Hybrid DeepONet Surrogates for Multiphase Flow in Porous Media

TL;DR

This paper addresses the computational burden of surrogate modeling for multiphase flow in porous media by proposing Hybrid DeepONet architectures that decouple spatial and temporal learning. By integrating Fourier Neural Operators (FNOs), Multi-Layer Perceptrons (MLPs), and Kolmogorov–Arnold Networks (KANs) within a DeepONet framework, the authors create branch and trunk networks that handle space and time separately, reducing memory requirements. Through numerical experiments on a progression from 2D steady Darcy flow to 3D SPE10 reservoir models, the hybrids achieve accurate predictions with significantly fewer parameters than traditional neural operators, with FNO-based branches often delivering the best performance, including for post-processed scalars like oil production rates and water cuts. The work demonstrates scalability to large-scale reservoir simulations and highlights the potential for efficient digital-twin workflows in petroleum engineering.

Abstract

The solution of partial differential equations (PDEs) plays a central role in numerous applications in science and engineering, particularly those involving multiphase flow in porous media. Complex, nonlinear systems govern these problems and are notoriously computationally intensive, especially in real-world applications and reservoirs. Recent advances in deep learning have spurred the development of data-driven surrogate models that approximate PDE solutions with reduced computational cost. Among these, Neural Operators such as Fourier Neural Operator (FNO) and Deep Operator Networks (DeepONet) have shown strong potential for learning parameter-to-solution mappings, enabling the generalization across families of PDEs. However, both methods face challenges when applied independently to complex porous media flows, including high memory requirements and difficulty handling the time dimension. To address these limitations, this work introduces hybrid neural operator surrogates based on DeepONet models that integrate Fourier Neural Operators, Multi-Layer Perceptrons (MLPs), and Kolmogorov-Arnold Networks (KANs) within their branch and trunk networks. The proposed framework decouples spatial and temporal learning tasks by splitting these structures into the branch and trunk networks, respectively. We evaluate these hybrid models on multiphase flow in porous media problems ranging in complexity from the steady 2D Darcy flow to the 2D and 3D problems belonging to the $10$th Comparative Solution Project from the Society of Petroleum Engineers. Results demonstrate that hybrid schemes achieve accurate surrogate modeling with significantly fewer parameters while maintaining strong predictive performance on large-scale reservoir simulations.

Figures (20)

• Figure 1: DeepONet Architecture. The model is comprised of two subnetworks: a branch and a trunk network. The branch network takes as input the function $v$, represented by the vector $\textbf{v}$, and the trunk network takes as input the query point $\xi$. The outputs of the networks are combined through a merge operation (inner product, Hadamard product, linear/non-linear transformation), and the model outputs the result of the operator $\mathcal{G}(\textbf{v})$ in query point $\xi$, that is, $\mathcal{G}(\textbf{v})(\xi)$.
• Figure 2: Fourier Neural Operator architecture. Figure (a) shows the overall structure, in which the input is first lifted to a higher dimension through operator $\textbf{P}$, then passed through Fourier layers, and lastly projected back into the output space dimension through operation $\textbf{Q}$. Figure (b) shows the structure of a Fourier layer, which comprises of the application of the FFT ($\mathcal{F}$), a linear transformation $\textbf{R}$ and the reverse FFT ($\mathcal{F}^{-1}$), then the result is summed with a local linear transformation $\textbf{W}$, and lastly a non-linear activation function $\sigma$ is applied, generating the layer output.
• Figure 3: Hybrid DeepONet model. The input features are subdivided into spatial $\textbf{v}$ and temporal $\boldsymbol{\xi}$ information. The spatial information $\textbf{v}$ is processed by the branch network, which can encapsulate an FNO, a KAN, or an MLP model, while the temporal information $\boldsymbol{\xi}$ is processed by the trunk network, which comprises a KAN or an MLP model. The outputs of each subnetwork are then combined through a Hadamard product to generate the model output $\mathcal{G}(\mathbf{v})(\boldsymbol{\xi})$.
• Figure 4: Training (a) and test (b) relative error $2-$norms of the proposed hybrid models throughout 500 training epochs in the Darcy flow problem.
• Figure 5: Test results for 2D Darcy flow. The column on the left contains the ground truth pressure for comparison, the middle column the predicted pressure fields, and the right column the pointwise abolute error for: (a) DeepONet (FNO + KAN), $||p||_{\infty} = 0.201$ and $||p||_{rel} = 0.0495$; (b) DeepONet (FNO + MLP), $||p||_{\infty} = 0.184$ and $||p||_{rel} = 0.0476$; (c) DeepONet (KAN), $||p||_{\infty} = 0.190$ and $||p||_{rel} = 0.0545$; and (d) DeepONet (MLP), $||p||_{\infty}= 0.234$ and $||p||_{rel} = 0.0635$.

Theorems & Definitions (8)

• Lemma A.1: Spline Approximation
• proof
• Theorem A.1: MLP-KAN Equivalence
• proof
• Corollary A.1
• proof
• Theorem A.2: Uniform Convergence
• proof