Variational Quantum Physics-Informed Neural Networks for Hydrological PDE-Constrained Learning with Inherent Uncertainty Quantification

Prasad Nimantha Madusanka Ukwatta Hewage; Midhun Chakkravarthy; Ruvan Kumara Abeysekara

quant-phApr 10, 2026

Variational Quantum Physics-Informed Neural Networks for Hydrological PDE-Constrained Learning with Inherent Uncertainty Quantification

Authors: Prasad Nimantha Madusanka Ukwatta Hewage, Midhun Chakkravarthy, Ruvan Kumara Abeysekara

Organizations: Faculty of Computer Science and Multimedia, Lincoln University College, Petaling Jaya, Selangor, Malaysia · BCAS Campus, Colombo, Sri Lanka

Abstract

We propose a Hybrid Quantum-Classical Physics-Informed Neural Network (HQC-PINN) that integrates parameterized variational quantum circuits into the PINN framework for hydrological PDE-constrained learning. Our architecture encodes multi-source remote sensing features into quantum states via trainable angle encoding, processes them through a hardware-efficient variational ansatz with entangling layers, and constrains the output using the Saint-Venant shallow water equations and Manning's flow equation as differentiable physics loss terms. The inherent stochasticity of quantum measurement provides a natural mechanism for uncertainty quantification without requiring explicit Bayesian inference machinery. We further introduce a quantum transfer learning protocol that pre-trains on multi-hazard disaster data before fine-tuning on flood-specific events. Numerical simulations on multi-modal satellite and meteorological data from the Kalu River basin, Sri Lanka, show that the HQC-PINN achieves convergence in ~3x fewer training epochs and uses ~44% fewer trainable parameters compared to an equivalent classical PINN, while maintaining competitive classification accuracy. Theoretical analysis indicates that hydrological physics constraints narrow the effective optimization landscape, providing a natural mitigation against barren plateaus in variational quantum circuits. This work establishes the first application of quantum-enhanced physics-informed learning to hydrological prediction and demonstrates a viable path toward quantum advantage in environmental science.

Figures & tables

Model	Qubits	Layers	Epochs to target
cPINN	—	4	94
HQC-PINN	4	2	42
HQC-PINN	4	3	36
HQC-PINN	8	2	33
HQC-PINN	8	3	26
HQC-PINN	8	4	28

Table 1: Convergence comparison: epochs to reach target validation loss

\mathcal{L}_{\text{val}}\leq 0.40

. HQC-PINN configurations converge 2.6–3.6

\times

faster than the classical PINN, consistent with prior qPINN convergence results [ 10 ] .

Model	Acc. (%)	F1-macro	Params	Acc./kP
Random Forest	90.3	0.899	N/A	—
cPINN	69.4	0.705	33,793	2.05
Classical BNN	91.96	—	28,672	3.21
VQC-only (8q, 3L)	67.8	0.682	12,480	5.43
HQC-PINN (4q, 3L)	70.2	0.714	14,468	4.85
HQC-PINN (8q, 3L)	71.8	0.731	18,944	3.79

Table 2: Test set classification performance. Accuracy, macro-averaged F1 score, parameter count, and parameter efficiency ratio (accuracy per 1000 parameters). Best quantum result in bold.

Component	cPINN	HQC-PINN
Classical layers	33,793	18,896
Quantum parameters (VQC)	0	48
Total	33,793	18,944
Reduction	—	43.9%

Table 3: Parameter breakdown for HQC-PINN (8q, 3L) versus classical PINN. The VQC replaces two classical hidden layers while the pre-processing network performs learned dimensionality reduction to the qubit register.

Model	Coverage (%)	Entropy	$\sigma^{2}_{\text{aleatoric}}$
Classical BNN	92.3	0.320	0.028
HQC-PINN (8q, 3L)	88.7	0.341	0.033
QTL (8q, 3L)	90.1	0.312	0.026

Table 4: Uncertainty quantification comparison. Coverage at the 90% nominal level, average predictive entropy, and mean aleatoric uncertainty.

Configuration	Acc. (%)	Epochs
Full HQC-PINN	71.8	26
$-$ Saint-Venant loss ( $\lambda_{\text{SV}}=0$ )	68.3	41
$-$ Manning loss ( $\lambda_{\text{M}}=0$ )	70.4	30
$-$ Both physics losses	67.8	51
$-$ Quantum layer (classical only)	69.4	94
$-$ Entanglement (product state only)	69.1	38

Table 5: Ablation study on the HQC-PINN (8q, 3L) configuration. Removing physics constraints or the quantum layer degrades performance.

Work	Physics	UQ	Multi-modal	Transfer
Grzesiak et al. [ 12 ]	×	×	×	×
Dutta et al. [ 11 ]	✓ a	×	×	×
Klement et al. [ 10 ]	✓ b	×	×	×
Rodriguez-Grasa et al. [ 19 ]	×	×	×	×
This work	✓ c	✓	✓	✓

Table 6: Comparison with prior quantum approaches to flood/environmental prediction. Our work is the first to combine quantum circuits with hydrological PDE constraints and multi-modal remote sensing data.

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

Xiang Rao, Yina Liu, Yuxuan Shen

School of Petroleum Engineering, Yangtze University, Wuhan 430100, China · College of Future Technology, Yangtze University, Wuhan 430100, China · State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University), Wuhan 430100, China+2

PDE Solving Physics-Informed ML Hybrid Quantum-Classical ML

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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.

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