Hybrid DeepONet Architectures for Porous Media Flow Simulation

The solution of partial differential equations (PDEs) plays a central role in several areas of science and engineering. With the advancement of deep learning and the growing interest in data-driven methods, new approaches have emerged that aim to capture the behavior of PDE solutions. As surrogate models, Neural Operators [1] have gained notoriety for learning parameter-to-solution operators, enabling the learning and representation of families of PDE solutions. On the other hand, Kolmogorov–Arnold Networks (KANs) [2] represent an alternative to traditional MLP-type neural networks, replacing fixed linear connections with compositions of learnable univariate functions. This work presents hybrid models of Deep Operator Networks (DeepONet) [3]. As a distinguishing feature, we integrated different DeepONet architectures within the NVIDIA PhysicsNeMo framework, exploring both native models, such as the Fourier Neural Operator (FNO) and MLP, as well as a custom implementation of KAN. We applied these hybrid models to the context of porous media, where the partial differential equations are governed by system characteristics that model fluid flow, as in the Tenth SPE Comparative Solution Project [4], which consists of a transient two-phase flow application in a 2D spatial porous medium domain. The entire modeling process, from construction to training and inference, was carried out entirely within the PhysicsNeMo environment, with the goal of investigating the impact of combining different architectures.   REFERENCES  [1] Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A. and Anandkumar, A. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020.   [2] Liu, Z., Wang, Y., Vaidya, S., Ruehle, F., Halverson, J., Soljačić, M., Hou, TY. and Tegmark, M. Kan: Kolmogorov-arnold networks. arXiv preprint arXiv:2404.19756, 2024. [3] Lu, L., Jin, P., and Karniadakis, G. E. Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. arXiv preprint arXiv:1910.03193, 2019. [4] Christie, M. A., and Blunt, M. J. Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. SPE Reservoir Evaluation & Engineering 4(04): 308–317, 2001.