QEKI: A Quantum–Classical Framework for Efficient Bayesian Inversion of PDEs

Solving Bayesian inverse problems efficiently stands as a major bottleneck in scientific computing. Although Bayesian Physics-Informed Neural Networks (B-PINNs) have introduced a robust way to quantify uncertainty, the high-dimensional parameter spaces inherent in deep learning often lead to prohibitive sampling costs. Addressing this, our work introduces Quantum-Encodable Bayesian PINNs trained via Classical Ensemble Kalman Inversion (QEKI), a framework that pairs Quantum Neural Networks (QNNs) with Ensemble Kalman Inversion (EKI). The core advantage lies in the QNN's ability to act as a compact surrogate for PDE solutions, capturing complex physics with significantly fewer parameters than classical networks. By adopting the gradient-free EKI for training, we mitigate the barren plateau issue that plagues quantum optimization. Through several benchmarks on 1D and 2D nonlinear PDEs, we show that QEKI yields precise inversions and substantial parameter compression, even in the presence of noise. While large-scale applications are constrained by current quantum hardware, this research outlines a viable hybrid framework for including quantum features within Bayesian uncertainty quantification.