# TorchDyn Quickstart¶

TorchDyn is the toolkit for continuous models in PyTorch. Play with state-of-the-art architectures or use its powerful libraries to create your own.

Central to the torchdyn approach are continuous neural networks, where width, depth (or both) are taken to their infinite limit. On the optimization front, we consider continuous “data-stream” regimes and gradient flow methods, where the dataset represents a time-evolving signal processed by the neural network to adapt its parameters.

By providing a centralized, easy-to-access collection of model templates, tutorial and application notebooks, we hope to speed-up research in this area and ultimately contribute to turning neural differential equations into an effective tool for control, system identification and common machine learning tasks.

[1]:

import sys ; sys.path.append('../')
from torchdyn.models import *
from torchdyn.datasets import *
from torchdyn import *


## Generate data from a static toy dataset¶

We’ll be generating data from toy datasets. In torchdyn, we provide a wide range of datasets often use to benchmark and understand Neural ODEs. Here we will use the classic moons dataset and train a Neural ODE for binary classification

[2]:

d = ToyDataset()
X, yn = d.generate(n_samples=512, noise=1e-1, dataset_type='moons')

[3]:

import matplotlib.pyplot as plt

colors = ['orange', 'blue']
fig = plt.figure(figsize=(3,3))
for i in range(len(X)):
ax.scatter(X[i,0], X[i,1], s=1, color=colors[yn[i].int()])


Generated data can be easily loaded in the dataloader with standard PyTorch calls

[4]:

import torch
import torch.utils.data as data
device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")

X_train = torch.Tensor(X).to(device)
y_train = torch.LongTensor(yn.long()).to(device)
train = data.TensorDataset(X_train, y_train)


We utilize Pytorch Lightning to handle training loops, logging and general bookkeeping. This allows torchdyn and Neural Differential Equations to have access to modern best practices for training and experiment reproducibility.

In particular, we combine modular torchdyn models with LightningModules via a Learner class:

[5]:

import torch.nn as nn
import pytorch_lightning as pl

class Learner(pl.LightningModule):
def __init__(self, model:nn.Module):
super().__init__()
self.model = model

def forward(self, x):
return self.model(x)

def training_step(self, batch, batch_idx):
x, y = batch
y_hat = self.model(x)
loss = nn.CrossEntropyLoss()(y_hat, y)
logs = {'train_loss': loss}
return {'loss': loss, 'log': logs}

def configure_optimizers(self):



## Define a Neural ODE¶

Analogously to most forward neural models we want to realize a map

$x \mapsto \hat y$

where $$\hat y$$ becomes the best approximation of a true output $$y$$ given an input $$x$$. In torchdyn you can define very simple Neural ODE models of the form

\begin{split}\left\{ \begin{aligned} \dot{z}(s) &= f(z(s), \theta)\\ z(0) &= x\\ \hat y & = z(1) \end{aligned} \right. \quad s\in[0,1]\end{split}

by just specifying a neural network $$f$$ and giving some simple settings.

Note: This Neural ODE model is of depth-invariant type as neither $$f$$ explicitly depend on $$s$$ nor the parameters $$\theta$$ are depth-varying. Together with their depth-variant counterpart with $$s$$ concatenated in the vector field was first proposed and implemented by [Chen T. Q. et al, 2018]

### Define the vector field (DEFunc)¶

The first step is to define any PyTorch torch.nn.Module. This takes the role of the Neural ODE vector field $$f(h,\theta)$$

[6]:

f = nn.Sequential(
nn.Linear(2, 16),
nn.Tanh(),
nn.Linear(16, 2)
)


In this case we chose $$f$$ to be a simple MLP with one hidden layer and $$\tanh$$ activation

### Define the NeuralDE¶

The final step to define a Neural ODE is to instantiate the torchdyn’s class NeuralDE passing some customization arguments and f itself.

In this case we specify: * we compute backward gradients with the 'adjoint' method. * we will use the 'dopri5' (Dormand-Prince) ODE solver from torchdiffeq;

[7]:

model = NeuralDE(f, sensitivity='adjoint', solver='dopri5').to(device)


## Train the Model¶

[8]:

learn = Learner(model)
trainer = pl.Trainer(min_epochs=200, max_epochs=300)
trainer.fit(learn)

GPU available: True, used: False
TPU available: False, using: 0 TPU cores

| Name  | Type     | Params
-----------------------------------
0 | model | NeuralDE | 82



[8]:

1


With the method trajectory of NeuralDE objects you can quickly evaluate the entire trajectory of each data point in X_train on an interval s_span

[9]:

s_span = torch.linspace(0,1,100)
trajectory = model.trajectory(X_train, s_span).detach().cpu()


### Plot the Training Results¶

We can first plot the trajectories of the data points in the depth domain $$s$$

[10]:

color=['orange', 'blue']

fig = plt.figure(figsize=(10,2))
for i in range(500):
ax0.plot(s_span, trajectory[:,i,0], color=color[int(yn[i])], alpha=.1);
ax1.plot(s_span, trajectory[:,i,1], color=color[int(yn[i])], alpha=.1);
ax0.set_xlabel(r"$s$ [Depth]") ; ax0.set_ylabel(r"$h_0(s)$")
ax1.set_xlabel(r"$s$ [Depth]") ; ax1.set_ylabel(r"$z_1(s)$")
ax0.set_title("Dimension 0") ; ax1.set_title("Dimension 1")

[10]:

Text(0.5, 1.0, 'Dimension 1')


Then the trajectory in the state-space

As you can see, the Neural ODE steers the data-points into regions of null loss with a continuous flow in the depth domain. Finally, we can also plot the learned vector field $$f$$

[11]:

# evaluate vector field
n_pts = 50
x = torch.linspace(trajectory[:,:,0].min(), trajectory[:,:,0].max(), n_pts)
y = torch.linspace(trajectory[:,:,1].min(), trajectory[:,:,1].max(), n_pts)
X, Y = torch.meshgrid(x, y) ; z = torch.cat([X.reshape(-1,1), Y.reshape(-1,1)], 1)
f = model.defunc(0,z.to(device)).cpu().detach()
fx, fy = f[:,0], f[:,1] ; fx, fy = fx.reshape(n_pts , n_pts), fy.reshape(n_pts, n_pts)
# plot vector field and its intensity
fig = plt.figure(figsize=(4, 4)) ; ax = fig.add_subplot(111)
ax.streamplot(X.numpy().T, Y.numpy().T, fx.numpy().T, fy.numpy().T, color='black')
ax.contourf(X.T, Y.T, torch.sqrt(fx.T**2+fy.T**2), cmap='RdYlBu')

[11]:

<matplotlib.contour.QuadContourSet at 0x1f3458fbe08>