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Adaptive Error-Bounded Hierarchical Matrices for Efficient Neural Network Compression

John Mango, Ronald Katende

TL;DR

This work tackles the computational and memory bottlenecks of Physics-Informed Neural Networks (PINNs) caused by large dense weight matrices. It introduces a dynamic, error-bounded hierarchical matrix ($H$-matrix) compression that adaptively refines block structure while preserving Neural Tangent Kernel ($NTK$) properties, ensuring stable and efficient training. The approach combines error-estimated adaptive refinement, convergence acceleration, and a block-rank regularization mechanism, demonstrating superior performance over traditional compression techniques (e.g., Singular Value Decomposition, pruning, and quantization) and enabling real-time inference. The results indicate improved convergence, robustness to perturbations, and scalable deployment of PINNs in complex scientific and engineering PDE problems.

Abstract

This paper introduces a dynamic, error-bounded hierarchical matrix (H-matrix) compression method tailored for Physics-Informed Neural Networks (PINNs). The proposed approach reduces the computational complexity and memory demands of large-scale physics-based models while preserving the essential properties of the Neural Tangent Kernel (NTK). By adaptively refining hierarchical matrix approximations based on local error estimates, our method ensures efficient training and robust model performance. Empirical results demonstrate that this technique outperforms traditional compression methods, such as Singular Value Decomposition (SVD), pruning, and quantization, by maintaining high accuracy and improving generalization capabilities. Additionally, the dynamic H-matrix method enhances inference speed, making it suitable for real-time applications. This approach offers a scalable and efficient solution for deploying PINNs in complex scientific and engineering domains, bridging the gap between computational feasibility and real-world applicability.

Adaptive Error-Bounded Hierarchical Matrices for Efficient Neural Network Compression

TL;DR

This work tackles the computational and memory bottlenecks of Physics-Informed Neural Networks (PINNs) caused by large dense weight matrices. It introduces a dynamic, error-bounded hierarchical matrix (-matrix) compression that adaptively refines block structure while preserving Neural Tangent Kernel () properties, ensuring stable and efficient training. The approach combines error-estimated adaptive refinement, convergence acceleration, and a block-rank regularization mechanism, demonstrating superior performance over traditional compression techniques (e.g., Singular Value Decomposition, pruning, and quantization) and enabling real-time inference. The results indicate improved convergence, robustness to perturbations, and scalable deployment of PINNs in complex scientific and engineering PDE problems.

Abstract

This paper introduces a dynamic, error-bounded hierarchical matrix (H-matrix) compression method tailored for Physics-Informed Neural Networks (PINNs). The proposed approach reduces the computational complexity and memory demands of large-scale physics-based models while preserving the essential properties of the Neural Tangent Kernel (NTK). By adaptively refining hierarchical matrix approximations based on local error estimates, our method ensures efficient training and robust model performance. Empirical results demonstrate that this technique outperforms traditional compression methods, such as Singular Value Decomposition (SVD), pruning, and quantization, by maintaining high accuracy and improving generalization capabilities. Additionally, the dynamic H-matrix method enhances inference speed, making it suitable for real-time applications. This approach offers a scalable and efficient solution for deploying PINNs in complex scientific and engineering domains, bridging the gap between computational feasibility and real-world applicability.
Paper Structure (9 sections, 3 theorems, 23 equations, 6 figures, 1 algorithm)

This paper contains 9 sections, 3 theorems, 23 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integration of Hierarchical Matrices with PINNs
  4. Adaptive Hierarchical Matrix Construction with Error Estimation
  5. Improved Convergence and Regularization in PINNs via Adaptive Hierarchical Matrices
  6. Example
  7. Adaptive Construction Algorithm for Integration of Hierarchical Matrices with PINNs
  8. Numerical Results
  9. Conclusion

Key Result

Theorem 1

Consider an ill-conditioned matrix $A$ approximated by an adaptive H-matrix $H$ within a Physics-Informed Neural Network (PINN). Let $\tau$ be the local error threshold, $\epsilon$ the perturbation error, $\delta$ the adversarial perturbation bound, and $T$ the number of adaptive refinement steps. T

Figures (6)

  • Figure 1: Flowchart of how to incorporate H-matrices into PINNs
  • Figure 2: Accuracy and Compression ratio over epochs
  • Figure 3: Real-time inference speed with compressed models
  • Figure 4: Error propagation for dynamic H-matrix vs. SVD
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 1: Hierarchical Matrix
  • Theorem 1: Comprehensive Convergence of Adaptive H-Matrix PINNs
  • proof
  • Theorem 2: Preservation of NTK Properties in Hierarchical Matrix Integration with PINNs
  • proof
  • Definition 2: Neural Tangent Kernel (NTK)
  • Theorem 3: Condition Number Bound of the Adaptively Constructed Hierarchical Matrix
  • Definition 3: Condition Number
  • proof