Add functional parameter example (#442)

* Create functional_parameters.py

* Define parameter set first

* Update CHANGELOG.md

* Apply suggestions from code review
Co-authored-by: Brady Planden <55357039+BradyPlanden@users.noreply.github.com>

CHANGELOG.md

@@ -2,6 +2,7 @@
 
 ## Features
 
+- [#441](https://github.com/pybop-team/PyBOP/issues/441) - Adds an example for estimating constants within a `pybamm.FunctionalParameter`.
 - [#405](https://github.com/pybop-team/PyBOP/pull/405) - Adds frequency-domain based EIS prediction methods via `model.simulateEIS` and updates to `problem.evaluate` with examples and tests.
 - [#460](https://github.com/pybop-team/PyBOP/pull/460) - Notebook example files added for ECM and folder structure updated.
 - [#450](https://github.com/pybop-team/PyBOP/pull/450) - Adds support for IDAKLU with output variables, and corresponding examples, tests.

examples/scripts/functional_parameters.py

@@ -0,0 +1,97 @@
+import numpy as np
+import pybamm
+
+import pybop
+
+# This example demonstrates how to use a pybamm.FunctionalParameter to
+# optimise functional parameters using PyBOP.
+
+# Method: Define a new scalar parameter for use in a functional parameter
+# that already exists in the model, for example an exchange current density.
+
+
+# Load parameter set
+parameter_set = pybop.ParameterSet.pybamm("Chen2020")
+
+
+# Define a new function using pybamm parameters
+def positive_electrode_exchange_current_density(c_e, c_s_surf, c_s_max, T):
+    # New parameters
+    j0_ref = pybamm.Parameter(
+        "Positive electrode reference exchange-current density [A.m-2]"
+    )
+    alpha = pybamm.Parameter("Positive electrode charge transfer coefficient")
+
+    # Existing parameters
+    c_e_init = pybamm.Parameter("Initial concentration in electrolyte [mol.m-3]")
+
+    return (
+        j0_ref
+        * ((c_e / c_e_init) * (c_s_surf / c_s_max) * (1 - c_s_surf / c_s_max)) ** alpha
+    )
+
+
+# Give default values to the new scalar parameters and pass the new function
+parameter_set.update(
+    {
+        "Positive electrode reference exchange-current density [A.m-2]": 1,
+        "Positive electrode charge transfer coefficient": 0.5,
+    },
+    check_already_exists=False,
+)
+parameter_set["Positive electrode exchange-current density [A.m-2]"] = (
+    positive_electrode_exchange_current_density
+)
+
+# Model definition
+model = pybop.lithium_ion.SPM(
+    parameter_set=parameter_set, options={"contact resistance": "true"}
+)
+
+# Fitting parameters
+parameters = pybop.Parameters(
+    pybop.Parameter(
+        "Positive electrode reference exchange-current density [A.m-2]",
+        prior=pybop.Gaussian(1, 0.1),
+    ),
+    pybop.Parameter(
+        "Positive electrode charge transfer coefficient",
+        prior=pybop.Gaussian(0.5, 0.1),
+    ),
+)
+
+# Generate data
+sigma = 0.001
+t_eval = np.arange(0, 900, 3)
+values = model.predict(t_eval=t_eval)
+corrupt_values = values["Voltage [V]"].data + np.random.normal(0, sigma, len(t_eval))
+
+# Form dataset
+dataset = pybop.Dataset(
+    {
+        "Time [s]": t_eval,
+        "Current function [A]": values["Current [A]"].data,
+        "Voltage [V]": corrupt_values,
+    }
+)
+
+# Generate problem, cost function, and optimisation class
+problem = pybop.FittingProblem(model, parameters, dataset)
+cost = pybop.RootMeanSquaredError(problem)
+optim = pybop.SciPyMinimize(cost, max_iterations=125)
+
+# Run optimisation
+x, final_cost = optim.run()
+print("Estimated parameters:", x)
+
+# Plot the timeseries output
+pybop.quick_plot(problem, problem_inputs=x, title="Optimised Comparison")
+
+# Plot convergence
+pybop.plot_convergence(optim)
+
+# Plot the parameter traces
+pybop.plot_parameters(optim)
+
+# Plot the cost landscape with optimisation path
+pybop.plot2d(optim, steps=15)