On the representation of energy-preserving quadratic operators with application to Operator Inference
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| Format: | Preprint |
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2025
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| _version_ | 1866912274198822912 |
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| author | Gkimisis, Leonidas Duff, Igor Pontes Goyal, Pawan Benner, Peter |
| author_facet | Gkimisis, Leonidas Duff, Igor Pontes Goyal, Pawan Benner, Peter |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10824 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the representation of energy-preserving quadratic operators with application to Operator Inference Gkimisis, Leonidas Duff, Igor Pontes Goyal, Pawan Benner, Peter Numerical Analysis 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. |
| title | On the representation of energy-preserving quadratic operators with application to Operator Inference |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2503.10824 |