A Statistical Machine Learning Approach for Adapting Reduced-Order Models using Projected Gaussian Process

Abstract: The Proper Orthogonal Decomposition (POD) computes the optimal basis modes that span a low-dimensional subspace where the Reduced-Order Models (ROMs) reside. Because a governing equation is often parameterized by a set of parameters, challenges immediately arise when one would like to investigate how systems behave differently over the parameter space (in design, control, uncertainty quantification and real-time operations). In this case, the POD basis needs to be updated so as to adapt ROM that accurately captures the variation of a system's behavior over its parameter space. This paper proposes a Projected Gaussian Process (pGP) and formulate the problem of adapting POD basis as a supervised statistical learning problem, for which the goal is to learn a mapping from the parameter space to the Grassmann Manifold that contains the optimal vector subspaces. A mapping is firstly found between the Euclidean space and the horizontal space of an orthogonal matrix that spans a reference subspace in the Grassmann Manifold. Then, a second mapping from the horizontal space to the Grassmann Manifold is established through the Exponential/Logarithm maps between the manifold and its tangent space. Finally, given a new parameter, the conditional distribution of a vector can be found in the Euclidean space using the Gaussian Process (GP) regression, and such a distribution is projected to the Grassmann Manifold that yields the optimal subspace for the new parameter. The proposed statistical learning approach allows us to optimally estimate model parameters given data (i.e., the prediction/interpolation becomes problem-specific), and quantify the uncertainty associated with the prediction. Numerical examples are presented to demonstrate the advantages of the proposed pGP for adapting POD basis against parameter changes.

• The paper introduces a pGP framework that adapts POD bases via a supervised mapping from Euclidean space to the Grassmann Manifold.
• It integrates Gaussian Process regression with uncertainty quantification to improve accuracy in numerical simulations of PDE systems.
• Numerical examples in fluid dynamics and heat transfer demonstrate enhanced efficiency and precision over traditional global POD methods.

A Statistical Machine Learning Approach for Adapting Reduced-Order Models using Projected Gaussian Process

The paper presents an innovative methodology for addressing the adaptation of Reduced-Order Models (ROMs) to variations in parameterized systems. By leveraging the capabilities of Gaussian Processes (GP) and exploring the Grassmann Manifold, the authors introduce the Projected Gaussian Process (pGP) to enable efficient updates to the Proper Orthogonal Decomposition (POD) basis in a computational framework, ensuring accuracy across different parameter settings.

Core Contributions

Model Conceptualization

The authors begin by addressing the challenges inherent in maintaining accurate ROMs when subject to varying parameters. In systems described by Partial Differential Equations (PDEs), capturing dynamic behavior across a multi-dimensional parameter space poses significant computational difficulties. The proposed pGP model formulates the POD adaptation as a supervised statistical learning task, aiming to create a functional map from the parameter space to the Grassmann Manifold. This manifold, by containing optimal vector subspaces, allows for an injective mapping where data is projected to accurately refine ROMs for updated parameter conditions.

Key Methodological Developments

The methodology is outlined through multi-step mappings:

• Mapping from Euclidean to Grassmann Space: The authors derive an efficient injective mapping to transfer data from the high-dimensional Euclidean space to the more analytically compatible Grassmann Manifold.
• Projected Gaussian Process (pGP) Framework: By incorporating GP regression into the manifold space, the authors propose that predictions of new POD bases become not only data-driven but also inherently equipped with uncertainty quantification. This is achieved by optimizing hyperparameters specific to each problem domain, thus fine-tuning the model to its context.
• Numerical Illustration: Example domains such as fluid dynamics and heat transfer demonstrate the pGP’s robustness and flexibility, showcasing improved accuracy in adaptive ROM updates compared to traditional interpolation methods and even global POD models.

Implications and Further Research

The implications of this research reach across both practical applications and theoretical study. Practically, the approach supports enhanced accuracy and efficiency in simulations necessary for complex system design and control, such as aeroelasticity computations or real-time digital twins. Theoretically, it opens avenues in manifold-based machine learning within scientific computing, pointing toward a future path where parameter dimensionality can be selectively reduced, enhancing both model performance and computational cost.

Conclusion

With the proposed pGP approach, the authors establish a sophisticated methodology that not only adapts ROMs to varying parameters effectively but also integrates statistical learning capabilities, such as parameter selection and uncertainty quantification. This integration marks a significant step forward in creating adaptable and reliable ROMs for high-complexity systems and sets the stage for future innovations in model adaptation within simulation sciences.