatment. Advances in Neural Information Processing Systems, 36, 37068-37088.

Ma, M., Xie, P., Teng, F., Wang, B., Ji, S., Zhang, J., & Li, T. (2023). HiSTGNN: Hierarchical spatio-temporal graph neural network for weather forecasting. Information Sciences, 648, 119580.

Wang, H., & Yamamoto, N. (2020). Using a partial differential equation with Google Mobility data to predict COVID-19 in Arizona. arXiv preprint arXiv:2006.16928.

Depina, I., Jain, S., Mar Valsson, S., & Gotovac, H. (2022). Application of physics-informed neural networks to inverse problems in unsaturated groundwater flow. Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 16(1), 21-36.

Huang, S., Feng, W., Tang, C., He, Z., Yu, C., & Lv, J. (2025). Partial differential equations meet deep neural networks: A survey. IEEE Transactions on Neural Networks and Learning Systems.

Shukla, K., Xu, M., Trask, N., & Karniadakis, G. E. (2022). Scalable algorithms for physics-informed neural and graph networks. Data-Centric Engineering, 3, e24.

Nadeem, M., He, J. H., & Islam, A. (2021). The homotopy perturbation method for fractional differential equations: part 1 Mohand transform. International Journal of Numerical Methods for Heat & Fluid Flow, 31(11), 3490-3504.

Li, Z., Meidani, K., & Farimani, A. B. (2022). Transformer for partial differential equations' operator learning. arXiv preprint arXiv:2205.13671.

Rackauckas, C., Ma, Y., Martensen, J., Warner, C., Zubov, K., Supekar, R., ... & Edelman, A. (2020). Universal differential equations for scientific machine learning. arXiv preprint arXiv:2001.04385.

Bolin, D., & Wallin, J. (2020). Multivariate type G Matérn stochastic partial differential equation random fields. Journal of the Royal Statistical Society Series B: Statistical Methodology, 82(1), 215-239.

Hao, Z., Liu, S., Zhang, Y., Ying, C., Feng, Y., Su, H., & Zhu, J. (2022). Physics-informed machine learning: A survey on problems, methods and applications. arXiv preprint arXiv:2211.08064.

Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422-440.

Liu, X. Y., Zhu, M., Lu, L., Sun, H., & Wang, J. X. (2024). Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics. Communications Physics, 7(1), 31.

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Published

2026-03-22

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Issue

Vol. 3 No. 1 (2026): Vol.3, Assue 1, March 2026

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Articles