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Leveraging SPD Matrices on Riemannian Manifolds in Quantum Classical Hybrid Models for Structural Health Monitoring

Azadeh Alavi, Sanduni Jayasinghe

TL;DR

The paper addresses real-time FEM-based structural health monitoring of bridges by mapping a $7$-dimensional input to a $1017$-dimensional output under tight time constraints. It introduces a hybrid quantum-classical multilayer perceptron that operates on Symmetric Positive Definite (SPD) matrices within the Riemannian manifold and uses polynomial feature expansion for nonlinear representation, followed by amplitude encoding for quantum processing. Through eigenvector decomposition of SPD matrices and normalization, data are prepared for quantum circuits, and three architectures are compared, with the SPD-Enhanced Hybrid achieving the best performance at $MSE = 0.00031$. The results support SPD-based representations as a viable path to improved real-time SHM fem analyses in structural engineering, suggesting further exploration of advanced hybrid networks for practical deployment.

Abstract

Realtime finite element modeling of bridges assists modern structural health monitoring systems by providing comprehensive insights into structural integrity. This capability is essential for ensuring the safe operation of bridges and preventing sudden catastrophic failures. However, FEM computational cost and the need for realtime analysis pose significant challenges. Additionally, the input data is a 7 dimensional vector, while the output is a 1017 dimensional vector, making accurate and efficient analysis particularly difficult. In this study, we propose a novel hybrid quantum classical Multilayer Perceptron pipeline leveraging Symmetric Positive Definite matrices and Riemannian manifolds for effective data representation. To maintain the integrity of the qubit structure, we utilize SPD matrices, ensuring data representation is well aligned with the quantum computational framework. Additionally, the method leverages polynomial feature expansion to capture nonlinear relationships within the data. The proposed pipeline combines classical fully connected neural network layers with quantum circuit layers to enhance model performance and efficiency. Our experiments focused on various configurations of such hybrid models to identify the optimal structure for accurate and efficient realtime analysis. The best performing model achieved a Mean Squared Error of 0.00031, significantly outperforming traditional methods.

Leveraging SPD Matrices on Riemannian Manifolds in Quantum Classical Hybrid Models for Structural Health Monitoring

TL;DR

The paper addresses real-time FEM-based structural health monitoring of bridges by mapping a -dimensional input to a -dimensional output under tight time constraints. It introduces a hybrid quantum-classical multilayer perceptron that operates on Symmetric Positive Definite (SPD) matrices within the Riemannian manifold and uses polynomial feature expansion for nonlinear representation, followed by amplitude encoding for quantum processing. Through eigenvector decomposition of SPD matrices and normalization, data are prepared for quantum circuits, and three architectures are compared, with the SPD-Enhanced Hybrid achieving the best performance at . The results support SPD-based representations as a viable path to improved real-time SHM fem analyses in structural engineering, suggesting further exploration of advanced hybrid networks for practical deployment.

Abstract

Realtime finite element modeling of bridges assists modern structural health monitoring systems by providing comprehensive insights into structural integrity. This capability is essential for ensuring the safe operation of bridges and preventing sudden catastrophic failures. However, FEM computational cost and the need for realtime analysis pose significant challenges. Additionally, the input data is a 7 dimensional vector, while the output is a 1017 dimensional vector, making accurate and efficient analysis particularly difficult. In this study, we propose a novel hybrid quantum classical Multilayer Perceptron pipeline leveraging Symmetric Positive Definite matrices and Riemannian manifolds for effective data representation. To maintain the integrity of the qubit structure, we utilize SPD matrices, ensuring data representation is well aligned with the quantum computational framework. Additionally, the method leverages polynomial feature expansion to capture nonlinear relationships within the data. The proposed pipeline combines classical fully connected neural network layers with quantum circuit layers to enhance model performance and efficiency. Our experiments focused on various configurations of such hybrid models to identify the optimal structure for accurate and efficient realtime analysis. The best performing model achieved a Mean Squared Error of 0.00031, significantly outperforming traditional methods.
Paper Structure (15 sections, 6 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 15 sections, 6 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Mathematical Background
  4. Symmetric Positive Definite (SPD) Matrices
  5. Riemannian Manifold
  6. Method
  7. Step 1: Data Preparation and Polynomial Feature Expansion
  8. Step 2: SPD Matrix Generation
  9. Step 3: Eigenvector Decomposition and Projection
  10. Step 4: Normalization for Quantum State Preparation
  11. Step 5: Quantum Processing
  12. Model Architectures
  13. Results and Discussion
  14. Detailed Analysis of Results
  15. Conclusion

Figures (2)

  • Figure 1: Illustration of the quantum circuit used in the hybrid model. The circuit consists of multiple layers of $R_x$, $R_y$, and $R_z$ gates interspersed with CNOT gates for entanglement. This structure is designed to process the normalized vector inputs effectively by leveraging the quantum computational framework.
  • Figure 2: Quantum Circuit