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

TL;DR

This paper introduces a hybrid quantum-classical neural network for hydrological PDE learning, leveraging quantum measurement stochasticity for uncertainty quantification and demonstrating improved efficiency over classical methods.

Contribution

It presents the first quantum-enhanced physics-informed neural network for hydrological modeling, integrating variational quantum circuits and a transfer learning protocol.

Findings

Achieves convergence in ~3x fewer epochs than classical PINNs.

Uses ~44% fewer trainable parameters while maintaining accuracy.

Quantum physics constraints help mitigate barren plateaus in training.

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.