examples/notebooks/pouch_cell_identification.ipynb

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   "source": [
    "## Pouch Cell Model Parameter Identification\n",
    "\n",
    "In this notebook, we present the single particle model with a two dimensional current collector. This is achieved via the potential-pair models introduced in Marquis et al. [[1]](https://doi.org/10.1149/1945-7111/abbce4) as implemented in PyBaMM. At a high-level this is accomplished as a potential-pair model which is resolved across the discretised spatial locations.\n",
    "\n",
    "### Setting up the Environment\n",
    "\n",
    "Before we begin, we need to ensure that we have all the necessary tools. We will install PyBOP and upgrade dependencies:"
   ]
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  {
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    "outputId": "0d785b07-7cff-4aeb-e60a-4ff5a669afbf"
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   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "/Users/engs2510/Documents/Git/PyBOP/.nox/notebooks-overwrite/bin/python3: No module named pip\r\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Note: you may need to restart the kernel to use updated packages."
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "/Users/engs2510/Documents/Git/PyBOP/.nox/notebooks-overwrite/bin/python3: No module named pip\r\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Note: you may need to restart the kernel to use updated packages.\n"
     ]
    }
   ],
   "source": [
    "%pip install --upgrade pip ipywidgets -q\n",
    "%pip install pybop -q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "jAvD5fk104p0"
   },
   "source": [
    "### Importing Libraries\n",
    "\n",
    "With the environment set up, we can now import PyBOP alongside other libraries we will need:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "SQdt4brD04p1"
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "import pybop\n",
    "\n",
    "go = pybop.PlotlyManager().go\n",
    "pybop.PlotlyManager().pio.renderers.default = \"notebook_connected\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "5XU-dMtU04p2"
   },
   "source": [
    "## Generating Synthetic Data\n",
    "\n",
    "To demonstrate parameter estimation, we first need some data. We will generate synthetic data using a forward model, which requires defining a parameter set and the model itself.\n",
    "\n",
    "### Defining Parameters and Model\n",
    "\n",
    "We start by creating an example parameter set and then instantiate the single-particle model (SPM):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "parameter_set = pybop.ParameterSet.pybamm(\"Marquis2019\")\n",
    "parameter_set.update(\n",
    "    {\n",
    "        \"Negative electrode active material volume fraction\": 0.495,\n",
    "        \"Positive electrode active material volume fraction\": 0.612,\n",
    "    }\n",
    ")\n",
    "model = pybop.lithium_ion.SPM(\n",
    "    parameter_set=parameter_set,\n",
    "    options={\"current collector\": \"potential pair\", \"dimensionality\": 2},\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Updating the Spatial Grid\n",
    "\n",
    "Next, we update the number of spatial locations to solve the potential-pair model,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model.var_pts[\"y\"] = 5\n",
    "model.var_pts[\"z\"] = 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulating the Forward Model\n",
    "\n",
    "We can then simulate the model using the `predict` method, with a default constant current to generate voltage data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "sBasxv8U04p3"
   },
   "outputs": [],
   "source": [
    "t_eval = np.arange(0, 900, 3)\n",
    "values = model.predict(t_eval=t_eval)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Adding Noise to Voltage Data\n",
    "\n",
    "To make the parameter estimation more realistic, we add Gaussian noise to the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigma = 0.001\n",
    "corrupt_values = values[\"Voltage [V]\"].data + np.random.normal(0, sigma, len(t_eval))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "X8-tubYY04p_"
   },
   "source": [
    "## Identifying the Parameters"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "PQqhvSZN04p_"
   },
   "source": [
    "We will now set up the parameter estimation process by defining the datasets for optimisation and selecting the model parameters we wish to estimate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Creating a Dataset\n",
    "\n",
    "The dataset for optimisation is composed of time, current, and the noisy voltage data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "zuvGHWID04p_"
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   "outputs": [],
   "source": [
    "dataset = pybop.Dataset(\n",
    "    {\n",
    "        \"Time [s]\": t_eval,\n",
    "        \"Current function [A]\": values[\"Current [A]\"].data,\n",
    "        \"Voltage [V]\": corrupt_values,\n",
    "    }\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ffS3CF_704qA"
   },
   "source": [
    "### Defining Parameters to Estimate\n",
    "\n",
    "We select the parameters for estimation and set up their prior distributions and bounds:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "WPCybXIJ04qA"
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   "outputs": [],
   "source": [
    "parameters = pybop.Parameters(\n",
    "    pybop.Parameter(\n",
    "        \"Negative electrode active material volume fraction\",\n",
    "        prior=pybop.Gaussian(0.7, 0.05),\n",
    "        bounds=[0.45, 0.9],\n",
    "    ),\n",
    "    pybop.Parameter(\n",
    "        \"Positive electrode active material volume fraction\",\n",
    "        prior=pybop.Gaussian(0.58, 0.05),\n",
    "        bounds=[0.5, 0.8],\n",
    "    ),\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For plotting purposes, we want additional variables to be stored in the problem class. These are defined as,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "additional_variables = [\n",
    "    \"Negative current collector potential [V]\",\n",
    "    \"Positive current collector potential [V]\",\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "n4OHa-aF04qA"
   },
   "source": [
    "### Setting up the Optimisation Problem\n",
    "\n",
    "With the datasets and parameters defined, we can set up the optimisation problem, its cost function, and the optimiser."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "etMzRtx404qA"
   },
   "outputs": [],
   "source": [
    "problem = pybop.FittingProblem(\n",
    "    model, parameters, dataset, additional_variables=additional_variables\n",
    ")\n",
    "cost = pybop.SumSquaredError(problem)\n",
    "optim = pybop.CMAES(cost, max_iterations=30)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "caprp-bV04qB"
   },
   "source": [
    "### Running the Optimisation\n",
    "\n",
    "We proceed to run the CMA-ES optimisation algorithm to estimate the parameters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "-9OVt0EQ04qB"
   },
   "outputs": [],
   "source": [
    "x, final_cost = optim.run()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "-4pZsDmS04qC"
   },
   "source": [
    "### Viewing the Estimated Parameters\n",
    "\n",
    "After the optimisation, we can examine the estimated parameter values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "Hgz8SV4i04qC",
    "outputId": "e1e42ae7-5075-4c47-dd68-1b22ecc170f6"
   },
   "outputs": [
    {
     "data": {
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       "array([0.49949096, 0.60907043])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "x  # This will output the estimated parameters"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "KxKURtH704qC"
   },
   "source": [
    "## Plotting and Visualisation\n",
    "\n",
    "PyBOP provides various plotting utilities to visualise the results of the optimisation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "-cWCOiqR04qC"
   },
   "source": [
    "### Comparing System Response\n",
    "\n",
    "We can quickly plot the system's response using the estimated parameters compared to the target:"
   ]
  },
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   "execution_count": null,
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    "pybop.quick_plot(problem, problem_inputs=x, title=\"Optimised Comparison\");"
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   "source": [
    "### Spatial Plotting\n",
    "\n",
    "We can now plot the spatial variables from the solution object. First, the final negative current collector potential can be displayed. In this example, this is just a reference variable, but could be used for fitting or optimisation in the correct workflows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sol = problem.evaluate(parameters.as_dict(x))\n",
    "\n",
    "go.Figure(\n",
    "    [\n",
    "        go.Contour(\n",
    "            x=np.arange(0, model.var_pts[\"y\"] - 1, 1),\n",
    "            y=np.arange(0, model.var_pts[\"z\"] - 1, 1),\n",
    "            z=sol[\"Negative current collector potential [V]\"][:, :, -1],\n",
    "            colorscale=\"Viridis\",\n",
    "        )\n",
    "    ],\n",
    "    layout=dict(\n",
    "        title=\"Negative current collector potential [V]\",\n",
    "        xaxis_title=\"x node\",\n",
    "        yaxis_title=\"y node\",\n",
    "        width=600,\n",
    "        height=600,\n",
    "    ),\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We plot can then plot the positive current collector potential,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "                            \n",
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "go.Figure(\n",
    "    [\n",
    "        go.Contour(\n",
    "            x=np.arange(0, model.var_pts[\"y\"] - 1, 1),\n",
    "            y=np.arange(0, model.var_pts[\"z\"] - 1, 1),\n",
    "            z=sol[\"Positive current collector potential [V]\"][:, :, -1],\n",
    "            colorscale=\"Viridis\",\n",
    "        )\n",
    "    ],\n",
    "    layout=dict(\n",
    "        title=\"Positive current collector potential [V]\",\n",
    "        xaxis_title=\"x node\",\n",
    "        yaxis_title=\"y node\",\n",
    "        width=600,\n",
    "        height=600,\n",
    "    ),\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Convergence and Parameter Trajectories\n",
    "\n",
    "To assess the optimisation process, we can plot the convergence of the cost function and the trajectories of the parameters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "N5XYkevi04qD"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "            Plotly.purge(gd);\n",
       "            observer.disconnect();\n",
       "        }}\n",
       "}});\n",
       "\n",
       "// Listen for the removal of the full notebook cells\n",
       "var notebookContainer = gd.closest('#notebook-container');\n",
       "if (notebookContainer) {{\n",
       "    x.observe(notebookContainer, {childList: true});\n",
       "}}\n",
       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
       "}}\n",
       "\n",
       "                        })                };                });            </script>        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pybop.plot_convergence(optim)\n",
    "pybop.plot_parameters(optim);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cost Landscape\n",
    "\n",
    "Finally, we can visualise the cost landscape and the path taken by the optimiser:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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