# Sentiment analysis in your own PDF documents with ChatGPT

While preparing some documents for a forthcoming academic review, I was asked by my department chair to go through all of the student comments from every course I've taught in the last 6 years and find a few "positive" comments that he could quote in his summary writeup.

As I usually teach 3 courses per year, sometimes with cross-listings in multiple departments and/or undergrad/grad sections. All in, this resulted in 32 PDF documents each with many student comments that I needed to draw from. Doing it manually, would require opening each document, reading and/or compiling the comments and then evaluating and choosing the "most positive" comments. I decided to use ChatGPT with a text embedding to assit me in this task, and thought the code might be useful to others so I am sharing it below along with some comments and documentation.

First we start with the package imports. I'm heavily using langchian which provides abstractions for using large language models (LLMs) and tooling to easily "chain together" tasks such as reading in text, creating a vectorstore of the text embedding to be passed to a LLM along with a prompt which a specific question or instruction.

```
import os
from PyPDF2 import PdfReader
from langchain.text_splitter import CharacterTextSplitter
from langchain.embeddings import OpenAIEmbeddings
from langchain.vectorstores import FAISS
from langchain.memory import ConversationBufferMemory
from langchain.chains import ConversationalRetrievalChain
from langchain.chat_models import ChatOpenAI
from langchain.prompts import ChatPromptTemplate
from langchain.prompts.chat import SystemMessage, HumanMessagePromptTemplate
from IPython.display import display, Markdown
```

We're going to use ChatGPT from OpenAI so we'll need to supply an API key. This video tutorial demonstrates how to aquire an API key. If you'd like to use the code below, you'll need to uncomment and paste your API key in the string to the right of the equals sign below.

```
#os.environ['OPENAI_API_KEY'] = "<your API key here>"
```

Now we'll define a few helper functions to help us find PDFs in a given directory and then parse the text out of them. The `get_pdf_text`

function below combines all the text from all the PDFs into a single text string.

```
def find_pdf_files(directory_path):
pdf_files = []
for root, dirs, files in os.walk(directory_path):
for file in files:
if file.endswith('.pdf'):
pdf_files.append(file)
return pdf_files
def get_pdf_text(pdf_files):
text = ""
for pdf in pdf_files:
pdf_reader = PdfReader(pdf)
for page in pdf_reader.pages:
text += page.extract_text()
return text
```

Next we'll use the `CharacterTextSpitter`

class from langchain to take the continuous string of text from all the PDFs and turn them into chunks of texts which is needed to create a vectorstore text embedding. Text ebeddings measure the relatedness of text strings.

```
def get_text_chunks(raw_text):
text_splitter = CharacterTextSplitter(
separator = '\n',
chunk_size = 2000,
chunk_overlap = 500,
length_function = len
)
chunks = text_splitter.split_text(raw_text)
return chunks
```

Here we create the vectorstore using the `OpenAIEmbeddings`

class. While we are using an OpenAI embedding here, it's not required. Langchain provides nice abstractions that allow for using different embeddings model with ChatGPT.

```
def get_vectorstore(chunks):
embeddings = OpenAIEmbeddings()
vectorstore = FAISS.from_texts(texts=chunks, embedding=embeddings)
return vectorstore
```

Below we create the converstation chiain which combines a prompt template with the text embedding and allows for users querys. Prompt templates can increase the accuracy of LMM responses. Special thanks to Jeremy Howard and his Twitter thread from which I took his custom instructions as a system prompt template.

```
def get_conversation_chain(vectorstore):
template = ChatPromptTemplate.from_messages(
[
SystemMessage(
content=(
"You are an autoregressive language model that has been fine-"
"tuned with instruction-tuning and RLHF. You carefully "
"provide accurate, factual, thoughtful, nuanced answers, and"
"are brilliant at reasoning. If you think there might not be "
"a correct answer, you say so. Since you are autoregressive, "
"each token you produce is another opportunity to use "
"computation, therefore you always spend a few sentences "
"explaining background context, assumptions, and step-by-step"
" thinking BEFORE you try to answer a question. Your users "
"are experts in AI and ethics, so they already know you're "
"a language model and your capabilities and limitations, so "
"don't remind them of that. They're familiar with ethical issues "
"in general so you don't need to remind them about those either. "
"Don't be verbose in your answers, but do provide details and "
"examples where it might help the explanation."
)
),
HumanMessagePromptTemplate.from_template("{text}"),
]
)
llm = ChatOpenAI()#model_name="gpt-4")
memory = ConversationBufferMemory(memory_key='chat_history', return_messages=True)
conversation_chain = ConversationalRetrievalChain.from_llm(
llm=llm,
condense_question_prompt=template,
retriever=vectorstore.as_retriever(),
memory=memory
)
return conversation_chain
```

Finally, the function below combines all the helper functions and returns a chat chain that we can ask questions based on the content contained in our PDFs.

```
def create_chat_from_pdfs(directory):
pdfs = find_pdf_files(directory)
raw_text = get_pdf_text(pdfs)
chunks = get_text_chunks(raw_text)
vs = get_vectorstore(chunks)
chain = get_conversation_chain(vs)
return chain
```

Here we initialize the chain to read in any PDFs in the current working directory.

```
chat_chain = create_chat_from_pdfs(".")
```

Now we use the chain and our custom text embedding to find 15 of the "most postive and supportive" student comments giving preference to those that use the phrase "Dr. Foster". Finally we print out the results of the query using Markdown to format the list in rich text.

```
result = chat_chain.run("In the given text, there are many student comments "
"preceeded with the word RESPONSE in capital letters. "
"Choose 15 of the most positive and supportive "
"student comments for the instructor and course. "
"Give preference to the comments that call out "
"Dr. Foster by name")
display(Markdown(result))
```

While you don't have access to the original PDFs, I can assure you that these comments have been parsed from them and quite responsive to the prompt I used. They are not all "perfectly" responsive, but I really only need 2-3 comments, so I asked for 15 so that I can then inspect them to downselect and choose the ones I'd like to provide.

You should be able to adapt this code to your own use cases.

# What I might have done in a recent hackathon

### and why domain expertise matters in machine learning...¶

Recently, my colleague Prof. Michael Pyrcz (@GeostatsGuy) and myself hosted a data science hackathon open to students at The University of Texas at Austin with support from The Hildebrand Department of Petroleum and Geosystems Engineering. Given the COVID pandemic, we hosted the event virtually over two weekends in April 2021. The first weekend consisted of a workshop followed by an explanation of the problem dataset, then the students broke into teams and worked on their solutions that were ultimately submitted for scoring, code review, and presentation judging by a panel of industry experts the following weekend.

This was a really fun experience for me personally and the feedback has been incredible from all involved. Given that I have been teaching data science skills and challenging students in my classes with difficult coding exercises for years, it was great to see them showcase their capabilities.

I'd encourage you to take a look at the detailed problem statement and datasets linked to above, but briefly, the challenge was: given petrophysical data from well logs and spatial maps interpreted from seismic on 73 oil wells that have been in production for some time, predict the cumulative 3 year production for 10 new wells recently drilled, and having well log data, but not yet put into production. In addition to their "best estimate" the students where asked to provide 100 additional realizations such that an uncertainty model could be evaluated. The objective scoring consisted of comparisons of the mean squared error with respect to their "best estimate" and the true 3 year production (which we know, but was withheld from the students) as well as a scoring of their uncertainty model with a "goodness measure" proposed by Deutsch (1996).

I thought it would be a fun exercise to consider what I might have done to complete this challenge myself. In the spirit of the hackathon, I'll limit my time working on the project to what I can accomplish in a single day. Of course, since I am already familiar with the dataset, I have a head start over the teams in the competition who generally spent a few hours just investigating the data the first day. But given my short timeline, I won't be training any complicated neural network architectures or really any complicated machine learning models at all. I want to see how I can use my domain expertise to engineer meaningful features to get an answer quickly. I also have a surprise in store with respect to how I handled the uncertainty model. So let's get started...

### Feature Imputation¶

First, like all work in Python, I'll start with the module imports used during this work.

```
import pandas as pd
import seaborn as sns
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib import colors
from scipy.optimize import curve_fit
from sklearn.cluster import KMeans
from sklearn.neighbors import KNeighborsClassifier
from sklearn.model_selection import GridSearchCV, KFold
from sklearn.decomposition import PCA
from sklearn.preprocessing import (PolynomialFeatures, MaxAbsScaler,
MinMaxScaler, StandardScaler, RobustScaler)
from sklearn.linear_model import LinearRegression, ElasticNet
from sklearn.tree import DecisionTreeRegressor
from sklearn.ensemble import BaggingRegressor
from sklearn.pipeline import Pipeline
from sklearn.metrics import mean_absolute_percentage_error
sns.set(rc={'text.usetex' : True})
```

And I'll read in the well log datasets from both the wells that have been in production as well as what we call the *preproduction* wells, i.e. those we are trying to predict the cumulative production for after 3 years.

```
production_wells_df = pd.read_csv('wellbore_data_producer_wells.csv',
index_col='Well_ID')
preproduction_wells_df = pd.read_csv('wellbore_data_preproduction_well.csv',
index_col='Well_ID')
```

Now we'll combine those two dataframes into a single dataframe.

```
wells_df = pd.concat([production_wells_df, preproduction_wells_df])
wells_df.head()
```

And inspect some overall statistics on the dataset.

```
wells_df.describe().T
```

We can see that there is quite a bit of missing data. Of particular concern is the large amount of missing permeabilities as this will be one of the strongest indicators of production. So our first task is going to be to impute the missing features, especially working toward a good model for permeability. To start, let's plot some of the provided map data. Here we are showing the spatial distributions of percentage facies in the reservoir. There is a fault in the reservoir which we only know the location of, but no other information. The white circles represent wells we have production for, the red circles are wells we are trying to predict production. The fault is indicated by the red line.

```
production_well_locations = \
(production_wells_df[['X, m', 'Y, m']]
.groupby(production_wells_df.index)
.mean())
preproduction_well_locations = \
(preproduction_wells_df[['X, m', 'Y, m']]
.groupby(preproduction_wells_df.index)
.mean())
px = production_well_locations['X, m'].to_numpy()[::1]
py = -production_well_locations['Y, m'].to_numpy() + 10000
ppx = preproduction_well_locations['X, m'].to_numpy()[::1]
ppy = -preproduction_well_locations['Y, m'].to_numpy() + 10000
fx = np.linspace(0, 1e4, num=5)
fy = fx - 1750
fig = plt.figure(constrained_layout=True, figsize=(8,5))
ax = fig.subplot_mosaic(
[['sand', 'sandy shale', 'colorbar'],
['shale', 'shaly sand', 'colorbar']],
gridspec_kw={"width_ratios": [1, 1, 0.1]})
cmap = cm.get_cmap('viridis', 100)
bounds = np.linspace(0,1.0,num=100)
vals = bounds#[:-1]
norm = colors.BoundaryNorm(bounds, cmap.N)
ax['sand'].imshow(np.load('2d_sand_propotion.npy'), cmap=cmap, norm=norm,
extent=[0, 10000, 0, 10000])
ax['sand'].scatter(px, py, color='w', marker='o', alpha=0.5)
ax['sand'].scatter(ppx, ppy, color='r', marker='o', alpha=0.5)
ax['sand'].plot(fx, fy, color='r')
ax['sand'].set_ylim([0,1e4])
ax['sand'].set_title('Sand')
ax['sand'].set_aspect(1)
ax['sand'].grid(None)
ax['sandy shale'].imshow(np.load('2d_sandy_shale_propotion.npy'), cmap=cmap,
norm=norm, extent=[0, 10000, 0, 10000])
ax['sandy shale'].scatter(px, py, color='w', marker='o', alpha=0.5)
ax['sandy shale'].scatter(ppx, ppy, color='r', marker='o', alpha=0.5)
ax['sandy shale'].plot(fx, fy, color='r')
ax['sandy shale'].set_ylim([0,1e4])
ax['sandy shale'].set_title('Sandy shale')
ax['sandy shale'].set_aspect(1)
ax['sandy shale'].grid(None)
ax['shale'].imshow(np.load('2d_shale_propotion.npy'), cmap=cmap, norm=norm,
extent=[0, 10000, 0, 10000])
ax['shale'].scatter(px, py, color='w', marker='o', alpha=0.5)
ax['shale'].scatter(ppx, ppy, color='r', marker='o', alpha=0.5)
ax['shale'].plot(fx, fy, color='r')
ax['shale'].set_ylim([0,1e4])
ax['shale'].set_title('Shale')
ax['shale'].set_aspect(1)
ax['shale'].grid(None)
im = ax['shaly sand'].imshow(np.load('2d_shaly_sand_propotion.npy'), cmap=cmap,
norm=norm, extent=[0, 10000, 0, 10000])
ax['shaly sand'].scatter(px, py, color='w', marker='o', alpha=0.5)
ax['shaly sand'].scatter(ppx, ppy, color='r', marker='o', alpha=0.5)
ax['shaly sand'].plot(fx, fy, color='r')
ax['shaly sand'].set_ylim([0,1e4])
ax['shaly sand'].set_title('Shaly sand')
ax['shaly sand'].set_aspect(1)
ax['shaly sand'].grid(None)
cbar = fig.colorbar(im, cax=ax['colorbar'], boundaries=bounds, values=vals);
cbar.set_ticks([0.0, 0.25, 0.5, 0.75, 1.0])
cbar.set_ticklabels([r'$0$', r'$\frac{1}{4}$', r'$\frac{1}{2}$',
r'$\frac{3}{4}$', r'$1$'])
```

It doesn't look like the fault produces any offset or discontinuity in the spatial facies information, that's good news, so we'll proceed with imputing the missing facies information using spatial location information. First, we'll subset the dataframe and replace the string labels for facies with numerical ones, so 0 will indicated Sandstone, 1 will indicate Sandy shale, and so on...

```
facies_df = (wells_df[['X, m', 'Y, m', 'Depth, m', 'Rock facies']]
.replace({'Sandstone': 0,
'Sandy shale': 1,
'Shale': 2,
'Shaly sandstone': 3})
)
missing_facies = facies_df['Rock facies'].isnull()
facies_df[~missing_facies].head()
```

Now we'll build a k-nearest neighbors classifier, do a little hyperparameter tuning with scikit-learn's builtin GridSearchCV.

```
parameters = {'weights': ('uniform', 'distance'),
'n_neighbors':[4, 6, 8, 10]}
knn = KNeighborsClassifier()
gcv = GridSearchCV(knn, parameters, cv=KFold(random_state=2, shuffle=True))
gcv.fit(facies_df.loc[~missing_facies, 'X, m':'Depth, m'],
facies_df.loc[~missing_facies, 'Rock facies'])
gcv.best_params_
```

Using the hyperparameter settings above, we can now predict (and impute) the missing facies values.

```
facies_df.loc[missing_facies, 'Rock facies'] = \
gcv.predict(facies_df.loc[missing_facies, 'X, m':'Depth, m'])
wells_df['Rock facies'] = facies_df['Rock facies'].astype('int')
wells_df.describe().T.loc[['X, m', 'Y, m', 'Rock facies']]
```

Given that we'd expect rocks of the same facies to have a similar density and acoustic impedance, we'll impute those missing features with the averages from each facies.

```
wells_df['Density, g/cm3'] = \
(wells_df.groupby('Rock facies')['Density, g/cm3']
.apply(lambda df: df.fillna(df.mean()))
.reset_index()
.set_index(['Well_ID'])
)
wells_df['Acoustic Impedance, kg/s-m^2'] = \
(wells_df.groupby('Rock facies')['Acoustic Impedance, kg/s-m^2']
.apply(lambda df: df.fillna(df.mean()))
.reset_index()
.set_index(['Well_ID'])
)
```

Now we'll subset the dataframe and use the features shown to impute the missing porosity values using polynomial regression.

```
missing_porosity = wells_df['Porosity, fraction'].isnull()
porosity_df = (wells_df.loc[:, 'X, m':'Density, g/cm3']
.drop('Permeability, mD', axis=1))
porosity_df.head()
```

We'll setup a pipeline and use `GridSearchCV`

again to find the best hyperparameters.

```
scalers = [MaxAbsScaler(), MinMaxScaler(), StandardScaler(), RobustScaler()]
pipe = Pipeline([
('scaler', StandardScaler()),
('pca', PCA()),
('poly', PolynomialFeatures()),
('reg', LinearRegression())
])
params = {
'scaler': scalers,
'poly__degree': [1, 2, 3],
'pca__n_components': [1, 2, 3, 4, 5, 6],
}
gcv = GridSearchCV(pipe, params, cv=KFold(random_state=5, shuffle=True))
gcv.fit(porosity_df[~missing_porosity].drop('Porosity, fraction', axis=1),
porosity_df.loc[~missing_porosity, 'Porosity, fraction'])
gcv.best_params_
```

The best parameters are shown above. Now we'll use this model to impute the missing porosity.

```
porosity_df.loc[missing_porosity, 'Porosity, fraction'] = \
gcv.predict(porosity_df[missing_porosity].drop('Porosity, fraction', axis=1))
```

Below we'll plot the imputed and given porosities. Nothing looks too strange here, none of the imputed values are outliers with respect to the ranges of the given data.

```
fig, ax = plt.subplots()
ax.scatter(porosity_df[~missing_porosity &
(porosity_df['Rock facies'] == 0)]['Depth, m'],
porosity_df[~missing_porosity &
(porosity_df['Rock facies'] == 0)]['Porosity, fraction'],
color='r', facecolors='none', label='Sandstone')
ax.scatter(porosity_df[missing_porosity &
(porosity_df['Rock facies'] == 0)]['Depth, m'],
porosity_df[missing_porosity &
(porosity_df['Rock facies'] == 0)]['Porosity, fraction'],
color='r', label='Sandstone (Imputed)')
ax.scatter(porosity_df[~missing_porosity &
(porosity_df['Rock facies'] == 1)]['Depth, m'],
porosity_df[~missing_porosity &
(porosity_df['Rock facies'] == 1)]['Porosity, fraction'],
color='b', facecolors='none', label='Sandy shale')
ax.scatter(porosity_df[missing_porosity &
(porosity_df['Rock facies'] == 1)]['Depth, m'],
porosity_df[missing_porosity &
(porosity_df['Rock facies'] == 1)]['Porosity, fraction'],
color='b', label='Sandy shale (Imputed)')
ax.scatter(porosity_df[~missing_porosity &
(porosity_df['Rock facies'] == 2)]['Depth, m'],
porosity_df[~missing_porosity &
(porosity_df['Rock facies'] == 2)]['Porosity, fraction'],
color='g', facecolors='none', label='Shale')
ax.scatter(porosity_df[missing_porosity &
(porosity_df['Rock facies'] == 2)]['Depth, m'],
porosity_df[missing_porosity &
(porosity_df['Rock facies'] == 2)]['Porosity, fraction'],
color='g', label='Shale (Imputed)')
ax.scatter(porosity_df[~missing_porosity &
(porosity_df['Rock facies'] == 3)]['Depth, m'],
porosity_df[~missing_porosity &
(porosity_df['Rock facies'] == 3)]['Porosity, fraction'],
color='k', facecolors='none', label='Shaly sand')
ax.scatter(porosity_df[missing_porosity &
(porosity_df['Rock facies'] == 3)]['Depth, m'],
porosity_df[missing_porosity &
(porosity_df['Rock facies'] == 3)]['Porosity, fraction'],
color='k', label='Shaly sand (Imputed)')
ax.set_xlabel('Depth, m')
ax.set_ylabel('Porosity, fraction')
ax.legend(bbox_to_anchor=(1.5, 1), loc='upper right', ncol=1);
```

We can also look at the distribution of the imputed porosities and compare to the given values. The imputation preserves the bimodal distribution of the original data.

```
fig, ax = plt.subplots(1, 2)
sns.distplot(porosity_df.loc[~missing_porosity, 'Porosity, fraction'], ax=ax[0])
ax[0].set_title('Given values')
sns.distplot(porosity_df.loc[missing_porosity, 'Porosity, fraction'], ax=ax[1]);
sns.distplot(porosity_df.loc[missing_porosity, 'Porosity, fraction'], ax=ax[1]);
ax[1].set_title('Imputed values');
```

Now we'll add this imputed data to our `wells_df`

.

```
wells_df['Porosity, fraction'] = porosity_df['Porosity, fraction']
```

To impute the missing permeabilities, we'll use some knowledge of petrophysics to do a little feature engineering. There is a widely used correlation between porosity and permeability called the Kozeny-Carmen relationship which models permeability as

$$ \kappa \sim \frac{\phi^3}{(1-\phi)^2} $$

where $\kappa$ is the permeability and $\phi$ is the porosity. We can quickly take a look at this relationship using seaborn's builtin `regplot`

function, i.e.

```
missing_perms = wells_df['Permeability, mD'].isnull()
ϕ = wells_df.loc[~missing_perms, 'Porosity, fraction'].to_numpy()
fig, ax = plt.subplots()
sns.regplot(ϕ ** 3 / (1 - ϕ) ** 2,
wells_df.loc[~missing_perms, 'Permeability, mD'], ax=ax);
ax.set_xlabel(r'$\frac{\phi^3}{(1 - \phi)^2}$');
```

While not the best model, we can use this data to condition our more complex prediction to come. First we find the slope and intercept of the blue line above.

```
fphi = lambda ϕ, m, kappa0: m * ϕ ** 3 / (1 - ϕ) ** 2 + kappa0
popt, _ = curve_fit(fphi, ϕ, wells_df.loc[~missing_perms, 'Permeability, mD'])
```

Now we'll use the model to create a feature we'll call `'KC permeability, mD'`

.

```
perm_df = wells_df.loc[:, 'X, m':'Density, g/cm3']
perm_df['KC permeability, mD'] = fphi(wells_df['Porosity, fraction'],
popt[0], popt[1])
```

Using the data show above, we'll build a model to impute the missing permeabilities. Again, using `GridSearchCV`

to hyperparameter tune, we have

```
scalers = [MaxAbsScaler(), MinMaxScaler(), StandardScaler(), RobustScaler()]
pipe = Pipeline([
('scaler', StandardScaler()),
('pca', PCA()),
('poly', PolynomialFeatures()),
('reg', LinearRegression())
])
params = {
'scaler': scalers,
'poly__degree': [2, 3, 4],
'pca__n_components': [1, 2, 3, 4, 5, 6, 7],
}
gcv = GridSearchCV(pipe, params, cv=KFold(random_state=5, shuffle=True))
gcv.fit(perm_df[~missing_perms].drop('Permeability, mD', axis=1),
perm_df.loc[~missing_perms, 'Permeability, mD'])
gcv.best_params_
```

With these parameters, we can predict the missing permeabilities.

```
perm_df.loc[missing_perms, 'Permeability, mD'] = \
gcv.predict(perm_df[missing_perms].drop('Permeability, mD', axis=1))
```

Visualizing the results of the prediction against the given data, this model appears to perform well.

```
fig, ax = plt.subplots()
ax.scatter(perm_df[~missing_perms &
(perm_df['Rock facies'] == 0)]['Porosity, fraction'],
perm_df[~missing_perms &
(perm_df['Rock facies'] == 0)]['Permeability, mD'],
color='r', facecolors='none', label='Sandstone')
ax.scatter(perm_df[missing_perms &
(perm_df['Rock facies'] == 0)]['Porosity, fraction'],
perm_df[missing_perms &
(perm_df['Rock facies'] == 0)]['Permeability, mD'],
color='r', label='Sandstone (Imputed)')
ax.scatter(perm_df[~missing_perms &
(perm_df['Rock facies'] == 1)]['Porosity, fraction'],
perm_df[~missing_perms &
(perm_df['Rock facies'] == 1)]['Permeability, mD'],
color='b', facecolors='none', label='Sandy shale')
ax.scatter(perm_df[missing_perms &
(perm_df['Rock facies'] == 1)]['Porosity, fraction'],
perm_df[missing_perms &
(perm_df['Rock facies'] == 1)]['Permeability, mD'],
color='b', label='Sandy shale (Imputed)')
ax.scatter(perm_df[~missing_perms &
(perm_df['Rock facies'] == 2)]['Porosity, fraction'],
perm_df[~missing_perms &
(perm_df['Rock facies'] == 2)]['Permeability, mD'],
color='g', facecolors='none', label='Shale')
ax.scatter(perm_df[missing_perms &
(perm_df['Rock facies'] == 2)]['Porosity, fraction'],
perm_df[missing_perms &
(perm_df['Rock facies'] == 2)]['Permeability, mD'],
color='g', label='Shale (Imputed)')
ax.scatter(perm_df[~missing_perms &
(perm_df['Rock facies'] == 3)]['Porosity, fraction'],
perm_df[~missing_perms &
(perm_df['Rock facies'] == 3)]['Permeability, mD'],
color='k', facecolors='none', label='Shaly sand')
ax.scatter(perm_df[missing_perms &
(perm_df['Rock facies'] == 3)]['Porosity, fraction'],
perm_df[missing_perms &
(perm_df['Rock facies'] == 3)]['Permeability, mD'],
color='k', label='Shaly sand (Imputed)')
ax.set_xlabel('Porosity, fraction')
ax.set_ylabel('Permeability, mD')
ax.legend(bbox_to_anchor=(1.5, 1), loc='upper right', ncol=1);
```

The `'KC permeability, mD'`

is redundant now, so we'll drop it from the dataframe.

```
perm_df.drop('KC permeability, mD', axis=1, inplace=True)
```

### Feature engineering¶

Since the fault runs right through both the produced wells and the wells we are trying to predict production for, let's engineer a few features related to the fault. First we'll engineer a feature we'll call `'Distance to fault'`

which computes the perpendicular distance to the fault. Also, we don't know if the fault compartmentalizes the reservoir in any way, so we'll also create a feature called `'left/right fault'`

to indicate which side of the fault a well lies.

```
perm_df['Distance to fault'] = np.sqrt((perm_df['X, m'] + perm_df['Y, m'] - 11750) ** 2) / np.sqrt(2)
perm_df['left/right fault'] = np.sign(perm_df['X, m'] * (-11750) - (perm_df['Y, m'] - 11750) * (11750))
```

Because all of the prediction wells are "near" the fault, we'll create another boolean feature that gives some importance to wells near the fault. We'll define "near" as any well that is closer than the mean preproduction (i.e. prediction) well distance from the fault.

```
preproduction_wells = perm_df.index.isin(preproduction_wells_df
.index.to_numpy())
max_distance = perm_df.loc[preproduction_wells, 'Distance to fault'].mean()
perm_df['close to fault'] = np.where(
perm_df['Distance to fault'] < max_distance, 1, 0)
```

Give that we anticipate wells that are "similar" to have "similar" production, we'll use `KMeans`

clustering to find similar wells in our dataframe

```
scalers = [MaxAbsScaler(), MinMaxScaler(), StandardScaler(), RobustScaler()]
pipe = Pipeline([
('scaler', StandardScaler()),
('cluster', KMeans())
])
params = {
'scaler': scalers,
'cluster__n_clusters': [10, 12, 14],
}
gcv = GridSearchCV(pipe, params, cv=KFold(random_state=3, shuffle=True))
gcv.fit(perm_df.loc[~preproduction_wells])
gcv.best_params_
```

And use the model to predict which cluster the preproduction wells would fall into.

```
perm_df['Cluster'] = gcv.predict(perm_df)
```

Now we'll read in the production histories and merge them with our working dataframe.

```
production_history_df = pd.read_csv('production_history.csv',
index_col='Well_ID')
production_df = perm_df.merge(production_history_df,
how='left',
left_index=True,
right_index=True)
production_df.head()
```

Now we'll compute the average 1, 2, and 3 year productions for each cluster and assign a new feature with this average value as the expected average production at each well location.

```
avg_production_df = (production_df
.groupby('Cluster')
.mean()
.loc[:, 'Cumulative oil production (1 yr), MSTB' \
:'Cumulative Water production (3 yr), MSTB']
.reset_index()
.rename(columns={'Cumulative oil production (1 yr), ' \
'MSTB':'Avg. oil prod (1 yr)',
'Cumulative oil production (2 yr), ' \
'MSTB':'Avg. oil prod (2 yr)',
'Cumulative oil production (3 yr), ' \
'MSTB':'Avg. oil prod (3 yr)',
'Cumulative Water production (1 yr), ' \
'MSTB':'Avg. water prod (1 yr)',
'Cumulative Water production (2 yr), ' \
'MSTB':'Avg. water prod (2 yr)',
'Cumulative Water production (3 yr), ' \
'MSTB':'Avg. water prod (3 yr)'})
)
```

Merging these averaged production values into our working dataframe, we have the final dataframe which we'll use to make predictions with.

```
final_df = (production_df
.reset_index()
.merge(avg_production_df,
how='left',
on='Cluster')
.set_index(['Well_ID'])
)
final_df.head()
```

### Predictions¶

In order to hyperparameter tune our forthcoming models, we're going to use the "goodness" measure of Deutsch, 1996, i.e.

$$ \mbox{goodness} = 1 - \int_0^1 \left(2 a(p) - 2 \right) \left(\overline{\xi(p)} - p \right) \mathrm{d}p $$

in addition to the `mean_absolute_percentage_error`

from scikit-learn to score our models. The idea will be to use the `BaggingRegressor`

to create an ensemble of models that we average over to create our "best estimate" as well using each individual estimator as a realization for our uncertainty model. We'll score our model over each fold in the cross-validation using `GridSearchCV`

. Credit goes to Honggeun Jo for creating the original version of this function which I only slightly modified here.

```
def goodness_score(y_true, y_realizations):
goodness_score_array = []
# Define upper/lower boundary of "within-percentile" ranges
list_percentile_lower = 50 - 5 * np.arange(0, 11, dtype=np.int32)
list_percentile_upper = 50 + 5 * np.arange(0, 11, dtype=np.int32)
for i in range(11): # 0%, 10%, 20%, 30%, ... 100% percentiles ranges
num_within = 0 # Counts for predictions within the range
for (j, realization) in enumerate(y_realizations.T):
min_ = np.percentile(realization, list_percentile_lower[i])
max_ = np.percentile(realization, list_percentile_upper[i])
if y_true[j] > min_ and y_true[j] < max_:
num_within += 1
goodness_score_array.append(num_within)
goodness_score_upNdown = (np.array(goodness_score_array) -
np.arange(0,11, dtype=np.double))
a_interval_index = [1 if goodness_score_array[i+1] >= i+1
else 0 for i in range(10)]
goodness_score_ = 1
for i in range(10):
if a_interval_index[i] == 1:
goodness_score_ -= 0.5 * goodness_score_upNdown[i+1] / 45
else:
goodness_score_ -= -goodness_score_upNdown[i+1] / 55
return np.abs(goodness_score_)
def scorer(estimator, X, y):
mape = 1 - mean_absolute_percentage_error(y[:,-1], estimator.predict(X)[:, -1])
pipe = estimator[:-1]
Xt = pipe.transform(X)
realizations = []
features = estimator[-1].estimators_features_
for (i, e) in enumerate(estimator[-1].estimators_):
realizations.append(e.predict(Xt[:, features[i]])[:, -1])
realizations = np.array(realizations)
goodness = goodness_score(y[:,-1], realizations)
return 0.5 * mape + 0.5 * goodness
```

Separating our predictor and response features labels.

```
response_features = ['Cumulative Water production (1 yr), MSTB',
'Cumulative Water production (2 yr), MSTB',
'Cumulative Water production (3 yr), MSTB',
'Cumulative oil production (1 yr), MSTB',
'Cumulative oil production (2 yr), MSTB',
'Cumulative oil production (3 yr), MSTB']
predictor_features = final_df.columns[~final_df.columns
.isin(response_features)].to_numpy()
```

Since there are several data samples from each well, but we are only asked to report our prediction for the entire well, there are several options on how to proceed. This simplest is just to average all the samples over each well, we'll use this approach in the interest of time.

```
avg_final_df = final_df.groupby(final_df.index).mean()
```

Getting a boolean indexer for our training wells (the wells that have been in production).

```
train_wells = ~avg_final_df.index.isin(preproduction_wells_df.index)
```

Now we'll set up a pipeline and pass that to `GridSearchCV`

for hyperparameter tuning. I iterated this a few times to narrow down the final set of parameters to search, so that the final notebook/solution runs in a timely manner. Here I'm using 7 folds in the k-fold cross-validation because that leaves about 10 wells in the test set, similar to the number we need to predict.

```
scalers = [MaxAbsScaler(), RobustScaler()]
pipe = Pipeline([
('scaler', StandardScaler()),
('pca', PCA()),
('poly', PolynomialFeatures()),
('bag', BaggingRegressor(base_estimator=LinearRegression(),
n_estimators=100, n_jobs=-1, random_state=7)),
])
params = [{
'scaler': scalers,
'pca__n_components': [8, 9],
'poly__degree': [1, 2],
'bag__max_samples': [0.75, 1.0],
'bag__max_features': [0.75, 1.0],
'bag__bootstrap': [True, False],
'bag__base_estimator': [LinearRegression()]
}]
gcv = GridSearchCV(pipe, params, scoring=scorer,
cv=KFold(n_splits=7, shuffle=True, random_state=17)
)
gcv.fit(avg_final_df.loc[train_wells, predictor_features].to_numpy(),
avg_final_df.loc[train_wells, response_features].to_numpy())
print(gcv.best_score_)
gcv.best_params_
```

The "best score" and final set of hyperparameters are shown above. Now we'll write our best estimate and realizations to the solution file.

```
solution_df = pd.read_csv('solution.csv', index_col='Well_ID')
pipe = gcv.best_estimator_
#best estimate
solution_df['Prediction, MSTB'] = \
pipe.predict(avg_final_df.loc[~train_wells, predictor_features])[:, -1]
#realizations
Xt = pipe[:-1].transform(avg_final_df.loc[~train_wells, predictor_features])
features = pipe[-1].estimators_features_
for (i, estimator) in enumerate(pipe[-1].estimators_):
pt = estimator.predict(Xt[:, features[i]])[:, -1]
solution_df[f'R{i+1}, MSTB'] = pt
#write to file
solution_df.to_csv('solution.csv')
```

Submitting the solution file for scoring, we can generate the following plots. First we have the accuracy plot which compares the predictions to the actual 3 yr. production for the 10 wells.

```
from score import (create_accuracy_plot_and_return_mse,
create_realizations_plots,
create_goodness_plot_and_return_goodness_score)
truth_data = np.load('True_for_predrill_3yr.npy')
mse = create_accuracy_plot_and_return_mse(solution_df, truth_data);
print(f'MSE: {mse}')
```

Below the uncertainty models are shown along with the true values.

```
create_realizations_plots(solution_df, truth_data)
```

And finally the goodness plot. The goodness measure we used in our `scorer`

function above is related to the integral of the black dashed line with respect to the diagonal red line where the area below above the red line is weighted twice the area below.

```
goodness = create_goodness_plot_and_return_goodness_score(solution_df,
truth_data);
print(f'Goodness: {goodness}')
```

Finally, my rank in the competition among the other teams (note, this is only for the objective part of the scoring, in the actual Hackathon, there we had additional columns for code quality and presentations, therefore the final rankings of the other teams in this table does not reflect the actual final outcome of the competition). Not bad for a days work!

```
from IPython.display import Image
Image('results_table.png', width=500)
```

The final message here is that domain expertise, which leads to good imputation and feature engineering strategies is far more useful than fancy machine learning models. Additionally, understanding exactly the quantities of interest you are looking for so that you evaluate (i.e. score) your models well. In the end, I "won" the competition with a linear regression model.

# JAX implementation of FEA and efficient inverse problem solving with neural networks

If you haven't heard by now JAX is getting a lot of attention online as a "NumPy on steroids". At it's core, it can be thought of as a drop-in replacement for NumPy where the array calculations can be accelerated on GPUs or TPUs when available. This alone makes it worth looking at, especially if you have a lot of NumPy code that you would like to potentially speed up with GPU acceleration. Currently, most of the NumPy API is implemented in one-to-one correspondence, as well of some of the most used functions in SciPy.

The accelerated NumPy is just the beginning of the utility of JAX. All of the JAX NumPy data structures can be used in combination with most pure Python code to create functions which can be automatically differentiated. This includes computing the gradient of scalar functions, as well as Jacobian matrices of vector functions. These operations can be composed to compute gradients-of-gradients, etc. More information of the automatic differentiation capabilities are documented here.

Additionally, there is a built-in just-in-time compiler for compiling functions to be executed on CPUs/GPUs/TPUs, and support for automatic vectorization, i.e. functions written for scalar arguments can be easily mapped across arrays. These can be used with the automatic differentiation functions previously mentioned.

Finally, there is a very thin neural network library associated with JAX called stax. Other, more fully-featured libraries like Haiku, Flax, or Trax are under development on top of JAX technologies.

In what follows, I'll highlight most of these features of JAX by implementing a finite element analysis (FEA) model and then using the finite element residual as part of the objective function when training a neural network in an inverse solution to a potentially unknown constitutive model.

### Problem definition¶

As a model problem, we'll start with the one-dimensional pressure diffusivity equation which governs single phase fluid flow in a porous media with fluid density $\rho$ and small compressibility $c$.

$$ \rho c \frac{\partial p}{\partial t} = \frac{\partial}{\partial x}\left(\frac{\kappa}{\mu} \frac{\partial p}{\partial x}\right) $$

Assuming steady state, multiplying by a test function $\delta p$ on the left and integrating by parts over the domain $(0, L)$ we have

\begin{align} 0 =& \int_0^L \lambda(x) \frac{\partial \left(\delta p\right)}{\partial x} \frac{\partial p}{\partial x} \textrm{d}x - \left[ \lambda(x) \delta p \frac{\partial p}{\partial x} \right]_0^L \\ =& \int_0^L \lambda(x) \frac{\partial \left(\delta p\right)}{\partial x} \frac{\partial p}{\partial x} \textrm{d}x - \left[ q \right]_0^L \end{align}

where

$$ \lambda(x) = \frac{\kappa}{\mu}, $$

$\kappa$ is the porous medium's permeability and $\mu$ is the fluid viscosity. $\lambda$ is known as the *mobility* and is assumed to be spatially varying.

Using a Galerkin appoximation, i.e. $p = N_J p_J$ and $\delta p = N_I$ for $I,J = 1, 2, \ldots$ basis functions and splitting the domain into $n$ intervals, we now have

\begin{align} 0 =& \sum_{k=0}^n p_J \left(\int_{x_k}^{x_{k+1}} \lambda(x) \frac{\partial N_I}{\partial x} \frac{\partial N_J}{\partial x} \textrm{d}x - \left[ q \right]_{x_k}^{x_{k+1}} \right) \end{align}

where summation over the $J$ basis functions are implied for those that have support on the $I^{th}$ node. The right-hand side above is our residual, i.e. $\vec{R}$

\begin{align} R_I \equiv \sum_{k=0}^n p_J \left(\int_{x_k}^{x_{k+1}} \lambda(x) \frac{\partial N_I}{\partial x} \frac{\partial N_J}{\partial x} \textrm{d}x - \left[ q \right]_{x_k}^{x_{k+1}} \right) \end{align}

below, we'll integrate this residual vector using Gauss integration and solve for the unknown nodal pressures $p_J$. Without loss of generality we'll only consider Dirchelet boundary conditions, i.e. $q(x) = 0$.

While this model problem is linear, we'll implement the FEA model to use the residual form of the equations, and solve for the unknowns using a nonlinear Newton-Raphson solver where the Jacobian matrix at each iteration is computed via automatic-differentiation with JAX. All of the computations are written in a way that they could be accelerated on GPUs/TPUs and are just-in-time compiled.

### FEA implementation¶

Below are the imports we need, note that we explicitly enable 64-bit floating point numbers for JAX as 32-bit is the default.

```
from functools import partial, partialmethod
import numpy as np
import matplotlib.pyplot as plt
from jax import jit, vmap, value_and_grad, random, flatten_util
import jax.numpy as jnp
import jax.scipy.optimize
import jax.ops
from jax.config import config
config.update("jax_enable_x64", True)
```

```
class FEAProblem():
def __init__(self, nodes, mobility=lambda x: 1, left_bc=1.0, right_bc=-1.0):
"""FEAProblem class
Args:
nodes (array-like): nodal spatial locations of unknowns
mobility (callable): function defining the mobility function
in space
left_bc (real): Dirchelet boundary on the left of the domain
right_bc (real): Dirchelet boundary on the right of the domain
"""
self.nodes = jnp.array(nodes, dtype='float64')
self.connect = jnp.array([ jnp.arange(i,i+2)
for i in range(self.nodes.shape[0] - 1)])
self.__lbc = left_bc
self.__rbc = right_bc
self.mobility = mobility
return
def setup_element_matrices(self):
"""Integrates the element stiffness matrices for linear basis functions
with 2 points Guass integration
"""
nodes = self.nodes
#We will use Gauss integration at the following points
t1 = np.sqrt(1.0 / 3.0)
t2 = -np.sqrt(1.0 / 3.0)
# Because the points above are defined on the domain -1 < x < 1 and our
# elements are defined on arbitrary domains, we can use a change of
# variables to rescale the integration bounds
ξ1 = ((nodes[1:] - nodes[:-1]) * t1 + nodes[1:] + nodes[:-1]) / 2.0
ξ2 = ((nodes[1:] - nodes[:-1]) * t2 + nodes[1:] + nodes[:-1]) / 2.0
# Compute the function 𝜆, at the integration points
𝜆ξ1 = self.mobility(ξ1)
𝜆ξ2 = self.mobility(ξ2)
# Since the derivatives of the shape functions are not dependent on ξ,
# we can # create an array containing all of the element stiffness
# matrices at once
dNdξ = jnp.array([1 / (nodes[:-1] - nodes[1:]),
1 / (nodes[1:] - nodes[:-1])])
# Computes the matrix dN_i * dN_j for every element (each will be
# identical for equally spaced nodes)
ke_temp = jnp.einsum('i...,j...', dNdξ, dNdξ)
# Now we perform the Gauss integration computing the integrand for each
# element
temp = (nodes[1:] - nodes[:-1]) / 2 * (𝜆ξ1 + 𝜆ξ2)
# We have to add two axis so the broadcasting is performed correctly,
# the result here is the fully integrated element matrix array
self.ke = temp[:, None, None] * ke_temp
return
@partial(jit, static_argnums=(0,))
def residual(self, p):
"""Compute the FE residual vector
Args:
p (array-like): possible solution to residual equations
Returns:
(array-like): the residual vector
"""
nodes = self.nodes
connect = self.connect
ke = self.ke
# Initialize the residual
res = jnp.zeros_like(nodes)
# Compute the residual, i.e. sum all of the element
# matrices with the nodal values, p
res_temp = jnp.einsum('...ij, ...j', ke, p[connect]).flatten()
# This is the JAX equivalent of
# res[-1:1] = ..., i.e. in place assignment
res = res.at[1:-1].set(res_temp[1:-1].reshape(-1, 2).sum(axis=1))
# Fix the residual values at the boundaries
res = res.at[0].set(p[0] - self.__lbc)
res = res.at[-1].set(p[-1] - self.__rbc)
return res
@partial(jit, static_argnums=(0,))
def loss(self, p):
"""Compute the loss function, i.e. the
l2 norm of the residual vector.
Args:
p (array-like): possible solution to the residual equations
Returns:
(real): discrete 2-norm of residual vector
"""
return jnp.linalg.norm(self.residual(p))
@partial(jit, static_argnums=(0,))
def newton_step(self, p):
"""Compute one step of a Newton-Raphson iteration, uses
JAX to compute the exact Jacobian via automatic differentiation
Args:
p (array-like): possible solution to the residual equations
Returns:
p (array-like): updated solution to the residual equations after
a single Newton-Raphson step
"""
# The jacobian
K = jax.jacfwd(self.residual)(p)
# Compute the update direction
Δp, _ = jax.scipy.sparse.linalg.gmres(lambda x: -K @ x,
self.residual(p))
# Update the unknowns and return
p += Δp
return p
@partial(jit, static_argnums=(0,))
def solve(self, tolerance=1.0e-4):
""" Solve via Newton-Raphson iteration
Args:
tolerance (float): the tolerence at which the Newton-Raphson
iteration stops
Returns:
p (array-like): the converged solution to the residual equations
"""
# Integrate the shape functions over each element
self.setup_element_matrices()
# Initial guess is linear between the boundary conditions
p = jnp.linspace(self.__lbc, self.__rbc, num=self.nodes.shape[0])
# The Newton-Raphson loop, can be compiled and automatically
# differentiated
p = jax.lax.while_loop(lambda x: self.loss(x) > tolerance,
self.newton_step, p)
# Return the solution
return p
```

### Generate reference data¶

Below we'll solve the forward problem via FEA using the implementation above to verify things are working correctly as well as generate some reference data that we'll use in the inverse problem in the sequal. Here, the mobility function is

$$ \lambda(x) = x^3 + 0.001 $$

```
nodes = jnp.linspace(0, 1, num=20)
problem1 = FEAProblem(nodes, mobility = lambda x: x ** 3 + 0.001,
left_bc=15, right_bc=5)
p1 = problem1.solve(tolerance=1.0e-4)
```

```
fig, ax1 = plt.subplots()
ax2 = ax1.twinx()
ax1.plot(nodes, p1, 'k')
ax2.plot(nodes, nodes ** 3 + 0.001, 'r-.')
ax1.set_xlabel(r'$x$')
ax1.set_ylabel(r'$p(x)$', color='k');
ax2.set_ylabel(r'$\lambda(x)$', color='r');
ax2.tick_params(axis='y', colors='r')
ax2.spines['right'].set_color('r')
```

### Inverse problem implementation¶

Here we will write our inverse problem solver. We will inherit from the `FEAProblem`

class above so we can reuse some of the functions already defined.

Our objective function here will be the $l_2$-norm of the finite element residual when the "known data" is supplied as training data. Because the problem we are solving is a steady-state problem, we'll need to provide the endpoints of constitutive model to the objective function, otherwise there are infinite valid solutions to learning the constitutive model that only differ by a constant. If we extended this technique to time-dependent problems, I believe the need to provide the boundary constraints can be avoided.

We'll use a few functions from the `jax.experimental.stax`

module, just to make the neural network construction easier. Our minimizer here will use the second-order `"BFGS"`

method from `jax.scipy.optimize.minimize`

.

Here we assume our data is supplied at the nodes of the FE model, but this restriction could be easily generalized by evaluating the residuals at any given spatial location via the FE shape functions.

```
from jax.experimental import stax
from jax.experimental.stax import Dense, Tanh
from jax.nn.initializers import glorot_normal, normal
```

```
class InverseProblem(FEAProblem):
def __init__(self, nodes, mobility = stax.serial(Dense(1)),
mobility_left_bc=0.001, mobility_right_bc=1.001):
"""Class to solve for (i.e. learn) the constitutive model of
the steady-state pressure diffusivity equation using a
neural network (nn).
Args:
nodes (array-like): the spatial locations corresponding to the
data points
mobility (stax.nn): a stax neural network definition
mobility_left_bc (real): the left boundary value of the
mobility function
mobility_right_bc (real): the right boundary value of the
mobility function
"""
super().__init__(nodes, mobility = mobility)
self.__mlbc = mobility_left_bc
self.__mrbc = mobility_right_bc
# unpack the functions to initialize and apply the network parameters
net_init, self.net_apply = self.mobility
# initialize the network parameters with random numbers
seed = random.PRNGKey(10)
_, net_params = net_init(seed, (-1, 1))
# flatten the nn parameters into an single array
self.initial_net_params_flat, self.unravel_pytree = flatten_util.ravel_pytree(net_params)
@partial(jit, static_argnums=(0,))
def residual(self, u, data):
"""Compute the FE residual vector
Args:
u (array-like): the unknown parameters of the nn
data (array-like): the user supplied data (i.e. pressures)
at the nodes
Returns:
(array-like): residual vector
"""
# Put the parameters back into the stax nn data structure
net_params = self.unravel_pytree(u)
# Set the mobility function to be the current nn
self.mobility = lambda x: self._predict(x, net_params)
mobility = self.mobility(self.nodes)
# Integrate the element shape functions with the current
# mobility function
self.setup_element_matrices()
# Return the FE residual vector with the supplied data
return super().residual(data)
@partial(jit, static_argnums=(0,))
def loss(self, u, data):
"""Compute the squared error of the FE residual vector
along with the mobility at the end points
Args:
u (array-like): the unknown parameters of the nn
data (array-like): the user supplied data (i.e. pressures) at
the nodes
Returns:
(real): squared error of the FE residual and mobility function
end points
"""
# Compute FE residual with supplied data
residual = self.residual(u, data)
# Put the unknowns back into the stax nn data structure
net_params = self.unravel_pytree(u)
# Use the nn parameters to compute the mobility at the nodes
mobility = self._predict(nodes, net_params)
# Compute squared error norm of mobility end points
mobility_squared_error = ((mobility[0] - self.__mlbc) ** 2 +
(mobility[-1] - self.__mrbc) ** 2)
# Return FE + mobility squared error
return jnp.linalg.norm(residual) ** 2 + mobility_squared_error
def fit(self, data):
"""Fit the parameters of the neurual network representing the mobility
function
Args:
data (array-like): known pressure data at the nodal locations
"""
self.__lbc = data[0]
self.__rbc = data[-1]
u0 = self.initial_net_params_flat
# minimize the loss function w.r.t. the uknown parameters of the nn
u = jit(lambda u, data: jax.scipy.optimize.minimize(self.loss, u,
args=(data,),
method='BFGS').x)(u0, data)
# assign the final nn params, such that we can use if for predicting
self.net_params = self.unravel_pytree(u)
return
def _predict(self, x, params):
"""Used internal to the class only - evaluates the nn function
at x with the given parameters
"""
return vmap(partial(self.net_apply, params))(x).flatten()
def predict(self, x):
"""Evaluates the nn function at x with the fit parameters
Args:
x (array-like): spatial locations to evalute the neural network
"""
return self._predict(x, self.net_params)
```

### Solve inverse problem using NN¶

Below we'll test out our inverse problem solver using the data generated earlier from the forward finite element solution. First we define our neural network architecture. This is a fairly simple function, so we don't need a large and/or complex neural network. Here we have an input layer with only 4 nodes and a $\tanh$ activation function feeding to a single node output. More complicated architectures also work, yielding the same result at more computational cost.

We also need to define the layer `Dense64`

which is the same as `stax.Dense`

, but initialized to use 64-bit floats to be consistant with our data structures in the FEA residual calculation.

```
Dense64 = lambda x: Dense(x, W_init=glorot_normal(dtype='float64'),
b_init=normal(dtype='float64'))
nn = stax.serial(
Dense64(4), Tanh,
Dense64(1)
)
```

Now we instantiate the model and solve the inverse problem, i.e. train the network. We do have to supply the endpoints of the constitutive model. Given the problem is parabolic, there are infinite solutions to the inverse problem (they all have the same shape, but differ by a constant scale factor). We could remove this restriction by considering a time-dependent problem and supplying the time depended training data which we'll leave for future work.

```
iproblem1 = InverseProblem(nodes, mobility = nn,
mobility_left_bc = 0.001, mobility_right_bc = 1.001)
iproblem1.fit(p1)
```

Plotting the neural network function over the range of the domain and comparing with the reference, we can see that the inverse solver has "learned" the mobility function well.

```
x = nodes
fig, ax = plt.subplots()
ax.plot(x, x ** 3 + 0.001, 'k-')
ax.plot(x, iproblem1.predict(x), 'r-.')
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$\lambda(x)$');
ax.legend([r'$\lambda(x) = x^3 + 0.001$', r'$\lambda(x) = \mathcal{NN}(x)$']);
```

### Use the trained NN in the forward problem (verification)¶

Just to verify, we'll use our neural network as the mobility function in our forward finite element solver to demonstrate the resulting pressures are also accurate.

```
vproblem1 = FEAProblem(nodes, mobility = iproblem1.predict,
left_bc=15, right_bc=5)
v1 = vproblem1.solve()
fig, ax = plt.subplots()
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$p(x)$');
ax.plot(nodes, p1, 'k-');
ax.plot(nodes, v1, 'r-.');
ax.legend([r'$\lambda(x) = x^3 + 0.001$', r'$\lambda(x) = \mathcal{NN}(x)$']);
```

### Validate for different BCs¶

A major advantage of this approach over say, physics-informed neural networks, is that we have only "learned" the constitutive model, i.e. the mobility function, not the solution of the partial differential equation with the supplied boundary conditions. Instead, we rely on our finite element implementation to compute the solution. Which means we can now use our "learned" constitutive model to solve problems with different boundary conditions accurately.

```
problem2 = FEAProblem(nodes, mobility = lambda x: x ** 3 + 0.001,
left_bc=5, right_bc=20)
p2 = problem2.solve(tolerance
```