# Quantum-Classical Physics-Informed Neural Networks for Solving Reservoir Seepage Equations

arXiv: https://arxiv.org/abs/2512.03923 · HTML: https://mlsanity.com/paper/2512.03923/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 6 references that are in the map.
- First posted on arXiv 3 Dec 2025.

## Builds on (most cited first, 1 of 1)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)

## Built on by (most cited first, 2 of 2)

- [Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs](https://mlsanity.com/paper/2606.20326/) (2026)
- [Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations](https://mlsanity.com/paper/2608.08782/) (2026)

## Abstract (arXiv)

Solving reservoir seepage partial differential equations (PDEs) is critical for hydrocarbon development optimization and production forecasting. Conventional numerical solvers suffer from mesh errors and high computational cost. Standard Physics-Informed Neural Networks (PINNs) are limited by inefficient parameters. They also face weak high-dimensional representation and poor nonlinear fitting ability. We adapt the Discrete Variable (DV)-Circuit Quantum-Classical Physics-Informed Neural Network (QCPINN) to tackle these drawbacks. This work applies the method to four representative reservoir seepage models. The models cover heterogeneous single-phase pressure diffusion. They also include nonlinear Buckley-Leverett two-phase waterflooding. Another case is adsorption-coupled convection-diffusion compositional flow. The last one refers to fully coupled pressure-saturation two-phase flow with exponential permeability heterogeneity. The QCPINN combines classical preprocessing and postprocessing networks with a DV quantum core. It leverages quantum entanglement and superposition to boost high-dimensional feature mapping. Physical constraints are embedded within the framework to guarantee physically consistent solutions. We evaluate three quantum circuit topologies including Cascade, Cross-mesh and Alternate. Numerical tests prove QCPINNs outperform classical PINNs in prediction accuracy, and the circuit topologies show case-specific performance with clear applicable ranges. This work verifies QCPINN's practicability for reservoir engineering. It bridges quantum computing research with oilfield industrial applications.