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Data-driven identification of partial differential equations for multi-physics systems using stochastic optimization

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Abstract

We develop a stochastic optimization framework to identify governing equations in multi-physics systems. The proposed approach discovers partial differential equations (PDEs) by combining user’s prior knowledge of the underlying physics of a target system and its observed data. The technique relies on evolutionary processes to randomly generate PDEs and stochastically optimize their structure and coefficients to the data by exploring the infinite model space. Furthermore, the method captures the spatiotemporal dynamics of physical system by direct evaluation of the candidate PDEs under physical constraints. To achieve significant computational speedup, the proposed stochastic optimization method relies on a series of novel modifications. These consist of the incorporation of a multi-purpose loss function into the parallel fitness test and bloat control techniques into the evolutionary processes. As such, these innovations lead to a significant improvement of the effectiveness and computational efficiency in identifying PDEs from given data. We demonstrate the applicability of methodology in two illustrative examples: the nonlinear Burgers’ equation and the linear/nonlinear advection-dispersion equation. The exact PDEs are successfully identified, even with significant data noise, and captures the underlying physics of the target system. Through series of simulations, we assess the accuracy, robustness, and limitations of the proposed approach. In particular, we show the impact of key dimensionless groups (that accounts for the competition between various physical phenomena) in controlling the accuracy of the identification process. This work shows that developed identification method is a promising effective and robust gray box modeling tool for identifying PDEs.

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Data availability

All the result data presented in the paper are available together with the code in Github (http://github.com/**woousc/GPSI).

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Acknowledgements

**woo Im acknowledges the Kwanjeong Educational Foundation (17AmB25D) for financial support.

Funding

Kwanjeong Educational Foundation (17AmB25D) for financial support.

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Authors and Affiliations

  1. Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, 3620 South Vermont Ave., Los Angeles, CA, 90089, USA

    **woo Im, Felipe P. J. de Barros & Sami F. Masri

Authors
  1. **woo Im
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  2. Felipe P. J. de Barros
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  3. Sami F. Masri
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Contributions

All authors contributed to the study conception and design. Material preparation and coding were performed by JI. Analysis of the results were performed by JI, FPJdB and SM. The first draft of the manuscript was written by JI and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Felipe P. J. de Barros.

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Conflict of interest

The authors declare that they have no conflict of interest.

Code availibility

Genetic Programming for System Identification (GPSI), is written in Python (version 3.7.4) with the SymPy library (version 1.7.1) and is available in Github (https://github.com/**woousc/GPSI).

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Im, J., de Barros, F.P.J. & Masri, S.F. Data-driven identification of partial differential equations for multi-physics systems using stochastic optimization. Nonlinear Dyn 111, 1987–2007 (2023). https://doi.org/10.1007/s11071-022-08182-z

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  • DOI: https://doi.org/10.1007/s11071-022-08182-z

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