Optimal Low-dimensional Approximation of Transfer Operators via Flow Matching: Computation and Error Analysis
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| Format: | Preprint |
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2024
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| _version_ | 1866917774896398336 |
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| author | Zhang, Zhicheng Guo, Ling Wu, Hao |
| author_facet | Zhang, Zhicheng Guo, Ling Wu, Hao |
| contents | Reaction coordinates (RCs) are low-dimensional representations of complex dynamical systems that capture their long-term dynamics. In this work, we focus on the criteria of lumpability and decomposability, previously established for assessing RCs, and propose a new flow matching approach for the analysis and optimization of reaction coordinates based on these criteria. This method effectively utilizes data to quantitatively determine whether a given RC satisfies these criteria and enables end-to-end optimization of the reaction coordinate mapping model. Furthermore, we provide a theoretical analysis of the relationship between the loss function used in our approach and the operator error induced by dimension reduction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15981 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal Low-dimensional Approximation of Transfer Operators via Flow Matching: Computation and Error Analysis Zhang, Zhicheng Guo, Ling Wu, Hao Numerical Analysis Dynamical Systems Reaction coordinates (RCs) are low-dimensional representations of complex dynamical systems that capture their long-term dynamics. In this work, we focus on the criteria of lumpability and decomposability, previously established for assessing RCs, and propose a new flow matching approach for the analysis and optimization of reaction coordinates based on these criteria. This method effectively utilizes data to quantitatively determine whether a given RC satisfies these criteria and enables end-to-end optimization of the reaction coordinate mapping model. Furthermore, we provide a theoretical analysis of the relationship between the loss function used in our approach and the operator error induced by dimension reduction. |
| title | Optimal Low-dimensional Approximation of Transfer Operators via Flow Matching: Computation and Error Analysis |
| topic | Numerical Analysis Dynamical Systems |
| url | https://arxiv.org/abs/2408.15981 |