[Wang and Dowling, 2022] Wang, Jialu, and Alexander W. Dowling.
“Pyomo.DOE: An open‐source package for model‐based design of experiments in Python.”
AIChE Journal 68.12 (2022): e17813. https://doi.org/10.1002/aic.17813
Methodology Overview
Model-based Design of Experiments (MBDoE) is a technique to maximize the information gain of experiments by directly using science-based models with physically meaningful parameters. It is one key component in the model calibration and uncertainty quantification workflow shown below:
Fig. 2 The exploratory analysis, parameter estimation, uncertainty analysis, and MBDoE are combined into an iterative framework to select, refine, and calibrate science-based mathematical models with quantified uncertainty. Currently, Pyomo.DoE focuses on increasing parameter precision.
Pyomo.DoE provides the exploratory analysis and MBDoE capabilities to the Pyomo ecosystem. The user provides one Pyomo model, a set of parameter nominal values,
the allowable design spaces for design variables, and the assumed observation error model.
During exploratory analysis, Pyomo.DoE checks if the model parameters can be inferred from the postulated measurements or preliminary data.
MBDoE then recommends optimized experimental conditions for collecting more data.
Parameter estimation packages such as Parmest can perform parameter estimation using the available data to infer values for parameters,
and facilitate an uncertainty analysis to approximate the parameter covariance matrix.
If the parameter uncertainties are sufficiently small, the workflow terminates and returns the final model with quantified parametric uncertainty.
If not, MBDoE recommends optimized experimental conditions to generate new data.
Below is an overview of the type of optimization models Pyomo.DoE can accommodate:
Pyomo.DoE is suitable for optimization models of continuous variables
Pyomo.DoE can handle equality constraints defining state variables
Pyomo.DoE supports (Partial) Differential-Algebraic Equations (PDAE) models via Pyomo.DAE
Pyomo.DoE also supports models with only algebraic constraints
The general form of a DAE problem that can be passed into Pyomo.DoE is shown below:
\[\begin{array}{l}
\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{y}(t), \mathbf{u}(t), \overline{\mathbf{w}}, \boldsymbol{\theta}) \\
\mathbf{g}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{y}(t), \mathbf{u}(t), \overline{\mathbf{w}},\boldsymbol{\theta})=\mathbf{0} \\
\mathbf{y} =\mathbf{h}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{u}(t), \overline{\mathbf{w}},\boldsymbol{\theta}) \\
\mathbf{f}^{\mathbf{0}}\left(\dot{\mathbf{x}}\left(t_{0}\right), \mathbf{x}\left(t_{0}\right), \mathbf{z}(t_0), \mathbf{y}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\theta})\right)=\mathbf{0} \\
\mathbf{g}^{\mathbf{0}}\left( \mathbf{x}\left(t_{0}\right),\mathbf{z}(t_0), \mathbf{y}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\theta}\right)=\mathbf{0}\\
\mathbf{y}^{\mathbf{0}}\left(t_{0}\right)=\mathbf{h}\left(\mathbf{x}\left(t_{0}\right),\mathbf{z}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\theta}\right)
\end{array}\]
where:
\(\boldsymbol{\theta} \in \mathbb{R}^{N_p}\) are unknown model parameters.
\(\mathbf{x} \subseteq \mathcal{X}\) are dynamic state variables which characterize trajectory of the system, \(\mathcal{X} \in \mathbb{R}^{N_x \times N_t}\).
\(\mathbf{z} \subseteq \mathcal{Z}\) are algebraic state variables, \(\mathcal{Z} \in \mathbb{R}^{N_z \times N_t}\).
\(\mathbf{u} \subseteq \mathcal{U}\) are time-varying decision variables, \(\mathcal{U} \in \mathbb{R}^{N_u \times N_t}\).
\(\overline{\mathbf{w}} \in \mathbb{R}^{N_w}\) are time-invariant decision variables.
\(\mathbf{y} \subseteq \mathcal{Y}\) are measurement response variables, \(\mathcal{Y} \in \mathbb{R}^{N_r \times N_t}\).
\(\mathbf{f}(\cdot)\) are differential equations.
\(\mathbf{g}(\cdot)\) are algebraic equations.
\(\mathbf{h}(\cdot)\) are measurement functions.
\(\mathbf{t} \in \mathbb{R}^{N_t \times 1}\) is a union of all time sets.
Based on the above notation, the form of the MBDoE problem addressed in Pyomo.DoE is shown below:
\begin{equation}
\begin{aligned}
\underset{\boldsymbol{\varphi}}{\max} \quad & \Psi (\mathbf{M}(\mathbf{\hat{y}}, \boldsymbol{\varphi})) \\
\text{s.t.} \quad & \mathbf{M}(\boldsymbol{\hat{\theta}}, \boldsymbol{\varphi}) = \sum_r^{N_r} \sum_{r'}^{N_r} \tilde{\sigma}_{(r,r')}\mathbf{Q}_r^\mathbf{T} \mathbf{Q}_{r'} + \mathbf{V}^{-1}_{\boldsymbol{\theta}}(\boldsymbol{\hat{\theta}}) \\
& \dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{y}(t), \mathbf{u}(t), \overline{\mathbf{w}}, \boldsymbol{\theta}) \\
& \mathbf{g}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{y}(t), \mathbf{u}(t), \overline{\mathbf{w}},\boldsymbol{\theta})=\mathbf{0} \\
& \mathbf{y} =\mathbf{h}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{u}(t), \overline{\mathbf{w}},\boldsymbol{\theta}) \\
& \mathbf{f}^{\mathbf{0}}\left(\dot{\mathbf{x}}\left(t_{0}\right), \mathbf{x}\left(t_{0}\right), \mathbf{z}(t_0), \mathbf{y}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\theta})\right)=\mathbf{0} \\
& \mathbf{g}^{\mathbf{0}}\left( \mathbf{x}\left(t_{0}\right),\mathbf{z}(t_0), \mathbf{y}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\theta}\right)=\mathbf{0}\\
&\mathbf{y}^{\mathbf{0}}\left(t_{0}\right)=\mathbf{h}\left(\mathbf{x}\left(t_{0}\right),\mathbf{z}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\theta}\right)
\end{aligned}
\end{equation}
where:
\(\boldsymbol{\varphi}\) are design variables, which are manipulated to maximize the information content of experiments. It should consist of one or more of \(\mathbf{u}(t), \mathbf{y}^{\mathbf{0}}({t_0}),\overline{\mathbf{w}}\). With a proper model formulation, the timepoints for control or measurements \(\mathbf{t}\) can also be degrees of freedom.
\(\mathbf{M}\) is the Fisher information matrix (FIM), estimated as the inverse of the covariance matrix of parameter estimates \(\boldsymbol{\hat{\theta}}\). A large FIM indicates more information contained in the experiment for parameter estimation.
\(\mathbf{Q}\) is the dynamic sensitivity matrix, containing the partial derivatives of \(\mathbf{y}\) with respect to \(\boldsymbol{\theta}\).
\(\Psi\) is the design criteria to measure FIM.
\(\mathbf{V}_{\boldsymbol{\theta}}(\boldsymbol{\hat{\theta}})^{-1}\) is the FIM of previous experiments.
Pyomo.DoE provides four design criteria \(\Psi\) to measure the size of FIM:
Table 6 Pyomo.DoE design criteria
Design criterion |
Computation |
Geometrical meaning |
|---|
A-optimality |
\(\text{trace}({\mathbf{M}})\) |
Dimensions of the enclosing box of the confidence ellipse |
D-optimality |
\(\text{det}({\mathbf{M}})\) |
Volume of the confidence ellipse |
E-optimality |
\(\text{min eig}({\mathbf{M}})\) |
Size of the longest axis of the confidence ellipse |
Modified E-optimality |
\(\text{cond}({\mathbf{M}})\) |
Ratio of the longest axis to the shortest axis of the confidence ellipse |
In order to solve problems of the above, Pyomo.DoE implements the 2-stage stochastic program. Please see Wang and Dowling (2022) for details.
Pyomo.DoE Usage Example
We illustrate the use of Pyomo.DoE using a reaction kinetics example (Wang and Dowling, 2022).
The Arrhenius equations model the temperature dependence of the reaction rate coefficient \(k_1, k_2\). Assuming a first-order reaction mechanism gives the reaction rate model. Further, we assume only species A is fed to the reactor.
\begin{equation}
\begin{aligned}
k_1 & = A_1 e^{-\frac{E_1}{RT}} \\
k_2 & = A_2 e^{-\frac{E_2}{RT}} \\
\frac{d{C_A}}{dt} & = -k_1{C_A} \\
\frac{d{C_B}}{dt} & = k_1{C_A} - k_2{C_B} \\
C_{A0}& = C_A + C_B + C_C \\
C_B(t_0) & = 0 \\
C_C(t_0) & = 0 \\
\end{aligned}
\end{equation}
\(C_A(t), C_B(t), C_C(t)\) are the time-varying concentrations of the species A, B, C, respectively.
\(k_1, k_2\) are the rates for the two chemical reactions using an Arrhenius equation with activation energies \(E_1, E_2\) and pre-exponential factors \(A_1, A_2\).
The goal of MBDoE is to optimize the experiment design variables \(\boldsymbol{\varphi} = (C_{A0}, T(t))\), where \(C_{A0},T(t)\) are the initial concentration of species A and the time-varying reactor temperature, to maximize the precision of unknown model parameters \(\boldsymbol{\theta} = (A_1, E_1, A_2, E_2)\) by measuring \(\mathbf{y}(t)=(C_A(t), C_B(t), C_C(t))\).
The observation errors are assumed to be independent both in time and across measurements with a constant standard deviation of 1 M for each species.
Step 0: Import Pyomo and the Pyomo.DoE module and create an Experiment class
>>> # === Required import ===
>>> import pyomo.environ as pyo
>>> from pyomo.contrib.doe import DesignOfExperiments
>>> import numpy as np
class ReactorExperiment(Experiment):
def __init__(self, data, nfe, ncp):
"""
Arguments
---------
data: object containing vital experimental information
nfe: number of finite elements
ncp: number of collocation points for the finite elements
"""
self.data = data
self.nfe = nfe
self.ncp = ncp
self.model = None
#############################
Step 1: Define the Pyomo process model
The process model for the reaction kinetics problem is shown below. We build the model without any data or discretization.
def create_model(self):
"""
This is an example user model provided to DoE library.
It is a dynamic problem solved by Pyomo.DAE.
Return
------
m: a Pyomo.DAE model
"""
m = self.model = pyo.ConcreteModel()
# Model parameters
m.R = pyo.Param(mutable=False, initialize=8.314)
# Define model variables
########################
# time
m.t = ContinuousSet(bounds=[0, 1])
# Concentrations
m.CA = pyo.Var(m.t, within=pyo.NonNegativeReals)
m.CB = pyo.Var(m.t, within=pyo.NonNegativeReals)
m.CC = pyo.Var(m.t, within=pyo.NonNegativeReals)
# Temperature
m.T = pyo.Var(m.t, within=pyo.NonNegativeReals)
# Arrhenius rate law equations
m.A1 = pyo.Var(within=pyo.NonNegativeReals)
m.E1 = pyo.Var(within=pyo.NonNegativeReals)
m.A2 = pyo.Var(within=pyo.NonNegativeReals)
m.E2 = pyo.Var(within=pyo.NonNegativeReals)
# Differential variables (Conc.)
m.dCAdt = DerivativeVar(m.CA, wrt=m.t)
m.dCBdt = DerivativeVar(m.CB, wrt=m.t)
########################
# End variable def.
# Equation definition
########################
# Expression for rate constants
@m.Expression(m.t)
def k1(m, t):
return m.A1 * pyo.exp(-m.E1 * 1000 / (m.R * m.T[t]))
@m.Expression(m.t)
def k2(m, t):
return m.A2 * pyo.exp(-m.E2 * 1000 / (m.R * m.T[t]))
# Concentration odes
@m.Constraint(m.t)
def CA_rxn_ode(m, t):
return m.dCAdt[t] == -m.k1[t] * m.CA[t]
@m.Constraint(m.t)
def CB_rxn_ode(m, t):
return m.dCBdt[t] == m.k1[t] * m.CA[t] - m.k2[t] * m.CB[t]
# algebraic balance for concentration of C
# Valid because the reaction system (A --> B --> C) is equimolar
@m.Constraint(m.t)
def CC_balance(m, t):
return m.CA[0] == m.CA[t] + m.CB[t] + m.CC[t]
########################
Step 2: Finalize the Pyomo process model
Here we add data to the model and finalize the discretization. This step is required before the model can be labeled.
def finalize_model(self):
"""
Example finalize model function. There are two main tasks
here:
1. Extracting useful information for the model to align
with the experiment. (Here: CA0, t_final, t_control)
2. Discretizing the model subject to this information.
"""
m = self.model
# Unpacking data before simulation
control_points = self.data["control_points"]
# Set initial concentration values for the experiment
m.CA[0].value = self.data["CA0"]
m.CB[0].fix(self.data["CB0"])
# Update model time `t` with time range and control time points
m.t.update(self.data["t_range"])
m.t.update(control_points)
# Fix the unknown parameter values
m.A1.fix(self.data["A1"])
m.A2.fix(self.data["A2"])
m.E1.fix(self.data["E1"])
m.E2.fix(self.data["E2"])
# Add upper and lower bounds to the design variable, CA[0]
m.CA[0].setlb(self.data["CA_bounds"][0])
m.CA[0].setub(self.data["CA_bounds"][1])
m.t_control = control_points
# Discretizing the model
discr = pyo.TransformationFactory("dae.collocation")
discr.apply_to(m, nfe=self.nfe, ncp=self.ncp, wrt=m.t)
# Initializing Temperature in the model
cv = None
for t in m.t:
if t in control_points:
cv = control_points[t]
m.T[t].setlb(self.data["T_bounds"][0])
m.T[t].setub(self.data["T_bounds"][1])
m.T[t] = cv
# Make a constraint that holds temperature constant between control time points
@m.Constraint(m.t - control_points)
def T_control(m, t):
"""
Piecewise constant temperature between control points
"""
neighbour_t = max(tc for tc in control_points if tc < t)
return m.T[t] == m.T[neighbour_t]
#########################
Step 4: Implement the get_labeled_model method
This method utilizes the previous 3 steps and is used by Pyomo.DoE to build the model to perform optimal experimental design.
def get_labeled_model(self):
if self.model is None:
self.create_model()
self.finalize_model()
self.label_experiment()
return self.model
Step 5: Exploratory analysis (Enumeration)
Exploratory analysis is suggested to enumerate the design space to check if the problem is identifiable,
i.e., ensure that D-, E-optimality metrics are not small numbers near zero, and Modified E-optimality is not a big number.
Pyomo.DoE can perform exploratory sensitivity analysis with the compute_FIM_full_factorial function.
The compute_FIM_full_factorial function generates a grid over the design space as specified by the user. Each grid point represents an MBDoE problem solved using compute_FIM method. In this way, sensitivity of the FIM over the design space can be evaluated.
The following code executes the above problem description:
DATA_DIR = pathlib.Path(__file__).parent
file_path = DATA_DIR / "result.json"
with open(file_path) as f:
data_ex = json.load(f)
# Put temperature control time points into correct format for reactor experiment
data_ex["control_points"] = {
float(k): v for k, v in data_ex["control_points"].items()
}
# Create a ReactorExperiment object; data and discretization information are part
# of the constructor of this object
experiment = ReactorExperiment(data=data_ex, nfe=10, ncp=3)
# Use a central difference, with step size 1e-3
fd_formula = "central"
step_size = 1e-3
# Use the determinant objective with scaled sensitivity matrix
objective_option = "determinant"
scale_nominal_param_value = True
# Create the DesignOfExperiments object
# We will not be passing any prior information in this example
# and allow the experiment object and the DesignOfExperiments
# call of ``run_doe`` perform model initialization.
doe_obj = DesignOfExperiments(
experiment,
fd_formula=fd_formula,
step=step_size,
objective_option=objective_option,
scale_constant_value=1,
scale_nominal_param_value=scale_nominal_param_value,
prior_FIM=None,
jac_initial=None,
fim_initial=None,
L_diagonal_lower_bound=1e-7,
solver=None,
tee=False,
get_labeled_model_args=None,
_Cholesky_option=True,
_only_compute_fim_lower=True,
)
# Make design ranges to compute the full factorial design
design_ranges = {"CA[0]": [1, 5, 9], "T[0]": [300, 700, 9]}
# Compute the full factorial design with the sequential FIM calculation
doe_obj.compute_FIM_full_factorial(design_ranges=design_ranges, method="sequential")
# Plot the results
doe_obj.draw_factorial_figure(
sensitivity_design_variables=["CA[0]", "T[0]"],
fixed_design_variables={
"T[0.125]": 300,
"T[0.25]": 300,
"T[0.375]": 300,
"T[0.5]": 300,
"T[0.625]": 300,
"T[0.75]": 300,
"T[0.875]": 300,
"T[1]": 300,
},
title_text="Reactor Example",
xlabel_text="Concentration of A (M)",
ylabel_text="Initial Temperature (K)",
figure_file_name="example_reactor_compute_FIM",
log_scale=False,
)
###########################
An example output of the code above, a design exploration for the initial concentration and temperature as experimental design variables with 9 values, produces the four figures summarized below:
A heatmap shows the change of the objective function, a.k.a. the experimental information content, in the design space. Horizontal and vertical axes are the two experimental design variables, while the color of each grid shows the experimental information content. For A optimality (top left subfigure), the figure shows that the most informative region is around \(C_{A0}=5.0\) M, \(T=300.0\) K, while the least informative region is around \(C_{A0}=1.0\) M, \(T=700.0\) K.