scholarly journals Function Space Optimization: A symbolic regression method for estimating parameter transfer functions for hydrological models

2020 ◽  
Author(s):  
Moritz Feigl ◽  
Mathew Herrnegger ◽  
Daniel Klotz ◽  
Karsten Schulz
Keyword(s):  
Function Space ◽  
Transfer Functions ◽  
Symbolic Regression ◽  
Regression Method ◽  
Hydrological Models

Catchment to model space mapping – learning transfer functions from data by symbolic regression

2021 ◽  
Author(s):  
Moritz Feigl ◽  
Robert Schweppe ◽  
Stephan Thober ◽  
Mathew Herrnegger ◽  
Luis Samaniego ◽  
...  
Keyword(s):  
Transfer Function ◽  
Function Space ◽  
Transfer Functions ◽  
Continuous Optimization ◽  
Model Space ◽  
Symbolic Regression ◽  
Regression Method ◽  
Hydrological Models ◽  
Distributed Hydrological Model ◽  
Distributed Hydrological Models

<p>The Function Space Optimization (FSO) method, recently developed by Feigl et al. (2020), automatically estimates the transfer function structure and coefficients to parameterize spatially distributed hydrological models. FSO is a symbolic regression method, searching for an optimal transfer function in a continuous optimization space, using a text generating neural network (variational autoencoder).</p><p>We apply our method to the distributed hydrological model mHM (www.ufz.de/mhm), which is based on a priori defined transfer functions. We estimate mHM transfer functions for the parameters “saturated hydraulic conductivity” and “field capacity”, which both influence a range of hydrologic processes, e.g. infiltration and evapotranspiration.</p><p>The FSO and standard mHM approach are compared using data from 229 basins, including 7 large training basins and 222 smaller validation basins, distributed across Germany. For training, 5 years of data of 7 gauging stations is used, while up to 35 years, with a median of 32 years, are used for validation. This setup is adopted from a previous study by Zink et al. (2017), testing mHM in the same basins and which is used as a benchmark. Maps of soil properties (sand/clay percentage, bulk density) and topographic properties (aspect, slope, elevation) are used as possible inputs for transfer functions.</p><p>FSO estimated transfer functions improved the mHM model performance in the validation catchments significantly when compared to the benchmark results, and only show a small decrease in performance compared to the training results. Results demonstrate that an automatic estimation of parameter transfer functions by FSO is beneficial for the parameterization of distributed hydrological models and allows for a robust parameter transfer to other locations.</p><p> </p><p>Feigl, M., Herrnegger, M., Klotz, D., & Schulz, K. (2020). Function Space Optimization: A symbolic regression method for estimating parameter transfer functions for hydrological models. Water resources research, 56(10), e2020WR027385.</p><p>Zink, M., Kumar, R., Cuntz, M., & Samaniego, L. (2017). A high-resolution dataset of water fluxes and states for Germany accounting for parametric uncertainty. Hydrol. Earth Syst. Sci, 21, 1769-1790.</p>

scholarly journals Symbolic Regression for the Estimation of Transfer Functions of Hydrological Models

2017 ◽  
Vol 53 (11) ◽  
pp. 9402-9423 ◽  
Author(s):  
D. Klotz ◽  
M. Herrnegger ◽  
K. Schulz
Keyword(s):  
Transfer Functions ◽  
Symbolic Regression ◽  
Hydrological Models

Automatic estimation of parameter transfer functions for distributed hydrological models - a case study with the mHM model

2020 ◽  
Author(s):  
Moritz Feigl ◽  
Stephan Thober ◽  
Mathew Herrnegger ◽  
Luis Samaniego ◽  
Karsten Schulz
Keyword(s):  
Transfer Function ◽  
Function Space ◽  
Hydrological Model ◽  
Transfer Functions ◽  
Dimensional Space ◽  
Model Parameters ◽  
Hydrological Models ◽  
Estimation Of Parameters ◽  
General Applicability ◽  
Distributed Hydrological Models

<p>The estimation of parameters for spatially distributed rainfall runoff models is a long-studied, complex and ill-posed problem. Relating parameters of distributed hydrological models to geophysical properties of catchments could potentially solve some of the major difficulties connected to it.</p><p>One way to define this relationship is by the use of explicit equations called parameter transfer functions, which relate geophysical catchment properties to the model parameters. Computing parameter fields using transfer functions would result in spatially consistent parameter fields and the potential to extrapolate to other catchments. A further advantage is that the dimensionality of the parameter space is reduced because the transfer function parameters are applied to all computational units (i.e., grid cells). However, the structure and parameterization of transfer functions is often only implicitly assumed or needs to be derived by a laborious literature guided trial and error process.</p><p>For this reason we use Function Space Optimization (FSO), a symbolic regression approach which automatically estimates the structure and parameterization of transfer functions from catchment data. FSO transfers the search of the optimal function to a searchable continuous vector space. To create this space, a text generating neural network with a variational autoencoder (VAE) architecture is used. It is trained to map possible transfer functions and their distributions to a 6-dimensional space. After training, a continuous optimization is applied to search for the optimal transfer function in this function space. FSO was already tested in a virtual experiment using a parsimonious hydrological model, where its ability to solve the problem of transfer function estimation was shown.</p><p>Here, we further test FSO by applying it in a real world setting to the mesoscale hydrological model (mHM). mHM is a widely applied distributed hydological model, which uses transfer functions for all its parameters. For this study, we estimate transfer functions for the parameters porosity and field capacity, which both influence a range of hydrologic processes, e.g. infiltration and evapotranspiration. We compare the FSO estimated transfer functions with the already existing mHM transfer functions and examine their influence on the model performance.</p><p>In summary, we show the general applicability of FSO for distributed hydrological models and the advantages and capabilities of automatically defining parameter transfer functions.</p>

scholarly journals Solution of the optimal control problem by symbolic regression method

2021 ◽  
Vol 186 ◽  
pp. 646-653
Author(s):  
A.I. Diveev ◽  
S.V. Konstantinov ◽  
A.M. Danilova
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Symbolic Regression ◽  
Regression Method

scholarly journals Method for the identification of explicit polynomial formulae for the friction in turbulent pipe flow

1999 ◽  
Vol 1 (2) ◽  
pp. 115-126 ◽  
Author(s):  
J. W. Davidson ◽  
D. Savic ◽  
G. A. Walters
Keyword(s):  
Least Squares ◽  
Symbolic Regression ◽  
Regression Method ◽  
Turbulent Pipe Flow ◽  
Model Complexity ◽  
Improve Performance ◽  
Rule Based ◽  
Least Squares Optimization ◽  
Starting Solutions ◽  
Crossover And Mutation

The paper describes a new regression method for creating polynomial models. The method combines numerical and symbolic regression. Genetic programming finds the form of polynomial expressions, and least squares optimization finds the values for the constants in the expressions. The incorporation of least squares optimization within symbolic regression is made possible by a rule-based component that algebraically transforms expressions to equivalent forms that are suitable for least squares optimization. The paper describes new operators of crossover and mutation that improve performance, and a new method for creating starting solutions that avoids the problem of under-determined functions. An example application demonstrates the trade-off between model complexity and accuracy of a set of approximator functions created for the Colebrook–White formula.

Access to Empirical Formulations of Combustion Efficiency for Gas Turbine Combustors with Improved Generalization Ability

2020 ◽  
pp. 1-32
Author(s):  
Zhuang Ma ◽  
Tingwei Ji ◽  
Tao Cui ◽  
Yao Zheng
Keyword(s):  
Experimental Data ◽  
Polynomial Regression ◽  
Burning Velocity ◽  
Velocity Model ◽  
Combustion Efficiency ◽  
Symbolic Regression ◽  
Regression Method ◽  
Generalization Ability ◽  
Operating Variables ◽  
Model Structures

Abstract Correlating combustion performance parameters to the main operating variables of combustors with mathematical expressions contributes to reducing the number of experiments and simplifying the design procedure of gas turbines. The application of empirical formulations meets the requirement with finite precision. The present study aims at adopting symbolic regression method to establish empirical formulations to correlate combustion efficiency with the main operating variables of gas turbine combustors. Differing from ordinary data modeling methods that search model parameters only with model structures fixed, symbolic regression method can search the structures and parameters of mathematical models simultaneously. In this article, attempts to correlate the experimental data of Combustor I using the mechanism model of burning velocity model, neural network, polynomial regression and symbolic regression are shown sequentially. Burning velocity model has not satisfactory accuracy by comparing the predictions with the experimental data which means its lower generalization ability. Comparatively, the predictions of the empirical formulation obtained by the present symbolic regression method are in good agreement with the experimental data, and also excel those of neural network and polynomial regression in generalization ability. Another two formulations are obtained by symbolic regression using the experimental data of Combustor II and III, and the different model structures of the two formulations indicate that there is still room for improvement in the present method.

Solving the Problem of the Optimal Control System General Synthesis Based on Approximation of a Set of Extremals using the Symbol Regression Method

2020 ◽  
pp. 59-74
Author(s):  
S.V. Konstantinov ◽  
A.I. Diveev
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Symbolic Regression ◽  
Regression Method ◽  
Synthesis Problem ◽  
Optimal Control System ◽  
Initial States

A new approach is considered to solving the problem of synthesizing an optimal control system based on the extremals' set approximation. At the first stage, the optimal control problem for various initial states out of a given domain is being numerically sold. Evolutionary algorithms are used to solve the optimal control problem numerically. At the second stage, the problem of approximating the found set of extremals by the method of symbolic regression is solved. Approach considered in the work makes it possible to eliminate the main drawback of the known approach to solving the control synthesis problem using the symbolic regression method, which consists in the fact that the genetic algorithm used in solving the synthesis problem does not provide information about proximity of the found solution to the optimal one. Here, control function is built on the basis of a set of extremals; therefore, any particular solution should be close to the optimal trajectory. Computational experiment is presented for solving the applied problem of synthesizing the four-wheel robot optimal control system in the presence of phase constraints. It is experimentally demonstrated that the synthesized control function makes it possible for any initial state from a given domain to obtain trajectories close to optimal in the quality functional. Initial states were considered during the experiment, both included in the approximating set of optimal trajectories and others from the same given domain. Approximation of the extremals set was carried out by the network operator method

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