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\begin{document}

\title{Data-driven discovery of governing differential equations for hydrologic
systems utilizing stochastic optimization}



\author[1]{Jinwoo Im}%
\author[1]{Sami Masri}%
\author[1]{Felipe de Barros}%
\affil[1]{University of Southern California}%


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  \date{June 13, 2024}


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\begin{abstract}
There has been progress in machine learning (ML) to produce accurate and
robust model predictions for hydrologic systems. This progress opens a
new opportunity to enhance our fundamental understanding of the
underlying physics of a given attribute in the subsurface environment.
In order to achieve such understanding through ML in addition to
reliable model predictions, we develop a general framework of system
identification, named by GPfSI, which provides interpretable models
representing system dynamics embedded in data. This approach aims to
discover multiple governing differential equations for a target
multi-physics system by combining user's prior knowledge about the
system dynamics and a data set of system excitation and responses. In
efforts to identify governing equations in the infinite model space in
an effective way, one of the machine learning methods, Genetic
Programming (GP), is employed. As a stochastic optimization method, GP
is utilized to optimize differential equations to a given data set
through evolutionary processes. We significantly enhance the
effectiveness and computational efficiency of the GP-based
identification process, by incorporating a multi-purpose loss function
and stochastic sampling into the parallelized fitness test, and bloat
control techniques into the evolution process. We demonstrate the
proposed framework, GPfSI, against simulated and experimental data sets.
In the demonstration case with the simulated data, the reference
canonical model, i.e., the advection-dispersion equation (ADE), was
successfully identified with a wide range of system characteristics and
data noise. In the application to the experimental data from a soil
column tracer study, GPfSI provided several nonlinear ADEs that are
capable of capturing the anomalous (i.e., non-Fickian) tailing behavior
observed in the solute breakthrough data. The inspection of the
identified models provides some insights into the underlying physics of
non-Fickian transport. Consequently, our results indicate that GPfSI is
an effective and robust gray box modeling tool to produce accurate model
predictions and enhance our fundamental understanding of hydrologic
systems.%
\end{abstract}%



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