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Enabling Equation-Free Modeling via Diffusion Maps

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Abstract

Equation-free modeling aims at extracting low-dimensional macroscopic dynamics from complex high-dimensional systems that govern the evolution of microscopic states. This algorithm relies on lifting and restriction operators that map macroscopic states to microscopic states and vice versa. Combined with simulations of the microscopic state, this algorithm can be used to apply Newton solvers to the implicitly defined low-dimensional macroscopic system or solve it more efficiently using direct numerical simulations. The key challenge is the construction of the lifting and restrictions operators that usually require a priori insight into the underlying application. In this paper, we design an application-independent algorithm that uses diffusion maps to construct these operators from simulation data. Code is available at https://doi.org/10.5281/zenodo.5793299.

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Data Availability

The datasets generated and analysed during the current study are available in the Zenodo repository at https://doi.org/10.5281/zenodo.5793299.

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Acknowledgements

Tracy Chin, Jacob Ruth, and Rebecca Santorella were supported by the NSF Grant DMS-1439786 through the Summer@ICERM program. Rebecca Santorella was also supported by the NSF through Grant 1644760. Bjorn Sandstede was supported by the NSF under Grants DMS-1408742, DMS-1714429 and CCF-1740741.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, USA

    Tracy Chin

  2. New York City, USA

    Jacob Ruth

  3. Department of Computer Science, Columbia University, New York City, USA

    Clayton Sanford

  4. Division of Applied Mathematics, Brown University, Providence, USA

    Rebecca Santorella & Björn Sandstede

  5. Department of Mathematics, University of California, Irvine, Irvine, USA

    Paul Carter

Authors
  1. Tracy Chin
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  2. Jacob Ruth
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  3. Clayton Sanford
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  4. Rebecca Santorella
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  6. Björn Sandstede
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Correspondence to Björn Sandstede.

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Chin, T., Ruth, J., Sanford, C. et al. Enabling Equation-Free Modeling via Diffusion Maps. J Dyn Diff Equat 36 (Suppl 1), 415–434 (2024). https://doi.org/10.1007/s10884-021-10127-w

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  • DOI: https://doi.org/10.1007/s10884-021-10127-w

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