data science
for (physical) scientists 14
dr.federica bianco | fbb.space | fedhere | fedhere
PINNs and Generative AI
this slide deck:
Deep Learning
0
recap
x1
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b1
b2
b3
b
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multilayer perceptron
w: weight
sets the sensitivity of a neuron
b: bias:
up-down weights a neuron
multilayer perceptron
output
layer of perceptrons
w: weight
sets the sensitivity of a neuron
b: bias:
up-down weights a neuron
f: activation function:
turns neurons on-off
multilayer perceptron
w: weight
sets the sensitivity of a neuron
b: bias:
up-down weights a neuron
f: activation function:
turns neurons on-off
layer connectivity
output
input layer
hidden layer
output layer
Fully connected: all nodes go to all nodes of the next layer.
output
input layer
hidden layer
output layer
Sparcely connected: all nodes go to all nodes of the next layer.
layer connectivity
output
input layer
hidden layer
output layer
Sparcely connected: all nodes go to all nodes of the next layer.
The last layer is always connected
layer connectivity
how does it relate to matrix multiplication
each layer is a matrix
Except this is a very misleading representation
there are no biases or activation functions
each layer should be a different shape
1x3
3x5
5x2
=
2x1
what we are doing is just a series of matrix multiplictions.
DeepNeuralNetwork
what we are doing is exactly a series of matrix multiplictions.
3x5
5x2
2x1
=
DeepNeuralNetwork
what we are doing is exactly a series of matrix multiplictions.
3x5
5x2
2x1
=
DeepNeuralNetwork
what we are doing is exactly a series of matrix multiplictions.
3x5
5x2
2x1
=
DeepNeuralNetwork
what we are doing is exactly a series of matrix multiplictions.
DeepNeuralNetwork
The purpose is to approximate a function φ
y = φ(x)
which (in general) is not linear with linear operations
DeepNeuralNetwork
The purpose is to approximate a function φ
y = φ(x)
which (in general) is not linear with linear operations
output
input layer
hidden layer
output layer
hidden layer
32 parameters and
?? hyperparameters
activation functions -
loss function - 1
optimization method - 1
architecture - M
how many hyperparameters?
Parameters and hyperparameters
Training models with this many parameters requires a lot of care:
. defining the metric
. optimization schemes
. training/validation/testing sets
But just like our simple linear regression case, the fact that small changes in the parameters leads to small changes in the output for the right activation functions.
define a cost function, e.g.
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proper care of your DNN
0
NN are a vast topics and we only have 2 weeks!
Some FREE references!
michael nielsen
better pedagogical approach, more basic, more clear
ian goodfellow
mathematical approach, more advanced, unfinished
michael nielsen
better pedagogical approach, more basic, more clear
Lots of parameters and lots of hyperparameters! What to choose?
cheatsheet
-
architecture - wide networks tend to overfit, deep networks are hard to train
- number of epochs - the sweet spot is when learning slows down, but before you start overfitting... it may take DAYS! jumps may indicate bad initial choices (like in all gradient descent)
- loss function - needs to be appropriate to the task, e.g. classification vs regression
-
activation functions - needs to be consistent with the loss function
- optimization scheme - needs to be appropriate to the task and data
- learning rate in optimization - balance speed and accuracy
- batch size - smaller batch size is faster but leads to overtraining
An article that compars various DNNs
An article that compars various DNNs
accuracy comparison
An article that compars various DNNs
accuracy comparison
An article that compars various DNNs
batch size
Lots of parameters and lots of hyperparameters! What to choose?
cheatsheet
- architecture - wide networks tend to overfit, deep networks are hard to train
-
number of epochs - the sweet spot is when learning slows down, but before you start overfitting... it may take DAYS! jumps may indicate bad initial choices
-
loss function - needs to be appropriate to the task, e.g. classification vs regression
-
activation functions - needs to be consistent with the loss function
- optimization scheme - needs to be appropriate to the task and data
- learning rate in optimization - balance speed and accuracy
- batch size - smaller batch size is faster but leads to overtraining
What should I choose for the loss function and how does that relate to the activation functiom and optimization?
Lots of parameters and lots of hyperparameters! What to choose?
Lots of parameters and lots of hyperparameters! What to choose?
cheatsheet
always check your loss function! it should go down smoothly and flatten out at the end of the training.
not flat? you are still learning!
too flat? you are overfitting...
loss (gallery of horrors)
jumps are not unlikely (and not necessarily a problem) if your activations are discontinuous (e.g. relu)
when you use validation you are introducing regularizations (e.g. dropout) so the loss can be smaller than for the training set
loss and learning rate (not that the appropriate learning rate depends on the chosen optimization scheme!)
Building a DNN
with keras and tensorflow
autoencoder for image recontstruction
What should I choose for the loss function and how does that relate to the activation functiom and optimization?
| loss | good for | activation last layer | size last layer |
|---|---|---|---|
| mean_squared_error | regression | linear | one node |
| mean_absolute_error | regression | linear | one node |
| mean_squared_logarithmit_error | regression | linear | one node |
| binary_crossentropy | binary classification | sigmoid | one node |
| categorical_crossentropy | multiclass classification | sigmoid | N nodes |
| Kullback_Divergence | multiclass classification, probabilistic inerpretation | sigmoid | N nodes |
On the interpretability of DNNs
Physics Informed NN
PiNN
1
Application regime:
PiNN
-infinity - 1950's
theory driven: little data, mostly theory, falsifiability and all that...
Application regime:
PiNN
-infinity - 1950's
theory driven: little data, mostly theory, falsifiability and all that...
-1980's - today
data driven: lots of data, drop theory and use associations, black-box modles
Application regime:
PiNN
-infinity - 1950's
theory driven: little data, mostly theory, falsifiability and all that...
-1980's - today
data driven: lots of data, drop theory and use associations, black-box modles
lots of data yet not enough for entirely automated decision making
complex theory that cannot be solved analytically
combine it with some theory
PiNN
Non Linear PDEs are hard to solve!
PiNN
- Raissi et al. Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. arXiv 1711.10561
- Raissi et al. Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations. arXiv 1711.10566
- Raissi et al. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comp. Phys. 378 pp. 686-707 DOI: 10.1016/j.jcp.2018.10.045
Non Linear PDEs are hard to solve!
Existence and uniqueness of solutions
A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions.
E.g.: Open problem of existence (and smoothness) of solutions to the Navier–Stokes equations is one of the seven Millennium Prize problems in mathematics.
PiNN
- Raissi et al. Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. arXiv 1711.10561
- Raissi et al. Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations. arXiv 1711.10566
- Raissi et al. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comp. Phys. 378 pp. 686-707 DOI: 10.1016/j.jcp.2018.10.045
Non Linear PDEs are hard to solve!
Linear approximation
The solutions in a neighborhood of a known solution can sometimes be studied by linearizing the PDE around the solution. This corresponds to studying the tangent space of a point of the moduli space of all solutions.
PiNN
- Raissi et al. Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. arXiv 1711.10561
- Raissi et al. Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations. arXiv 1711.10566
- Raissi et al. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comp. Phys. 378 pp. 686-707 DOI: 10.1016/j.jcp.2018.10.045
Non Linear PDEs are hard to solve!
Exact solutions
It is often possible to write down some special solutions explicitly in terms of elementary functions (though it is rarely possible to describe all solutions like this). One way of finding such explicit solutions is to reduce the equations to equations of lower dimension, preferably ordinary differential equations, which can often be solved exactly.
PiNN
- Raissi et al. Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. arXiv 1711.10561
- Raissi et al. Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations. arXiv 1711.10566
- Raissi et al. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comp. Phys. 378 pp. 686-707 DOI: 10.1016/j.jcp.2018.10.045
Non Linear PDEs are hard to solve!
Numerical solutions
Numerical solution on a computer is almost the only method that can be used for getting information about arbitrary systems of PDEs. There has been a lot of work done, but a lot of work still remains on solving certain systems numerically, especially for the Navier–Stokes and other equations related to weather prediction.
PiNN
Non Linear PDEs are hard to solve!
PiNN
Non Linear PDEs are hard to solve!
Domain
Boundary Conditions
PiNN
Non Linear PDEs are hard to solve!
How to solve analytically
https://www.youtube.com/watch?v=5ZrwxQr6aV4
PiNN
Non Linear PDEs are hard to solve!
- Provide training points at the boundary with calculated solution (trivial cause we have boundary conditions)
input layer
PiNN
Non Linear PDEs are hard to solve!
- Provide training points at the boundary with calculated solution (trivial cause we have boundary conditions)
- Provide the physical constraint: make sure the solution satisfies the PDE
???
PiNN
Non Linear PDEs are hard to solve!
- Provide training points at the boundary with calculated solution (trivial cause we have boundary conditions)
- Provide the physical constraint: make sure the solution satisfies the PDE
via a modified loss function that includes residuals of the prediction and residual of the PDE
PiNN
Non Linear PDEs are hard to solve!
- Provide training points at the boundary with calculated solution (trivial cause we have boundary conditions)
- Provide the physical constraint: make sure the solution satisfies the PDE
via a modified loss function that includes residuals of the prediction and residual of the PDE
PiNN
PiNN
Non Linear PDEs are hard to solve!
- Provide training points at the boundary with calculated solution (trivial cause we have boundary conditions)
- Provide the physical constraint: make sure the solution satisfies the PDE
via a modified loss function that includes residuals of the prediction and residual of the PDE
generative AI
2
Applications
-
Image Generation (and 3D Shape Generation)
-
Semantic Image-to-Photo Translation
-
Image Resolution Increase
-
Text-to-Speech Generator
-
Speech-to-Speech Conversion
-
Text Generation (Chat GP3)
-
Music Generation
-
Image-to-Image Conversion
GANs
GANs
VAE
Diffusion models
VAE
Autoencoders
3
Unsupervised learning with
Neural Networks
What do NN do? approximate complex functions with series of linear functions
.... so if my layers are smaller what I have is a compact representation of the data
Unsupervised learning with
Neural Networks
What do NN do? approximate complex functions with series of linear functions
To do that they extract information from the data
Each layer of the DNN produces a representation of the data a "latent representation" .
The dimensionality of that latent representation is determined by the size of the layer (and its connectivity, but we will ignore this bit for now)
.... so if my layers are smaller what I have is a compact representation of the data
Autoencoder Architecture
Feed Forward DNN:
the size of the input is 5,
the size of the last layer is 2
Autoencoder Architecture
- Encoder: outputs a lower dimensional representation z of the data x (similar to PCA, tSNE...)
- Decoder: Learns how to reconstruct x given z: learns p(x|z)
Autoencoder Architecture
Building a DNN
with keras and tensorflow
Trivial to build, but the devil is in the details!
Building a DNN
with keras and tensorflow
Trivial to build, but the devil is in the details!
from keras.models import Sequential
#can upload pretrained models from keras.models
from keras.layers import Dense, Conv2D, MaxPooling2D
#create model
model = Sequential()
#create the model architecture by adding model layers
model.add(Dense(10, activation='relu', input_shape=(n_cols,)))
model.add(Dense(10, activation='relu'))
model.add(Dense(1))
#need to choose the loss function, metric, optimization scheme
model.compile(optimizer='adam', loss='mean_squared_error')
#need to learn what to look for - always plot the loss function!
model.fit(x_train, y_train, validation_data=(x_test, y_test),
epochs=20, batch_size=100, verbose=1)
#note that the model allows to give a validation test,
#this is for a 3fold cross valiation: train-validate-test
#predict
test_y_predictions = model.predict(validate_X)Building a DNN
with keras and tensorflow
autoencoder for image recontstruction
encoder
This autoencoder model has a 64-neuron bottle neck. This means it will generate a compressed representation of the data out of that layer which is 16-dimensional (the original size is 784 pixels)
Building a DNN
with keras and tensorflow
autoencoder for image recontstruction
This autoencoder model has a 64-neuron bottle neck. This means it will generate a compressed representation of the data out of that layer which is 16-dimensional (the original size is 784 pixels)
Building a DNN
with keras and tensorflow
autoencoder for image recontstruction
decoder
This autoencoder model has a 64-neuron bottle neck. This means it will generate a compressed representation of the data out of that layer which is 16-dimensional (the original size is 784 pixels)
Building a DNN
with keras and tensorflow
autoencoder for image recontstruction
This autoencoder model has a 64-neuron bottle neck. This means it will generate a compressed representation of the data out of that layer which is 16-dimensional (the original size is 784 pixels)
bottle neck
Building a DNN
with keras and tensorflow
autoencoder for image recontstruction
This simple model has 200K parameters!
My original choice is to train it with "adadelta" with a mean squared loss function, all activation functions are relu, appropriate for a linear regression
Building a DNN
with keras and tensorflow
autoencoder for image recontstruction
What should I choose for the loss function and how does that relate to the activation functiom and optimization?
Building a DNN
with keras and tensorflow
autoencoder for image recontstruction
What should I choose for the loss function and how does that relate to the activation functiom and optimization?
| loss | good for | activation last layer | size last layer |
|---|---|---|---|
| mean_squared_error | regression | linear | one node |
| mean_absolute_error | regression | linear | one node |
| mean_squared_logarithmit_error | regression | linear | one node |
| binary_crossentropy | binary classification | sigmoid | one node |
| categorical_crossentropy | multiclass classification | sigmoid | N nodes |
| Kullback_Divergence | multiclass classification, probabilistic inerpretation | sigmoid | N nodes |
autoencoder for image recontstruction
model_digits64.add(Dense(ndim,
activation='linear'))
model_digits64_sig.compile(optimizer="adadelta",
loss="mean_squared_error") model_digits64_sig.add(Dense(ndim,
activation='sigmoid'))
model_digits64_sig.compile(optimizer="adadelta",
loss="mean_squared_error")
model_digits64_sig.add(Dense(ndim,
activation='sigmoid'))
model_digits64_bce.compile(optimizer="adadelta",
loss="binary_crossentropy")
loss function: did not finish learning, it is still decreasing rapidly
The predictions are far too detailed. While the input is not binary, it does not have a lot of details. Maybe approaching it as a binary problem (with a sigmoid and a binary cross entropy loss) will give better results
loss function: also did not finish learning, it is still decreasing rapidly
A sigmoid gives activation gives a much better result!
Binary cross entropy loss function: It is more appriopriate when the output layer is sigmoid
Even better results!
original
predicted
predicted
original
predicted
original
predicted
autoencoder for image recontstruction
A more ambitious model has a 16 neurons bottle neck: we are trying to extract 16 numbers to reconstruct the entire image! its pretty remarcable! those 16 number are extracted features from the data
predicted
original
latent
representation
models are neutral, the bias is in the data (or is it?)
Why does this AI model whitens Obama face?
Simple answer: the data is biased. The algorithm is fed more images of white people
models are neutral, the bias is in the data (or is it?)
Why does this AI model whitens Obama face?
Simple answer: the data is biased. The algorithm is fed more images of white people
But really, would the opposite have been acceptable? The bias is in society
Joy Boulamwini
models are neutral, the bias is in the data (or is it?)
comparing generative AI models
3
see also https://arxiv.org/pdf/2103.04922.pdf
The latent space is assumed to be Gaussian distributed - this causes inaccuracy (blurry) generation
similar to a VAE but with a NN in the middle that approximates the true distribution of the latent space
The latent space is assumed to be Gaussian distributed - this causes inaccuracy (blurry) generation
Normalizing Flows
have two networks trained at the same time that compete again each other in a minimax game.
The generator generates images, starting with pure noise.
The discriminator classifies the image from the generator as Real/Fake
trained not to be fooled by the generator.
generator is trained to make better images
Ian Goodfellow et al., 2014 Generative Adversarial Networks
GANs: Generative Adversarial NN
trained not to be fooled by the generator.
generator is trained to make better images
Minmax Loss Function:
minimize
maximize
GANs: Generative Adversarial NN
trained not to be fooled by the generator.
generator is trained to make better images
Minmax Loss Function:
minimize
maximize
log(D(G(z)))
change introduced to minimize geneerator saturation
GANs: Generative Adversarial NN
DDPM:Denoising Diffusion Probabilistic Model
Ho Jain Abbel 2006
Which generative AI is right for you??
Neural Networks: Transformers
Encoder + Decoder architecture
Attention mechanism
Multithreaded attention
Attention is all you need: transformer model
transformer generalized architecture elements
Attention is all you need
Encoder + Decoder architecture
Encodes the past
Encoder + Decoder architecture
decodes the past and predicts the future
MHA acting on encoder (1)
Attention is all you need (2017)
Attention is all you need (2017)
each attention head learns relationships between elements of the series (i.e. words/punctuation in the sentence)
resources
Neural Network and Deep Learning
an excellent and free book on NN and DL
http://neuralnetworksanddeeplearning.com/index.html
Deep Learning An MIT Press book in preparation
Ian Goodfellow, Yoshua Bengio and Aaron Courville
https://www.deeplearningbook.org/lecture_slides.html
History of NN
https://cs.stanford.edu/people/eroberts/courses/soco/projects/neural-networks/History/history2.html
DNN for time series
RNN
RNN architecture
input layer
output layer
hidden layers
Feed-forward architecture
RNN architecture
output layer
hidden layers
Feed-forward NN architecture
Recurrent NN architecture
input layer
output layer
RNN hidden layers
output layer
hidden layers
input layer
RNN architecture
input layer
output layer
RNN hidden layers
current state
previous state
Remember the state-space problem!
we want process a sequence of vectors x applying a recurrence formula at every time step:
RNN architecture
input layer
output layer
RNN hidden layers
Remember the state-space problem!
we want process a sequence of vectors x applying a recurrence formula at every time step:
current state
previous state
features
(can be time dependent)
function with parameters q
MLTSA:
state space model (from week 4)
A State-space model is a model to derive the value of a time-dependent variable x(t), the state, generated by a noisy Markovian process, from observations of a variable y(t), also subject to noise, linearly related to the target variable
Definition
RNN architecture
input layer
output layer
RNN hidden layers
Simplest possible RNN
Whh
Wxh
Why
RNN architecture
input layer
Alternative graphical representation of RNN
Whh
h(t-1)
h(t)
h(t+1)
h(t+2)
h(t+3)
h(t+4)
y(t)
y(t+1)
y(t+2)
y(t+4)
y(t+3)
y(t+5)
Why
Why
Why
Why
Why
Whh
Whh
Whh
Whh
Whh
Wxh
the weights are the same! always the same Whh and Why
RNN architecture
appllications
image captioning:
one image to a
sequence of worods
sentiment analysis
sequence of words to one sentiment
language translator
sequence of words to sequence of words
online: video classification frame by frame
RNN architecture
more complicated RNNs