Physics-informed Kolmogorov–Arnold networks to model flow in heterogeneous porous media with a mixed pressure-velocity formulation

Kolmogorov–Arnold networks (KANs), introduced in May 2024, present a novel network structure. Early research shows that they outperform multilayer perceptrons (MLPs) in computational efficiency, interpretability, and interaction. In MLP-based physics-informed neural networks (PINNs) for flow simulation in heterogeneous reservoirs, the mixed pressure-velocity formulation displays superior robustness and accuracy compared to the pure pressure formulation. This paper aims to create the first physics-informed KAN (PIKAN) by replacing MLP with KAN in the PINN and employing the mixed pressure-velocity formulation, assessing its computational performance in heterogeneous reservoir flow simulations. To build the PIKAN using a mixed pressure-velocity formulation, spatial coordinates serve as inputs, with pressure and velocity components as outputs. We use three neural networks to approximate pressure and the velocity components, respectively, and the model is referred to as P-V-3-PIKAN. The loss function, formulated by integrating the mixed formulation along with Dirichlet and Neumann boundary conditions, is meticulously optimized to facilitate the continuous refinement of PIKAN parameters. This mixed pressure-velocity formulation allows for automatic differentiation of the loss function, without evaluating discontinuous permeability distributions. Training and performance evaluation of the PIKANs conclude upon meeting accuracy criteria or reaching the maximum optimization steps. Four numerical experiments were conducted to assess the performance of P-V-3-PIKAN, as well as P-PIKAN using the pure pressure formulation, and P-V-3-PINN. Their efficacy was evaluated by comparing outcomes against high-fidelity benchmarks across various scenarios, encompassing unidirectional and multidirectional flows within heterogeneous reservoirs. The results indicate two key findings: First, P-V-3-PIKAN achieves superior convergence and significantly lower computational errors compared to P-V-3-PINN. This suggests that the PIKAN framework, which is predicated on the KAN model, outperforms the PINN framework, which is based on MLP. Second, when compared to P-V-3-PIKAN, which employs the mixed formulation, P-PIKAN, which uses a pure pressure formulation, exhibits notably higher computational errors. Particularly for seepage problems in reservoirs with zoned or discontinuous heterogeneity that cannot be expressed by smooth analytical functions, P-PIKAN fails to effectively capture this heterogeneity. This underscores the necessity of using mixed formulation over pure pressure formulation for handling seepage issues in heterogeneous reservoirs. This study introduces the promising KAN into flow simulation in porous media for the first time, and provides an initial reference for developing universal seepage simulation tools based on PIKAN.