Article

Boundary-integral Type Neural Network (BINN) for Flow Problems in Reservoirs

Liu Yina¹, Rao Xiang^2,3, He Xupeng^4,^*, Fu Qirun¹, Hussein Hoteit¹

¹1Physical Science and Engineering (PSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia
²School of Petroleum Engineering, Yangtze University, Wuhan, Hubei, China
³State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University), Wuhan 430100, China.

⁴Saudi Aramco, Dhahran, Saudi Arabia.

^*Correspondence: He Xupeng

Received: 2025-10-12Accepted: 2025-12-07Published: 2025-12-30

AESIG 2025, 1(1), 80-94; https://doi.org/10.58244/aesig.253463

Funding

This research was no funding provided.

Abstract

This study pioneers the application of Boundary-Integral Neural Networks (BINNs) to subsurface flow simulation, addressing steady-state single-phase flow governed by Laplace-type equations in hydrocarbon reservoirs. BINNs synergistically integrate boundary integral equations (BIEs) with deep learning to overcome limitations of traditional mesh-based methods and Physics-Informed Neural Networks (PINNs). By leveraging Green’s functions, BINNs transform partial differential equations into boundary-only formulations, where neural networks exclusively approximate boundary unknowns (pressure/flux), and interior solutions are reconstructed analytically. This approach achieves intrinsic dimensionality reduction, eliminating spatial domain discretization while ensuring mathematical consistency through exact boundary condition enforcement. Numerical validation demonstrates BINNs’ computational advantages: In a rectangular reservoir, BINNs achieve sub-0.01% relative error (4.80×10−4 MPa) at interior points. For a complex trapezoidal reservoir with geometric singularities, BINNs attain Boundary Element Method (BEM)-comparable accuracy (max error ~0.115MPa) without specialized singularity treatments. Furthermore, in a challenging non-convex domain featuring recessed boundaries and internal production wells, BINNs effectively resolve the pressure singularities near wellbores, achieving high-fidelity reconstruction with a maximum relative error of 0.32%. The method’s efficiency is evidenced by rapid convergence with minimal boundary sampling and moderate network sizes. BINNs provide a robust, meshless paradigm for reservoir-scale simulation, effectively handling irregular geometries while maintaining high fidelity and scalability, which uncovers its remarkable computational capabilities in the realm of single-phase flow, and offers an inaugural investigation and benchmark for its prospective extensive utilization in numerical reservoir simulations.

Keywords: Deep Learning; Boundary integral equations (BIEs); Physics-informed neural networks (PINNs); Boundary-Integral Neural Networks(BINNs); Meshless Reservoir Simulation; Dimensionality Reduction

Full Text

Download the full text PDF for viewing and using it according to the license of this paper.

References

Brebbia, C. A., Telles, J. C. F., & Wrobel, L. C. (2012). Boundary element techniques: theory and applications in engineering. Springer Science & Business Media.
Dai, Y., Li, Z., An, Y., & Deng, W. (2024). Numerical solutions of boundary problems in partial differential equations: A deep learning framework with Green's function. Journal of Computational Physics, 511, 113121
Damiano, S., & van Waterschoot, T. (2025). Sound Field Reconstruction Using Physics-Informed Boundary Integral Networks. arXiv preprint arXiv:2506.03917.
Hughes, T. J. (2003). The finite element method: linear static and dynamic finite element analysis. Courier Corporation.
Krishnapriyan, A., Gholami, A., Zhe, S., Kirby, R., & Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in neural information processing systems, 34, 26548-26560.
LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems (Vol. 31). Cambridge university press.
Lin, G., Hu, P., Chen, F., Chen, X., Chen, J., Wang, J., & Shi, Z. (2023). BINet: Learn to solve partial differential equations with boundary integral networks. CSIAM Transactions on Applied Mathematics, 4, 275–305.
Liu, B., Wei, J., Kang, L., Liu, Y., & Rao, X. (2025). Physics-informed neural network (PINNs) for convection equations in polymer flooding reservoirs. Physics of Fluids, 37(3).
Meng B.,Lu Y., &Jiang Y. (2024). Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations. arXiv:2406.02298.
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378, 686-707.
Rao, X. (2023). A generic workflow of projection-based embedded discrete fracture model for flow simulation in porous media. Computational Geosciences, 27(4), 561-590.
Rao, X., Cheng, L., Cao, R., Jiang, J., Fang, S., Jia, P., & Wang, L. (2018). A novel Green element method based on two sets of nodes. Engineering Analysis with Boundary Elements, 91, 124-131.
Rao, X., Cheng, L., Cao, R., Zhang, X., & Dai, D. (2019). A mimetic Green element method. Engineering Analysis with Boundary Elements, 99, 206-221.
Rao, X., He, X., Du, K., Kwak, H., Yousef, A., & Hoteit, H. (2024). A novel projection-based embedded discrete fracture model (pEDFM) for anisotropic two-phase flow simulation using hybrid of two-point flux approximation and mimetic finite difference (TPFA-MFD) methods. Journal of Computational Physics, 499, 112736.
Rao, X., Zhao, H., & Liu, Y. (2024). A novel meshless method based on the virtual construction of node control domains for porous flow problems. Engineering with Computers, 40(1), 171-211.
Rao, X., Liu, Y., He, X., & Hoteit, H. (2025). Physics-informed Kolmogorov–Arnold networks to model flow in heterogeneous porous media with a mixed pressure-velocity formulation. Physics of Fluids, 37(7).
Rao, X., Liu, Y., & Shen, Y. (2025). Quantum-Classical Physics-Informed Neural Networks for Solving Reservoir Seepage Equations. arXiv preprint arXiv:2512.03923.
Sun, J., Liu, Y., Wang, Y., Yao, Z., & Zheng, X. (2023). BINN: A deep learning approach for computational mechanics problems based on boundary integral equations. Computer Methods in Applied Mechanics and Engineering, 410, 116012.
Wang, S., Wang, H., & Perdikaris, P. (2021). On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 384, 113938.
Zhang, H., Anitescu, C., Bordas, S., Rabczuk, T., & Atroshchenko, E. (2022). Artificial neural network methods for boundary integral equations [Preprint]. TechRxiv.
Zhang, P., Xie, L., Gu, Y., Qu, W., Zhao, S., & Zhang, C. (2024). Boundary integrated neural networks for 2D elastostatic and piezoelectric problems. International Journal of Mechanical Sciences, 280, 109525.