Neural-Guided Domain Restriction to Accelerate Pseudospectra Computation for Structured Non-normal Banded Matrices

TL;DR

This paper introduces a neural network-guided method to efficiently compute pseudospectra of non-normal matrices by predicting sensitive regions, significantly reducing computational effort while maintaining accuracy.

Contribution

The authors develop a neural network approach that predicts sensitive regions for pseudospectra, enabling faster computation without exhaustive evaluation across the complex plane.

Findings

Substantial speedup in pseudospectra computation for non-normal banded matrices.

High accuracy in identifying sensitive regions with the neural network guidance.

Effective preprocessing strategy for large-scale matrix analysis.

Abstract

Computing pseudospectra of non-normal matrices is essential for understanding the stability and transient behavior of dynamical systems. Such analysis is critical in applications including fluid dynamics, control systems, and differential operators, where non-normality can lead to significant transient amplification and sensitivity to perturbations that are not captured by eigenvalue analysis alone. At large scales, commonly used numerical approaches for pseudospectra computation can become computationally demanding, as they require repeated auxiliary computations to identify spectrally sensitive regions in the complex plane. We present a neural network-based approach that predicts sensitive regions directly from matrix features, thereby avoiding exhaustive pseudospectra evaluation across the entire complex plane. We calibrate the prediction threshold on validation data to ensure reliable coverage of sensitive regions. The trained neural network guides the selection of grid points requiring full computation, enabling focused computation only where necessary. The approach provides a practical preprocessing strategy for efficient pseudospectra computation. Numerical experiments on non-normal banded matrices demonstrate substantial speedup compared to full grid-based numerical evaluation while maintaining high accuracy in identifying sensitive regions.