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

Area of Science:

• Scientific Computing
• Quantum Machine Learning
• Bayesian Inference

Background:

• Bayesian inverse problems are computationally intensive.
• Bayesian Physics-Informed Neural Networks (B-PINNs) offer uncertainty quantification but face high sampling costs due to large parameter spaces.
• Existing methods struggle with efficiency in high-dimensional parameter spaces.

Purpose of the Study:

• To develop a more efficient framework for solving Bayesian inverse problems.
• To reduce the computational cost associated with uncertainty quantification in complex systems.
• To explore the synergy between quantum computing and physics-informed neural networks.

Main Methods:

• Introduction of Quantum-Encodable Bayesian PINNs (QE-BPINNs).
• Integration of Quantum Neural Networks (QNNs) as surrogate models for Partial Differential Equation (PDE) solutions.
• Training the QE-BPINNs using Classical Ensemble Kalman Inversion (EKI) to avoid barren plateaus.
• Benchmarking on 1D and 2D nonlinear PDEs with noisy data.

Main Results:

• QE-BPINNs demonstrate significant parameter compression compared to classical networks.
• The QEKI framework achieves precise inversions even with noisy data.
• The hybrid approach effectively captures complex physics with fewer parameters.
• Successful application to nonlinear PDEs in 1D and 2D benchmarks.

Conclusions:

• QE-BPINNs offer a viable hybrid framework for Bayesian uncertainty quantification.
• The QEKI method provides an efficient alternative to traditional sampling techniques.
• This approach shows promise for reducing computational bottlenecks in scientific computing.
• Further development is needed for large-scale quantum hardware implementation.