Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference

Abstract: Operator inference learns low-dimensional dynamical-system models with polynomial nonlinear terms from trajectories of high-dimensional physical systems (non-intrusive model reduction). This work focuses on the large class of physical systems that can be well described by models with quadratic nonlinear terms and proposes a regularizer for operator inference that induces a stability bias onto quadratic models. The proposed regularizer is physics informed in the sense that it penalizes quadratic terms with large norms and so explicitly leverages the quadratic model form that is given by the underlying physics. This means that the proposed approach judiciously learns from data and physical insights combined, rather than from either data or physics alone. Additionally, a formulation of operator inference is proposed that enforces model constraints for preserving structure such as symmetry and definiteness in the linear terms. Numerical results demonstrate that models learned with operator inference and the proposed regularizer and structure preservation are accurate and stable even in cases where using no regularization or Tikhonov regularization leads to models that are unstable.

• The paper introduces a physics-informed regularizer that enhances stability by penalizing large norms in quadratic terms.
• It presents a non-intrusive operator inference method that preserves key physical structures like symmetry and negative definiteness in linear operators.
• Numerical experiments demonstrate improved model accuracy and stability over traditional techniques in problems including Burgers’ equation and reaction-diffusion systems.

Summary of "Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference"

Introduction

The paper addresses the challenge of learning low-dimensional models of dynamical systems using operator inference, especially for systems characterized by quadratic nonlinear terms. It introduces a novel physics-informed regularizer that leverages insights from the quadratic structure imposed by the underlying physics to enhance the stability of inferred models. This regularizer biases quadratic models towards stability by penalizing terms with large norms. The approach emphasizes the integration of data and physical insights over purely data-driven or model-based methodologies. Furthermore, it proposes a formulation that preserves structural properties such as symmetry and definiteness in the linear terms of the model, which are critical for maintaining the dynamical system's integrity.

Non-intrusive Model Reduction

Operator inference is emphasized as a non-intrusive model reduction technique that infers reduced-order models directly from trajectory data without requiring access to high-dimensional systems' internal operators. The method projects high-dimensional data onto a reduced space and applies least-squares regression to infer the model operators. This process is contrasted with classical intrusive methods, which rely on access to the full high-dimensional operator matrices. The authors highlight operator inference's ability to accommodate models with polynomial nonlinearities and its theoretical underpinnings that guarantee recovery under certain conditions.

Proposed Regularizer and Structure Preservation

Physics-informed Regularizer

The core innovation is the introduction of a physics-informed regularizer within the operator inference framework. This regularizer is rooted in Lyapunov theory, specifically the concept of stability radius, which measures the robustness of the system around an equilibrium point. By minimizing the Frobenius norm of the quadratic terms, the regularizer effectively enhances the system's stability radius, thereby promoting the learning of stable models. The authors show through mathematical formulation that the stability radius is inversely proportional to the norm of quadratic terms, reinforcing the need to control this norm during model inference.

Structure Preservation

In addition to the regularizer, the paper introduces constraints to preserve structural properties of the system within the reduced model. These constraints focus on ensuring properties like symmetry and negative definiteness in the linear operators, leveraging semi-definite programming techniques. By preserving these properties, the method ensures that the learned models retain key characteristics necessary for stability and accuracy.

Computational Procedure and Numerical Experiments

The paper outlines a computational algorithm for implementing the physics-informed operator inference (PIR-OpInf) and its structure-preserving variant (SPIR-OpInf). The procedure involves:

• Constructing a reduced basis from data.
• Applying parameter selection through cross-validation to choose regularization parameters.
• Using interpolation schemes that maintain structural properties for new parameter regimes.

Numerical experiments demonstrate the efficacy of PIR-OpInf and SPIR-OpInf across different application scenarios, including synthetic systems, Burgers' equation, and reaction-diffusion problems. The results consistently show that models derived using these methods exhibit enhanced stability and accuracy over traditional operator inference approaches with or without Tikhonov regularization.

Conclusions

The research contributes significantly to the field of scientific machine learning by advancing model reduction techniques that blend data-driven approaches with physics-based insights. The proposed physics-informed regularizer and structural constraints ensure that the learned models achieve a balance between stability, accuracy, and interpretability. The findings underscore the necessity of integrating domain knowledge into data-driven model inference, paving the way for robust and reliable reduced-order models in various scientific and engineering applications. Future work may explore extending these methods to a broader class of systems and integrating additional physical constraints for enhanced generalization.