A reduced-order model based on Gaussian process dynamical models for time-dependent parameterized partial...

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Machine learning

Background:

• Parameterized partial differential equations (PDEs) present significant high-dimensional challenges.
• Reduced-order modeling (ROM) is crucial for efficient simulation of complex systems.
• Existing ROMs struggle with nonlinear temporal dynamics and uncertainty quantification.

Purpose of the Study:

• To develop a novel reduced-order modeling framework for high-dimensional parameterized PDEs.
• To integrate tensor-train decomposition (TTD), Gaussian process regression (GPR), and Gaussian process dynamical models (GPDMs).
• To enable accurate time evolution prediction and uncertainty quantification for complex dynamics.

Main Methods:

• Tensor-train decomposition (TTD) for low-rank approximation of solution snapshots.
• Gaussian process regression (GPR) to map parameter space to TTD format.
• Gaussian process dynamical models (GPDMs) for temporal dynamics modeling and uncertainty quantification.

Main Results:

• The proposed framework effectively handles high-dimensional parameterized PDEs.
• Demonstrated superior accuracy in modeling nonlinear temporal dynamics compared to traditional methods.
• Achieved accurate time-domain interpolation and robust uncertainty quantification.

Conclusions:

• The integrated TTD-GPR-GPDM framework offers a powerful approach for complex parameterized PDEs.
• This method significantly advances reduced-order modeling capabilities for dynamic systems.
• The framework provides a reliable tool for prediction and uncertainty assessment in scientific computing.