The new model, called HG-KAN, short for heterogeneous graph Kolmogorov-Arnold network, was developed by Faming Gong, Junjie Xu, Yanpu Zhao, Xiaofeng Ji, and Chengze Du. Their goal was not to replace reservoir simulators entirely but to build a fast, reliable proxy that engineers could run over thousands of candidate injection strategies during operational screening, the stage at which the fate of a waterflood plan is often decided. The work was partially supported by the Natural Science Foundation of Shandong Province and the Qingdao Natural Science Foundation, and it reflects a growing trend in petroleum engineering: treating the subsurface as a network rather than a grid.

That network view is the first key ingredient. In HG-KAN, wells and geological layers are represented as different types of nodes in a heterogeneous graph, and the connections between them are not arbitrary. Instead, the graph construction is guided by physics, specifically by layer-wise transmissibility relations, which quantify how easily fluid can flow from one part of the reservoir to another through each geological layer. This matters because two injection wells sitting side by side on a map may be practically strangers in the subsurface if a fault or a low-permeability barrier separates them, while wells kilometers apart can be intimately connected through a highly transmissive sand layer. By encoding these layer-specific connectivity relationships directly into the graph structure, the model gives the neural network a head start: it does not have to rediscover the reservoir’s plumbing from production data alone.

The second key ingredient is the Kolmogorov-Arnold network, or KAN, an architecture that has generated considerable excitement since its introduction. Where conventional multi-layer perceptrons, the standard workhorse of deep learning, place fixed activation functions on nodes and learn weights on edges, KANs invert that arrangement: they place learnable, flexible activation functions on the edges themselves. This design, rooted in the Kolmogorov-Arnold representation theorem from mathematics, allows KANs to approximate complicated multivariate functions with potentially fewer parameters and, crucially for this application, with greater interpretability, because each learned edge function can be inspected and visualized. The authors argue that fixed nonlinear blocks like those in MLPs can reduce robustness and interpretability in strongly heterogeneous settings, precisely the conditions found in layered fault-block reservoirs where the relationship between injection rates and production responses shifts dramatically from one layer to the next.

HG-KAN combines these ideas in a dual-stream framework for injection-rate allocation. One stream processes the physics-guided heterogeneous graph, propagating information between wells and layers along transmissibility-weighted connections, while the architecture as a whole is designed to capture how fluid injected at one well propagates through specific layers and influences production at distant wells. The authors are careful to note an important nuance: while the graph construction is physics-guided, the current training objective does not explicitly include partial differential equation residual constraints of the kind found in fully physics-informed neural networks. In other words, the physics enters through the structure of the model rather than through the loss function, a design choice that keeps training simpler while still respecting the geology that governs subsurface flow.

The team evaluated the model on two proprietary field datasets drawn from a fault-block reservoir, a setting notorious for its geological complexity. The first was a 20-well pilot block, a smaller testbed where the model achieved a root mean square error of 37.31 cubic meters per day in predicting injection-production behavior. The second, more demanding dataset covered a 67-well management area, a scale at which inter-well interactions multiply combinatorially, and there HG-KAN reached an RMSE of 33.72 cubic meters per day. In both cases the model consistently outperformed strong heterogeneous graph baselines in the authors’ experiments, suggesting that the KAN-based nonlinear blocks, rather than the graph structure alone, contribute meaningfully to the accuracy gains.

Speed is where the results become genuinely striking for field operations. HG-KAN delivered inference times of 340 milliseconds per 100 samples on the pilot block and 884 milliseconds per 100 samples on the management area. Those figures translate into a practical capability that numerical simulation cannot match at screening scale: an engineer could evaluate thousands of injection-rate allocation scenarios in the time it takes a simulator to grind through a handful. In digital oilfield workflows, where operators increasingly rely on continuous data streams from smart wells to adjust waterflood strategies in near real time, that kind of throughput is not a luxury. It is the difference between reacting to reservoir behavior weeks after it happens and anticipating it.

The implications extend beyond one reservoir or one company. Waterflooding accounts for a substantial share of global oil production, and as fields mature, the fraction of water produced alongside oil tends to climb, driving up energy costs and environmental burdens. Studies of heterogeneous carbonate reservoirs have shown how uneven water injection can leave large volumes of oil unswept while water cycles uselessly through high-permeability channels. A surrogate model that respects layer-wise connectivity could help operators steer injected water into the layers that need it, improving sweep efficiency and reducing wasted injection. The same graph-based reasoning, moreover, echoes developments elsewhere in the geosciences, from graph neural networks for groundwater contaminant tracking to transformer models for production forecasting in unconventional reservoirs, indicating a broader convergence on relational, physics-aware machine learning for subsurface problems.

The authors are candid about the framework’s boundaries. The training data are proprietary and confidential field data that cannot be publicly shared, which means independent verification will depend on other operators testing the approach on their own assets. The absence of explicit PDE-residual constraints in the training objective leaves open the possibility that future versions could incorporate those residuals, potentially improving generalization in scenarios far from the training distribution. And like all data-driven surrogates, HG-KAN is only as trustworthy as the historical injection and production records it learns from, which in mature fields can be noisy, incomplete, and shaped by decades of changing operational practices.

Still, the study marks a notable step in a fast-moving field. Kolmogorov-Arnold networks have already been adapted for time series forecasting, anomaly detection, molecular property prediction, and physics-informed solutions of partial differential equations, and their marriage with heterogeneous graph structures for reservoir engineering appears here as one of the more consequential applications to date. If the accuracy and speed reported on these two fault-block datasets hold up across other geological settings, the humble decision of how much water to inject into which well, long a matter of simulation campaigns and engineering intuition, could become a task performed in milliseconds on a graph that knows the reservoir’s hidden architecture. For an industry under pressure to extract more with less, that is a compelling proposition, and it signals that the next generation of oilfield optimization may be written not in finite-difference solvers but in learnable functions along the edges of a graph.

Subject of Research: A physics-guided heterogeneous graph Kolmogorov-Arnold network for optimizing water injection rates in multi-layer oil reservoirs

Article Title: HG-KAN: heterogeneous graph kolmogorov-arnold network for multi-layer reservoir injection optimization

Article References: Gong, F., Xu, J., Zhao, Y., Ji, X., & Du, C. (2026). HG-KAN: heterogeneous graph kolmogorov-arnold network for multi-layer reservoir injection optimization. Cluster Computing, 29(15), Article 840. https://doi.org/10.1007/s10586-026-06623-7

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06623-7

Keywords: HG-KAN, Kolmogorov-Arnold networks, heterogeneous graph neural networks, reservoir optimization, waterflooding, injection-rate allocation, physics-guided machine learning, surrogate modeling, digital oilfield, fault-block reservoir, multi-layer reservoir, machine learning