Reduced Order Data-driven Twin Models for Nonlinear PDEs by Randomized Koopman Orthogonal Decomposition and Explainable Deep Learning

TL;DR

This paper presents a novel data-driven twin modeling framework for nonlinear PDEs that combines Koopman operator theory, randomized orthogonal projections, and explainable deep learning to achieve accurate, interpretable, and efficient reduced-order models demonstrated on shock wave phenomena.

Contribution

It introduces a new algorithm integrating Koopman theory with randomized projections and explainable deep learning for improved nonlinear PDE modeling.

Findings

Accurately captures nonlinear dynamics with reduced complexity.

Provides real-time adaptive calibration and prediction.

Demonstrates effectiveness on shock wave phenomena with three experiments.

Abstract

This study introduces a data-driven twin modeling framework based on modern Koopman operator theory, offering a significant advancement over classical modal decomposition by accurately capturing nonlinear dynamics with reduced complexity and no manual parameter adjustment. The method integrates a novel algorithm with Pareto front analysis to construct a compact, high-fidelity reduced-order model that balances accuracy and efficiency. An explainable NLARX deep learning framework enables real-time, adaptive calibration and prediction, while a key innovation-computing orthogonal Koopman modes via randomized orthogonal projections-ensures optimal data representation. This approach for data-driven twin modeling is fully self-consistent, avoiding heuristic choices and enhancing interpretability through integrated explainable learning techniques. The proposed method is demonstrated on shock wave phenomena using three experiments of increasing complexity, accompanied by a qualitative analysis of the resulting data-driven twin models.