Generating Radial Basis Function (RBF) models with PySMO#
Note
The IDAES surrogate API is a wrapper around the original PySMO surrogate interface.
The pysmo_surrogate.PysmoRBFTrainer method has the capability to generate different types of RBF surrogates from data based on the basis function selected. RBFs models are usually of the form where
where \(z_{j}\) are basis function centers (in this case, the training data points), \(w_{j}\) are the radial weights associated with each center \(z_{j}\), and \(\psi\) is a basis function transformation of the Euclidean distances.
PySMO offers a range of basis function transformations \(\psi\), as shown in the table below.
Transformation type |
PySMO option name |
\(\psi(d)\) |
|---|---|---|
Linear |
‘linear’ |
\(d\) |
Cubic |
‘cubic’ |
\(d^{3}\) |
Thin-plate spline |
‘spline’ |
\(d^{2}\ln(d)\) |
Gaussian |
‘gaussian’ |
\(e^{\left(-d^{2}\sigma^{2}\right)}\) |
Multiquadric |
‘mq’ |
\(\sqrt{1+\left(\sigma d\right)^{2}}\) |
Inverse mMultiquadric |
‘imq’ |
\(1/{\sqrt{1+\left(\sigma d\right)^{2}}}\) |
Selection of parametric basis functions increase the flexibility of the radial basis function but adds an extra parameter (\(\sigma\))to be estimated.
Basic Usage#
To generate an RBF model with PySMO, the pysmo_surrogate.PysmoRBFTrainer trainer is instantiated and initialized with the desired configuration arguments, and the training function train_surrogate is called:
# Required imports
>>> from idaes.core.surrogate.pysmo_surrogate import PysmoRBFTrainer
>>> import pandas as pd
# Load dataset from a csv file
>>> xy_data = pd.read_csv('data.csv', header=None, index_col=0)
# Define the input and output labels
input_labels = ['X1', 'X2']
output_labels = ['Y1','Y2']
# Create the RBF trainer object
>>> rbf_trainer = PysmoRBFTrainer(input_labels=input_labels, output_labels=output_labels, training_dataframe = data_training)
# Set PySMO options
>>> rbf_trainer.config.basis_function = 'cubic'
# Train the model
>>> rbf_train = rbf_trainer.train_surrogate()
Configuration Options#
PysmoRBFTrainer takes the following optional arguments:
Option |
Configuration argument |
Description |
|---|---|---|
basis_function |
PysmoRBFTrainer.config.basis_function |
Option to specify the type of basis function to be used in the RBF model. Default is ‘gaussian’. |
regularization |
PysmoRBFTrainer.config.regularization |
Boolean argument which determines whether model regularization is applied during training.
Default is True.
|
solution_method |
PysmoRBFTrainer.config.solution_method |
Method used to solve the parameter estimation problems for the RBF model:
BFGS (‘BFGS’), maximum likelihood (‘algebraic’) or Pyomo least squares minimization (‘pyomo’).
Default is ‘algebraic’.
|
Output#
The result of the pysmo_surrogate.PysmoRBFTrainer method is a python object containing information about the problem set-up, the optimal radial basis function weights \(w_{j}\) and different error and quality-of-fit metrics such as the mean-squared-error (MSE) and the \(R^{2}\) coefficient-of-fit.
Surrogate Visualization#
For visualizing PySMO-trained surrogates via parity and residual plots, see Visualizing Surrogate Model Results.
Building the IDAES Surrogate Object#
To add the model to an IDAES flowsheet or generate model predictions, the SurrogateTrainer object needs to be transformed into an IDAES SurrogateObject object. This is done by calling PySMOSurrogate and passing the generated surrogate expressions, along with variable labels and optionally the bounds:
>>> surr = PysmoSurrogate(rbf_train, input_labels, output_labels, input_bounds)
The resulting PysmoSurrogate object may be saved to (and reloaded from) a JSON file; for details, see the PySMO main page.
Prediction with PysmoRBFTrainer models#
Once the RBF model has been trained and the SurrogateObject object created, predictions for values at previously unsampled points x_unsampled can be evaluated by calling SurrogateObject’s evaluate_surrogate() function on the unsampled points:
>>> y_unsampled = surr.evaluate_surrogate(x_unsampled)
Flowsheet Integration#
The result of the RBF training process can be passed directly into a process flowsheet using the IDAES SurrogateBlock option. The following code snippet demonstrates how a saved RBF model may be integrated directly into an IDAES flowsheet:
# Required imports
>>> from pyomo.environ import Var, ConcreteModel, Constraint, SolverFactory, Objective, minimize
>>> from idaes.core import FlowsheetBlock
>>> from idaes.core.surrogate.pysmo_surrogate import PysmoSurrogate
>>> from idaes.core.surrogate.surrogate_block import SurrogateBlock
# Create a Pyomo model
>>> m = pyo.ConcreteModel()
>>> m.fs = FlowsheetBlock(default={"dynamic": False})
# create input and output variables
>>> m.fs.X1 = Var(initialize=0, bounds=(0, 5))
>>> m.fs.X2 = Var(initialize=0, bounds=(0, 5))
>>> m.fs.Y1 = Var(initialize=0)
>>> m.fs.Y2 = Var(initialize=0)
# create list of surrogate inputs and outputs for flowsheet
>>> inputs = [m.fs.X1, m.fs.X2]
>>> outputs = [m.fs.Y1, m.fs.Y2]
# create the Pyomo/IDAES block that corresponds to the surrogate
>>> m.fs.surrogate = SurrogateBlock(concrete=True)
>>> surrogates_obj =PysmoSurrogate.load_from_file('rbf_surrogate.json') # rbf_surrogate.json is an existing surrogate JSON file containing the rbf model
>>> m.fs.surrogate.build_model(surrogates_obj, input_vars=inputs, output_vars=outputs)
>>> m.fs.surrogate.pprint()
# Set the variable Y1 as the model objective
>>> m.fs.obj = Objective(expr=m.fs.Y1, sense=minimize)
# Solve the model
>>> solver = SolverFactory('ipopt')
>>> res = solver.solve(m, tee=True)
>>> m.fs.display()
For an example of optimizing a flowsheet containing a PySMO-trained RBF surrogate model, see the Autothermal reformer flowsheet optimization example.
References:#
[1] Forrester et al.’s book “Engineering Design via Surrogate Modelling: A Practical Guide”, https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470770801
[2] Hongbing Fang & Mark F. Horstemeyer (2006): Global response approximation with radial basis functions, https://www.tandfonline.com/doi/full/10.1080/03052150500422294
[3] Rippa, S. (1999): An algorithm for selecting a good value for the parameter c in radial basis function interpolation, https://doi.org/10.1023/A:1018975909870
[4] Mongillo M.A. (2011) Choosing Basis Functions and Shape Parameters for Radial Basis Function Methods, https://doi.org/10.1137/11S010840