Combine RNN with Neural ODEs

Intro to Neural ODEs

ResNets

Neural ODEs comes from ResNets

As these models grew to hundreds of layers deep, ResNets’ performance decreased. Deep learning had reached its limit. We need state-of-the-art performance to train deeper networks.

It also directly adds the input to the output, this shortcut connection improves the model since, at the worst, the residual block does not do anything.

One final thought. A ResNet can be described by the following equation:

$$h_{t+1}=h_t+f(h_t,{\theta }_t)$$

h - value of the hidden layer;
t - tell us which layer we are look at

the next hidden layer is the sum of the input and a function of the input as we have seen.

Find introduction to ResNets in Reference

Euler’s Method

How Neural ODEs work

Above equation seems like calculus, and if you don’t remember from calculus class, the Euler’s method is the simplest way to approximate the solution of a differential equation with initial value.

$$Initialvalueproblem:y^{\prime }(t)=f(t,y(t)),y(t_0)=y_0$$

$$Euler’sMethod:y_{n+1}=y_n+hf(t_n,y_n)$$

through this we can find numerical approximation

Euler’s method and ResNets equation are identical, the only difference being the step size $h$, that is multiplied by the function. Because of this similarity, we can think ResNets is underlying differential equation.

Instead of going from diffeq to Euler’s method, we can reverse engineer the problem. Starting from the ResNet, the resulting differential equation is

$$NeuralODE:\frac{dh(t)}{dt}=f(h(t),t,\theta )$$

which describes the dynamics of our model.

The Basics

How a Neural ODE works

The Neural ODEs combines two concepts: deep learning and differential equations, we use the most simple methods - Euler’s method to make predictions.

Q:
  How do we train it?
A:
  Adjoint method

Include using another numerical solver to run backwards through time (backpropagating) and updating the model’s parameters.

• Defining the architecture:

class ODEfunc(nn.Module):
	def __init__(self,dim):
		super(ODEfunc,self).__init__()
		#a 2d convolution that adds a time dimension to the output
		self.conv1 = ConcatConv2d(dim,dim,kernel_size=3,padding=1)
		self.norm2 = nn.BatchNorm2d(dim)
		self.conv2 = ConcatConv2d(dim,dim,kernel_size=3,padding=1)
		self.norm2 = nn.BatchNorm2d(dim)
		self.relu = nn.ReLU(inplace=True)
		# num of function evaluations for model depth
		self.nfe = 0
	def forward(self,t,x):
		self.nfe += 1
		out = self.conv1(t,x)
		out = self.norm1(out)
		out = self.relu(out)
		out = self.conv2(out)
		out = self.norm2(out)
		out = self.relu(out)
		return out

• Defining a neural ODE:

# numerically solves the diffeq thereby obtaining predictions
class NeuralODE(nn.Module):
	def __init__(self,odefunc):
		super(ODENet,self).__init__()
		self.odefunc = odefunc
		self.integraton_time = torch.tensor([0,1]).float()
	# backpropagates with the adjoint method to train model
	def forward(self,x):
		self