The paper is very well written, however, there are several questions about novelty of the work detailed below. The representational power of neural ODE models has not been studied much in the field. Although the negative examples and proof techniques are standard results in point-set topology and metric spaces, the appropriate application makes the idea very interesting.

The related works section seems adequate. Originality: The method is original in the deep learning literature. Though limitations of ODEs cannot cross paths is quite well-known, this paper views this deficiency from a modeling perspective and removes it while keeping within the ODE framework. The prose is very well written, and with many simple visualizations that support their claims.

If it makes sense to compare cross-entropy because it is objective that is being minimized for training e. But I think showing classification accuracy would be more meaningful and allow more follow-up works as it is the metric of interest. It is often the case in image classification that while validation cross entropy increases, the classification actually gets better. Right now, Figure 11 makes it seem like ANODE has a more significant overfitting problem, though I think the accuracy probably shouldn't increase very much even if the loss increases.A chain of residual blocks in a neural network is basically a solution of the ODE with the Euler method!

PDE-Net 2. Deep learning theory review: An optimal control and dynamical systems perspective. Neupde: Neural network based ordinary and partial differential equations for modeling time-dependent data. Introduction to Neural Ordinary Differential Equations. Alireza Afzal Aghaei M. Student Shahid Beheshti University. Mathematical Modeling of engineering problems leads to. Ordinary differential equations Partial differential equations Integral equations Optimal control.

Ordinary differential equations. A linear first order differential is defined as. Solving strategies for ODEs. Euler method for ODEs. Introduced at s. The solution to the following IVP. Modern ODE solvers.

### Neural ODEs

Euler method vs. Deep Neural Networks. Residual Neural Networks. Overcome vanishing gradient by skip connections Learn the residual function Much more powerful than traditional networks in most cases.

Skip connection. Traditional networks vs Resnets. In a general form, ResNets can be formulated as.We show that Neural Ordinary Differential Equations ODEs learn representations that preserve the topology of the input space and prove that this implies the existence of functions Neural ODEs cannot represent.

To address these limitations, we introduce Augmented Neural ODEs which, in addition to being more expressive models, are empirically more stable, generalize better and have a lower computational cost than Neural ODEs. Emilien Dupont. Arnaud Doucet. Yee Whye Teh.

**Lec 1 - MIT 18.03 Differential Equations, Spring 2006**

Neural Ordinary Differential Equations have been recently proposed as an We extend the framework of graph neural networks GNN to continuous tim The paper describes different approaches to generalize the trapezoidal m We introduce a provably stable variant of neural ordinary differential e To better conform to data geometry, recent deep generative modelling tec We generalize the notions of singularities and ordinary points from line Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

The relationship between neural networks and differential equations has been studied in several recent works. These models can be efficiently trained with backpropagation and have shown great promise on a number of tasks including modeling continuous time data and building normalizing flows with low computational cost. In this work, we explore some of the consequences of taking this continuous limit and the restrictions this might create compared with regular neural nets.

While it is often possible for NODEs to approximate these functions in practice, the resulting flows are complex and lead to ODE problems that are computationally expensive to solve. ANODEs augment the space on which the ODE is solved, allowing the model to use the additional dimensions to learn more complex functions using simpler flows see Fig.

### Augmented Neural ODEs

Our experiments also show that ANODEs generalize better, achieve lower losses with fewer parameters and are more stable to train.

The hidden state at time Ti. The analogy with ResNets can then be made more explicit. In ResNets, we map an input x to some output y by a forward pass of the neural network.

We then adjust the weights of the network to match y with some y true. We then adjust the dynamics of the system encoded by f such that the ODE transforms x to a y which is close to y true. ODE flows. We also define the flow associated to the vector field. NODEs for regression and classification. However, we are often interested in learning functions from R d to Re. To define a model from R d to Rwe follow the example given in linresnet for ResNets.

As shown in Fig. In this section, we introduce a simple function that ODE flows cannot represent, motivating many of the examples discussed later.The requirements that can be directly installed from PyPi can be found in requirements. Instructions for installing torchdiffeq can be found in this repo. More detailed examples and tutorials can be found in the augmented-neural-ode-example. This will log all the information about the experiments and generate plots for losses, NFEs and so on.

We also provide two demo notebooks that show how to reproduce some of the results and figures from the paper. The vector-field-visualizations. The augmented-neural-ode-example. This repository provides you with a labeling tool with little to no configuration needed!

The Tool was reduced in its functional scope to the most necessary. MMPose is an open-source toolbox for pose estimation based on PyTorch. It is a part of the OpenMMLab project. Python Awesome. Examples Requirements The requirements that can be directly installed from PyPi can be found in requirements.

Usage The usage pattern is simple Load some data Adam anode. Demos We also provide two demo notebooks that show how to reproduce some of the results and figures from the paper.

Vector fields The vector-field-visualizations. Easy to use labeling tool for State-of-the-art Deep Learning training purposes.Read this paper on arXiv. We show that Neural Ordinary Differential Equations ODEs learn representations that preserve the topology of the input space and prove that this implies the existence of functions Neural ODEs cannot represent.

To address these limitations, we introduce Augmented Neural ODEs which, in addition to being more expressive models, are empirically more stable, generalize better and have a lower computational cost than Neural ODEs.

The relationship between neural networks and differential equations has been studied in several recent works weinanproposal ; lubeyond ; haberstable ; ruthottodeep ; chenneural. In particular, it has been shown that Residual Networks hedeep can be interpreted as discretized ODEs. Taking the discretization step to zero gives rise to a family of models called Neural ODEs chenneural. These models can be efficiently trained with backpropagation and have shown great promise on a number of tasks including modeling continuous time data and building normalizing flows with low computational cost chenneural.

In this work, we explore some of the consequences of taking this continuous limit and which restrictions this might create compared with regular neural nets. While it is often possible for NODEs to approximate these functions in practice, the resulting flows are complex and lead to ODE problems that are computationally expensive to solve.

ANODEs augment the space on which the ODE is solved, allowing the model to use the additional dimensions to learn more complex functions using simpler flows see Fig. Our experiments also show that ANODEs generalize better, achieve lower losses with fewer parameters and are more stable to train. We can rearrange this equation as. The hidden state at time Ti.

The analogy with ResNets can then be made more explicit. In ResNets, we map an input x to some output y by a forward pass of the neural network.

We then adjust the weights of the network to match y with some y true. We then adjust the dynamics of the system encoded by f such that the ODE transforms x to a y which is close to y true. We note that f can be parameterized by any standard neural net architecture, including ones with activation functions that are not everywhere differentiable such as ReLU.

Existence and uniqueness of solutions to the ODE are still guaranteed and all results in this paper hold under these conditions see appendix for details. ODE flows. We also define the flow associated to the vector field f h tt of the ODE. The flow measures how the states of the ODE at a given time t depend on the initial conditions x.

NODEs for regression and classification. However, we are often interested in learning functions from R d to Re. To define a model from R d to Rwe follow the example given in linresnet for ResNets.

As shown in Fig. In this section, we introduce a simple function which ODE flows cannot represent, motivating many of the examples we will see later. The flow of an ODE cannot represent g 1 d x. A detailed proof is given in the appendix, however, the intuition behind the proof is simple. This simple observation is at the core of all the examples provided in this paper and forms the basis for many of the limitations of NODEs. We verify this behavior experimentally by training an ODE flow on the identity mapping and on g 1 d x.

The resulting flows are shown in Fig. As can be seen, the model easily learns the identity mapping but cannot represent g 1 d x. Indeed, since the trajectories cannot cross, the model maps all input points to zero to minimize the mean squared error. The reason for this is exactly because ResNets are a discretization of the ODE, allowing the trajectories to make discrete jumps to cross each other see Fig.

Indeed, the error arising when taking discrete steps allows the ResNet trajectories to cross. In this sense, ResNets can be interpreted as ODE solutions with large errors, with these errors allowing them to represent more functions.We will discuss a new family of neural networks models. These models can be viewed as continuous-depth architectures.

## Augmented Neural ODEs

Many neural network models such as residual networks, recurrent neural network decoders etc. For example, consider residual networks ResNets. In ResNets, the transformation of a hidden state from layer to layer is given by. Note that the iterative updates can be seen as an Euler discretization of a continuous transformation. Consider the ordinary differential equation ODE. Let be an initial valuechoose a step and set. Approximating the derivative by the incremental ratio and the right hand side of the ODE bywe obtain.

The backward Euler scheme has a better stability property than the forward Euler, though we need to solve a nonlinear equation at each step. Neural ODEs form a family of deep neural network models that can be interpreted as countinuous equivalent of residual networks. We start for example from the ResNet model. What happens as we add more layers and take smaller steps? We can rearrange this equation as.

Now lettingwe get. In the limit, we parameterize the continuous dynamics of hidden units using an ODE specified by a neural network:. We start from the input layer and define the output layer to be the solution of this ODE initial value problem at some time. The hidden state at timei. This value can be computed by a differential equation solver, which evaluates the hidden unit dynamics wherever necessary to determine the solution with the desired accuracy.

In ResNets we map an input to some output by a forward pass of the neural network. We then adjust the weights of the network to match with some. We then adjust the dynamics of the system encoded by such that the ODE transforms to a which is close to.

However, we will not delve into such details. How to train a Neural ODE? Consider a process described by an ODE and suppose that some observations are known. Can we find a way to approximate the function? Start from and track the evolution of the system using an ODE solver, ending up in some new state. Then evaluate the difference between and the observation ; we have to minimize this difference modifying parameters.Weather forecasting is a tricky problem.

Traditionally, it has been done by manually modelling weather dynamics using differential equations, but this approach is highly dependent on us getting the equations right. However, this approach requires huge amounts of data to reach good performance. Fortunately, there is a middle ground: What if we instead use machine learning to model the dynamics of the weather?

Instead of trying to model how the weather will look in the next time step, what if we instead model how the weather changes between time steps?

## Pytorch implementation of Augmented Neural ODEs

More concretely: What if we learn the differential equations that govern the change in weather? In this blog article we are going to use Julia and the SciML ecosystem to do just that. If the dynamics are constant, this has very powerful generalisation capabilities. This means that a single forward pass gives us an entire trajectory in contrast to e.

RNN s, where each forward pass through the model gives a single prediction. Figure 1: Solving a simple initial value problem using a trained neural ODE. Since the network has already been trained, it accurately model dynamics.

So how do we train a network inside an ODE? Fortunately, DiffEqFlux takes care of everything required to do this for us. There are several strategies that can be specified to compute gradients, and depending on the problem you might prefer one over the other. However, for this article the default InterpolatingAdjoint will be perfectly fine. The NeuralODE object itself has a few additional important hyper-parameters though.

Firstly, we have to specify an ODE solver and a time span to solve on. We will use the Tsit5 solver, which uses an explicit method. Secondly, the parameters reltol and abstol let us configure the solution error tolerance to trade off accuracy and training time.

Recall that a forward pass means solving an initial value problem, hence a lower tolerance gives a more accurate solution, and in turn better gradient estimates.

But of course, this requires more function evaluations and are consequently slower to compute. The dataset we are going to use comprises daily measurements of the climate in Delhi over several years.

The entire dataset is a single time series, where the last part is set aside for testing. Figure 2: Visualisation of the raw data. There is a clear seasonal trend, but there are some extreme outliers among the pressure measurements, which make it difficult to see any patterns. Something is off with the air pressure measurements, indicated by the extreme outliers after However, prior to the measurements show a nice periodic behavior.

Figure 3: Zooming in on the pressure before reveal the same pleasant seasonal behavior as in the other measurements.

## thoughts on “Augmented neural odes”