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On the representation of energy-preserving quadratic operators with application to Operator Inference

Leonidas Gkimisis, Igor Pontes Duff, Pawan Goyal, Peter Benner

TL;DR

This paper addresses the representation of energy-preserving quadratic nonlinearities in finite-dimensional dynamical systems. It proves that any energy-preserving quadratic operator $\mathbf{H}$ can be equivalently parameterized as $\tilde{\mathbf{H}}$ with skew-symmetric blocks, enabling a constructive algorithm for transformation. The authors integrate this representation into non-intrusive model reduction via Operator Inference, introducing a sequential LS formulation (Seq_OpInf_EP) that enforces energy preservation during inference. Numerical experiments on a 2D Burgers' equation demonstrate accurate dynamics and faithful energy properties with negligible overhead, highlighting the practical value of physics-informed quadratic representations in ROM inference.

Abstract

In this work, we investigate a skew-symmetric parameterization for energy-preserving quadratic operators. Earlier, [Goyal et al., 2023] proposed this parameterization to enforce energy-preservation for quadratic terms in the context of dynamical system data-driven inference. We here prove that every energy-preserving quadratic term can be equivalently formulated using a parameterization of the corresponding operator via skew-symmetric matrix blocks. Based on this main finding, we develop an algorithm to compute an equivalent quadratic operator with skew-symmetric sub-matrices, given an arbitrary energy-preserving operator. Consequently, we employ the skew-symmetric sub-matrix representation in the framework of non-intrusive reduced-order modeling (ROM) via Operator Inference (OpInf) for systems with an energy-preserving nonlinearity. To this end, we propose a sequential, linear least-squares (LS) problems formulation for the inference task, to ensure energy-preservation of the data-driven quadratic operator. The potential of this approach is indicated by the numerical results for a 2D Burgers' equation benchmark, compared to classical OpInf. The inferred system dynamics are accurate, while the corresponding operators are faithful to the underlying physical properties of the system.

On the representation of energy-preserving quadratic operators with application to Operator Inference

TL;DR

This paper addresses the representation of energy-preserving quadratic nonlinearities in finite-dimensional dynamical systems. It proves that any energy-preserving quadratic operator can be equivalently parameterized as with skew-symmetric blocks, enabling a constructive algorithm for transformation. The authors integrate this representation into non-intrusive model reduction via Operator Inference, introducing a sequential LS formulation (Seq_OpInf_EP) that enforces energy preservation during inference. Numerical experiments on a 2D Burgers' equation demonstrate accurate dynamics and faithful energy properties with negligible overhead, highlighting the practical value of physics-informed quadratic representations in ROM inference.

Abstract

In this work, we investigate a skew-symmetric parameterization for energy-preserving quadratic operators. Earlier, [Goyal et al., 2023] proposed this parameterization to enforce energy-preservation for quadratic terms in the context of dynamical system data-driven inference. We here prove that every energy-preserving quadratic term can be equivalently formulated using a parameterization of the corresponding operator via skew-symmetric matrix blocks. Based on this main finding, we develop an algorithm to compute an equivalent quadratic operator with skew-symmetric sub-matrices, given an arbitrary energy-preserving operator. Consequently, we employ the skew-symmetric sub-matrix representation in the framework of non-intrusive reduced-order modeling (ROM) via Operator Inference (OpInf) for systems with an energy-preserving nonlinearity. To this end, we propose a sequential, linear least-squares (LS) problems formulation for the inference task, to ensure energy-preservation of the data-driven quadratic operator. The potential of this approach is indicated by the numerical results for a 2D Burgers' equation benchmark, compared to classical OpInf. The inferred system dynamics are accurate, while the corresponding operators are faithful to the underlying physical properties of the system.

Paper Structure

This paper contains 9 sections, 2 theorems, 45 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Representation of energy-preserving quadratic matrices
  3. Algorithm for quadratic matrix representation
  4. Application to Operator Inference
  5. The OpInf framework
  6. Sequential Operator Inference formulation for energy-preserving nonlinearities
  7. Numerical Example: 2D Burgers' equation
  8. Discussion
  9. Alternative proof

Key Result

Theorem 1

For any energy-preserving quadratic matrix $\mathbf{H} \in \mathbb{R}^{n \times n^2}$, such that $\mathbf{x}^\top\mathbf{H} \left( \mathbf{x} \otimes \mathbf{x} \right)=0, \; \forall \mathbf{x} \in \mathbb{R}^n$, there always exists an equivalent $\tilde{\mathbf{H}}$ with a structure such that $\mathbf{H} \left( \mathbf{x} \otimes \mathbf{x} \right)=\tilde{\mathbf{H}} \left( \mathbf{x} \otimes \m

Figures (3)

  • Figure 1: Illustration of \ref{['alg:cap']} for an arbitrary, energy-preserving matrix $\mathbf{H} \in \mathbb{R}^{3 \times 9}$.
  • Figure 2: Alternative representation via \ref{['fact2']}. Note that both this representation as well as the one in \ref{['fig:subfinal']} leave the entries $h_{i_{j,i}}, \forall i,j \in \left[1, \dots, n\right]$ in \ref{['fig:sub1']} unchanged. These correspond to non-repeating terms in $\mathbf{H}\left( \mathbf{x} \otimes \mathbf{x} \right)$.
  • Figure 3: Numerical results for 2D Burgers' equation: Enforcing the physics-based energy preservation for the quadratic term through Seq_OpInf_EP produces accurate and more robust models in comparison to standard Seq_OpInf.

Theorems & Definitions (4)

  • Theorem
  • proof
  • Theorem
  • proof