# GPU-based MNIST Neural ODE Classifier

Training a classifier for **MNIST** using a neural ordinary differential equation **NN-ODE** on **GPUs** with **Minibatching**.

(Step-by-step description below)

```
using DiffEqFlux, DifferentialEquations, NNlib, MLDataUtils, Printf
using Flux.Losses: logitcrossentropy
using Flux.Data: DataLoader
using MLDatasets
using CUDA
CUDA.allowscalar(false)
function loadmnist(batchsize = bs, train_split = 0.9)
# Use MLDataUtils LabelEnc for natural onehot conversion
onehot(labels_raw) = convertlabel(LabelEnc.OneOfK, labels_raw,
LabelEnc.NativeLabels(collect(0:9)))
# Load MNIST
imgs, labels_raw = MNIST.traindata();
# Process images into (H,W,C,BS) batches
x_data = Float32.(reshape(imgs, size(imgs,1), size(imgs,2), 1, size(imgs,3)))
y_data = onehot(labels_raw)
(x_train, y_train), (x_test, y_test) = stratifiedobs((x_data, y_data),
p = train_split)
return (
# Use Flux's DataLoader to automatically minibatch and shuffle the data
DataLoader(gpu.(collect.((x_train, y_train))); batchsize = batchsize,
shuffle = true),
# Don't shuffle the test data
DataLoader(gpu.(collect.((x_test, y_test))); batchsize = batchsize,
shuffle = false)
)
end
# Main
const bs = 128
const train_split = 0.9
train_dataloader, test_dataloader = loadmnist(bs, train_split)
down = Chain(Flux.flatten, Dense(784, 20, tanh)) |> gpu
nn = Chain(Dense(20, 10, tanh),
Dense(10, 10, tanh),
Dense(10, 20, tanh)) |> gpu
nn_ode = NeuralODE(nn, (0.f0, 1.f0), Tsit5(),
save_everystep = false,
reltol = 1e-3, abstol = 1e-3,
save_start = false) |> gpu
fc = Chain(Dense(20, 10)) |> gpu
function DiffEqArray_to_Array(x)
xarr = gpu(x)
return reshape(xarr, size(xarr)[1:2])
end
# Build our overall model topology
model = Chain(down,
nn_ode,
DiffEqArray_to_Array,
fc) |> gpu;
# To understand the intermediate NN-ODE layer, we can examine it's dimensionality
img, lab = train_dataloader.data[1][:, :, :, 1:1], train_dataloader.data[2][:, 1:1]
x_d = down(img)
# We can see that we can compute the forward pass through the NN topology
# featuring an NNODE layer.
x_m = model(img)
classify(x) = argmax.(eachcol(x))
function accuracy(model, data; n_batches = 100)
total_correct = 0
total = 0
for (i, (x, y)) in enumerate(collect(data))
# Only evaluate accuracy for n_batches
i > n_batches && break
target_class = classify(cpu(y))
predicted_class = classify(cpu(model(x)))
total_correct += sum(target_class .== predicted_class)
total += length(target_class)
end
return total_correct / total
end
# burn in accuracy
accuracy(model, train_dataloader)
loss(x, y) = logitcrossentropy(model(x), y)
# burn in loss
loss(img, lab)
opt = ADAM(0.05)
iter = 0
cb() = begin
global iter += 1
# Monitor that the weights do infact update
# Every 10 training iterations show accuracy
if iter % 10 == 1
train_accuracy = accuracy(model, train_dataloader) * 100
test_accuracy = accuracy(model, test_dataloader;
n_batches = length(test_dataloader)) * 100
@printf("Iter: %3d || Train Accuracy: %2.3f || Test Accuracy: %2.3f\n",
iter, train_accuracy, test_accuracy)
end
end
# Train the NN-ODE and monitor the loss and weights.
Flux.train!(loss, Flux.params(down, nn_ode.p, fc), train_dataloader, opt, cb = cb)
```

## Step-by-Step Description

### Load Packages

```
using DiffEqFlux, DifferentialEquations, NNlib, MLDataUtils, Printf
using Flux.Losses: logitcrossentropy
using Flux.Data: DataLoader
using MLDatasets
```

### GPU

A good trick used here:

```
using CUDA
CUDA.allowscalar(false)
```

ensures that only optimized kernels are called when using the GPU. Additionally, the `gpu`

function is shown as a way to translate models and data over to the GPU. Note that this function is CPU-safe, so if the GPU is disabled or unavailable, this code will fallback to the CPU.

### Load MNIST Dataset into Minibatches

The preprocessing is done in `loadmnist`

where the raw MNIST data is split into features `x_train`

and labels `y_train`

by specifying batchsize `bs`

. The function `convertlabel`

will then transform the current labels (`labels_raw`

) from numbers 0 to 9 (`LabelEnc.NativeLabels(collect(0:9))`

) into one hot encoding (`LabelEnc.OneOfK`

).

Features are reshaped into format **[Height, Width, Color, BatchSize]** or in this case **[28, 28, 1, 128]** meaning that every minibatch will contain 128 images with a single color channel of 28x28 pixels. The entire dataset of 60,000 images is split into the train and test dataset, ensuring a balanced ratio of labels. These splits are then passed to Flux's DataLoader. This automatically minibatches both the images and labels. Additionally, it allows us to shuffle the train dataset in each epoch while keeping the order of the test data the same.

```
function loadmnist(batchsize = bs, train_split = 0.9)
# Use MLDataUtils LabelEnc for natural onehot conversion
onehot(labels_raw) = convertlabel(LabelEnc.OneOfK, labels_raw,
LabelEnc.NativeLabels(collect(0:9)))
# Load MNIST
imgs, labels_raw = MNIST.traindata();
# Process images into (H,W,C,BS) batches
x_data = Float32.(reshape(imgs, size(imgs,1), size(imgs,2), 1, size(imgs,3)))
y_data = onehot(labels_raw)
(x_train, y_train), (x_test, y_test) = stratifiedobs((x_data, y_data),
p = train_split)
return (
# Use Flux's DataLoader to automatically minibatch and shuffle the data
DataLoader(gpu.(collect.((x_train, y_train))); batchsize = batchsize,
shuffle = true),
# Don't shuffle the test data
DataLoader(gpu.(collect.((x_test, y_test))); batchsize = batchsize,
shuffle = false)
)
end
```

and then loaded from main:

```
# Main
const bs = 128
const train_split = 0.9
train_dataloader, test_dataloader = loadmnist(bs, train_split)
```

### Layers

The Neural Network requires passing inputs sequentially through multiple layers. We use `Chain`

which allows inputs to functions to come from previous layer and sends the outputs to the next. Four different sets of layers are used here:

```
down = Chain(Flux.flatten, Dense(784, 20, tanh)) |> gpu
nn = Chain(Dense(20, 10, tanh),
Dense(10, 10, tanh),
Dense(10, 20, tanh)) |> gpu
nn_ode = NeuralODE(nn, (0.f0, 1.f0), Tsit5(),
save_everystep = false,
reltol = 1e-3, abstol = 1e-3,
save_start = false) |> gpu
fc = Chain(Dense(20, 10)) |> gpu
```

`down`

: This layer downsamples our images into a 20 dimensional feature vector. It takes a 28 x 28 image, flattens it, and then passes it through a fully connected layer with `tanh`

activation

`nn`

: A 3 layers Deep Neural Network Chain with `tanh`

activation which is used to model our differential equation

`nn_ode`

: ODE solver layer

`fc`

: The final fully connected layer which maps our learned feature vector to the probability of the feature vector of belonging to a particular class

`|> gpu`

: An utility function which transfers our model to GPU, if it is available

### Array Conversion

When using `NeuralODE`

, this function converts the ODESolution's `DiffEqArray`

to a Matrix (CuArray), and reduces the matrix from 3 to 2 dimensions for use in the next layer.

```
function DiffEqArray_to_Array(x)
xarr = gpu(x)
return reshape(xarr, size(xarr)[1:2])
end
```

For CPU: If this function does not automatically fallback to CPU when no GPU is present, we can change `gpu(x)`

to `Array(x)`

.

### Build Topology

Next we connect all layers together in a single chain:

```
# Build our overall model topology
model = Chain(down,
nn_ode,
DiffEqArray_to_Array,
fc) |> gpu;
```

There are a few things we can do to examine the inner workings of our neural network:

```
img, lab = train_dataloader.data[1][:, :, :, 1:1], train_dataloader.data[2][:, 1:1]
# To understand the intermediate NN-ODE layer, we can examine it's dimensionality
x_d = down(img)
# We can see that we can compute the forward pass through the NN topology
# featuring an NNODE layer.
x_m = model(img)
```

This can also be built without the NN-ODE by replacing `nn-ode`

with a simple `nn`

:

```
# We can also build the model topology without a NN-ODE
m_no_ode = Chain(down,
nn,
fc) |> gpu
x_m = m_no_ode(img)
```

### Prediction

To convert the classification back into readable numbers, we use `classify`

which returns the prediction by taking the arg max of the output for each column of the minibatch:

`classify(x) = argmax.(eachcol(x))`

### Accuracy

We then evaluate the accuracy on `n_batches`

at a time through the entire network:

```
function accuracy(model, data; n_batches = 100)
total_correct = 0
total = 0
for (i, (x, y)) in enumerate(collect(data))
# Only evaluate accuracy for n_batches
i > n_batches && break
target_class = classify(cpu(y))
predicted_class = classify(cpu(model(x)))
total_correct += sum(target_class .== predicted_class)
total += length(target_class)
end
return total_correct / total
end
# burn in accuracy
accuracy(m, train_dataloader)
```

### Training Parameters

Once we have our model, we can train our neural network by backpropagation using `Flux.train!`

. This function requires **Loss**, **Optimizer** and **Callback** functions.

#### Loss

**Cross Entropy** is the loss function computed here which applies a **Softmax** operation on the final output of our model. `logitcrossentropy`

takes in the prediction from our model `model(x)`

and compares it to actual output `y`

:

```
loss(x, y) = logitcrossentropy(model(x), y)
# burn in loss
loss(img, lab)
```

#### Optimizer

`ADAM`

is specified here as our optimizer with a **learning rate of 0.05**:

`opt = ADAM(0.05)`

#### CallBack

This callback function is used to print both the training and testing accuracy after 10 training iterations:

```
cb() = begin
global iter += 1
# Monitor that the weights update
# Every 10 training iterations show accuracy
if iter % 10 == 1
train_accuracy = accuracy(model, train_dataloader) * 100
test_accuracy = accuracy(model, test_dataloader;
n_batches = length(test_dataloader)) * 100
@printf("Iter: %3d || Train Accuracy: %2.3f || Test Accuracy: %2.3f\n",
iter, train_accuracy, test_accuracy)
end
end
```

### Train

To train our model, we select the appropriate trainable parameters of our network with `params`

. In our case, backpropagation is required for `down`

, `nn_ode`

and `fc`

. Notice that the parameters for Neural ODE is given by `nn_ode.p`

:

```
# Train the NN-ODE and monitor the loss and weights.
Flux.train!(loss, Flux.params( down, nn_ode.p, fc), zip( x_train, y_train ), opt, cb = cb)
```

### Expected Output

```
Iter: 1 || Train Accuracy: 16.203 || Test Accuracy: 16.933
Iter: 11 || Train Accuracy: 64.406 || Test Accuracy: 64.900
Iter: 21 || Train Accuracy: 76.656 || Test Accuracy: 76.667
Iter: 31 || Train Accuracy: 81.758 || Test Accuracy: 81.683
Iter: 41 || Train Accuracy: 81.078 || Test Accuracy: 81.967
Iter: 51 || Train Accuracy: 83.953 || Test Accuracy: 84.417
Iter: 61 || Train Accuracy: 85.266 || Test Accuracy: 85.017
Iter: 71 || Train Accuracy: 85.938 || Test Accuracy: 86.400
Iter: 81 || Train Accuracy: 84.836 || Test Accuracy: 85.533
Iter: 91 || Train Accuracy: 86.148 || Test Accuracy: 86.583
Iter: 101 || Train Accuracy: 83.859 || Test Accuracy: 84.500
Iter: 111 || Train Accuracy: 86.227 || Test Accuracy: 86.617
Iter: 121 || Train Accuracy: 87.508 || Test Accuracy: 87.200
Iter: 131 || Train Accuracy: 86.227 || Test Accuracy: 85.917
Iter: 141 || Train Accuracy: 84.453 || Test Accuracy: 84.850
Iter: 151 || Train Accuracy: 86.063 || Test Accuracy: 85.650
Iter: 161 || Train Accuracy: 88.375 || Test Accuracy: 88.033
Iter: 171 || Train Accuracy: 87.398 || Test Accuracy: 87.683
Iter: 181 || Train Accuracy: 88.070 || Test Accuracy: 88.350
Iter: 191 || Train Accuracy: 86.836 || Test Accuracy: 87.150
Iter: 201 || Train Accuracy: 89.266 || Test Accuracy: 88.583
Iter: 211 || Train Accuracy: 86.633 || Test Accuracy: 85.550
Iter: 221 || Train Accuracy: 89.313 || Test Accuracy: 88.217
Iter: 231 || Train Accuracy: 88.641 || Test Accuracy: 89.417
Iter: 241 || Train Accuracy: 88.617 || Test Accuracy: 88.550
Iter: 251 || Train Accuracy: 88.211 || Test Accuracy: 87.950
Iter: 261 || Train Accuracy: 87.742 || Test Accuracy: 87.317
Iter: 271 || Train Accuracy: 89.070 || Test Accuracy: 89.217
Iter: 281 || Train Accuracy: 89.703 || Test Accuracy: 89.067
Iter: 291 || Train Accuracy: 88.484 || Test Accuracy: 88.250
Iter: 301 || Train Accuracy: 87.898 || Test Accuracy: 88.367
Iter: 311 || Train Accuracy: 88.438 || Test Accuracy: 88.633
Iter: 321 || Train Accuracy: 88.664 || Test Accuracy: 88.567
Iter: 331 || Train Accuracy: 89.906 || Test Accuracy: 89.883
Iter: 341 || Train Accuracy: 88.883 || Test Accuracy: 88.667
Iter: 351 || Train Accuracy: 89.609 || Test Accuracy: 89.283
Iter: 361 || Train Accuracy: 89.516 || Test Accuracy: 89.117
Iter: 371 || Train Accuracy: 89.898 || Test Accuracy: 89.633
Iter: 381 || Train Accuracy: 89.055 || Test Accuracy: 89.017
Iter: 391 || Train Accuracy: 89.445 || Test Accuracy: 89.467
Iter: 401 || Train Accuracy: 89.156 || Test Accuracy: 88.250
Iter: 411 || Train Accuracy: 88.977 || Test Accuracy: 89.083
Iter: 421 || Train Accuracy: 90.109 || Test Accuracy: 89.417
```