Wong Lik Hang Kenny

Wong Lik Hang Kenny

I am a senior year undergraduate student majoring in Computer Science, my research interests lie in Deep Learning, 3D Computer Vision, Embodied AI, Robotics

Neural ODEs

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Authors: Wong Lik Hang Kenny

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Neural Ordinary Differential Equations (Neural ODEs)

Neural Ordinary Differential Equations (Neural ODEs) are a powerful class of machine learning models that unify discrete, layer-based neural architectures (like residual networks) with continuous dynamical systems. They were introduced in the 2018 NeurIPS paper by Chen, Rubanova, Bettencourt, and Duvenaud, titled “Neural Ordinary Differential Equations.”

This blog post provides a gentle yet detailed introduction to Neural ODEs, explains why they generalize certain deep network architectures, how they can be trained, and illustrates their exciting properties. We will conclude with a discussion about how Neural ODEs can be applied in robotics and physics.

Table of Contents

1. Motivation: From Residual Networks to Continuous Depth

To appreciate Neural ODEs, let’s start with an important building block in modern deep learning: the residual block. A ResNet layer updates its hidden state \(\mathbf{h}_t\) as follows:

\[\mathbf{h}_{t+1} = \mathbf{h}_t + f(\mathbf{h}_t, \theta_t),\]

where \(f\) is a learnable function (often a neural network itself), and \(\theta_t\) are its parameters.

  • Interpretation as Euler’s method: If you squint at this update, you’ll notice it looks like a single “step” of the Euler method for solving a differential equation \(\frac{d\mathbf{h}}{dt} = f(\mathbf{h}, t, \theta)\).
  • As we stack more layers (or equivalently, take smaller and smaller steps), this residual update approaches a continuous transformation of \(\mathbf{h}\).

Neural ODEs extend this idea by directly modeling \(\frac{d\mathbf{h}(t)}{dt} = f(\mathbf{h}(t), t)\) and using ODE solvers instead of fixed discrete layers.

2. What Is an ODE and Why Make It Neural?

An Ordinary Differential Equation (ODE) describes how a variable \(\mathbf{x}(t)\) changes over continuous time \(t\). For example:

\[\frac{d\mathbf{x}(t)}{dt} \;=\; f\bigl(\mathbf{x}(t), t\bigr).\]
  • If you specify an initial condition \(\mathbf{x}(0)\), the ODE solutions are the unique trajectories \(\mathbf{x}(t)\) that satisfy this equation.
  • Solving an ODE often means using numerical ODE solvers (e.g., Runge-Kutta methods, Adams methods) to step forward in time.

Making it Neural means letting the function \(f\) be parameterized by a neural network—e.g.,

\[f(\mathbf{x}(t), t; \theta) \;=\; \text{NeuralNetwork}(\mathbf{x}(t), t; \theta).\]

We then rely on standard ODE solvers to integrate this neural network over time.

3. Defining a Neural ODE

Formally, a Neural ODE sets:

\[\frac{d\mathbf{h}(t)}{dt} = f(\mathbf{h}(t), t, \theta).\]

Starting from an input \(\mathbf{h}(0)\), we obtain an output \(\mathbf{h}(T)\) for some final time \(T\) by solving this initial value problem (IVP):

\[\mathbf{h}(T) = \mathbf{h}(0) + \int_{0}^{T} f\bigl(\mathbf{h}(t), t, \theta \bigr)\, dt.\]

In practice, we never compute that integral exactly by hand. Instead, we hand \(\mathbf{h}(0)\) and \(f\) to a black-box ODE solver, which:

  1. Dynamically decides time steps.
  2. Evaluates \(f\) at necessary points.
  3. Produces a numerical approximation to \(\mathbf{h}(T)\).

4. Training Neural ODEs via the Adjoint Method

One key challenge is how to train the parameters \(\theta\). Traditional backpropagation would require you to store every intermediate step in memory, which can be expensive or even infeasible for large or adaptive solvers.

The adjoint sensitivity method offers a clever solution:

  • Instead of storing every intermediate state, we run the ODE solver forward to obtain \(\mathbf{h}(T)\).
  • We compute the loss \(L(\mathbf{h}(T))\).
  • Then, we introduce an adjoint variable \(\mathbf{a}(t) = \frac{\partial L}{\partial \mathbf{h}(t)}\), which we solve backwards in time along with \(\mathbf{h}(t)\).

The reverse-time ODE for \(\mathbf{a}(t)\) involves Jacobian-vector products of \(f\). Crucially, automatic differentiation (as used in modern deep learning) can compute these products efficiently.

This approach:

  1. Keeps memory cost constant w.r.t. the number of forward solver steps.
  2. Allows the user to treat the ODE solver as a black box while still computing gradients for training.

5. Advantages and Properties of Neural ODEs

(a) Memory Efficiency

Because the adjoint method does not store intermediate activations for each ODE solver step, Neural ODEs scale to deeper/continuous architectures with constant memory cost.

  • In classical feed-forward or residual networks, memory grows with the number of layers (or steps).
  • In a Neural ODE, we effectively have infinitely many layers in principle, but do not pay a huge memory cost for that.

(b) Adaptive Computation

Traditional discrete networks use a fixed number of layers. By contrast, ODE solvers:

  • Adapt the step size to keep numerical error below a desired tolerance.
  • Potentially reduce computation by taking fewer steps in easy regions.
  • Provide finer resolution where the solution changes quickly.

(c) Continuous Normalizing Flows

A powerful application is Continuous Normalizing Flows (CNFs) for generative modeling. Standard normalizing flows rely on Jacobian determinants, which can be expensive for high-dimensional transformations. In CNFs, the log-probability evolves with a trace of the Jacobian, often simpler and more scalable.

(d) Continuous-Time Modeling of Data

Many real-world processes (especially physical or robotic systems) are inherently continuous in time. Neural ODEs let you:

  • Train on data sampled irregularly in time.
  • Generate predictions for any time, even times not in the training set.
  • Leverage advanced integrators to preserve domain-specific structure (e.g., symplectic integrators for Hamiltonian systems).

6. Applications in Robotics

In robotics, the dynamics of manipulators, drones, or walking robots often follow continuous-time physics:

  1. Learning Robot Dynamics:
    • Many robotics tasks require precise system identification—figuring out how states like position, velocity, and torque evolve over time.
    • With Neural ODEs, you can learn these continuous dynamics from partial or noisy measurements.
  2. Trajectory Optimization:
    • If you have a trained Neural ODE representing robot dynamics, you can integrate it to plan or optimize trajectories.
    • This is especially useful when you have data sampled at uneven intervals or the robot’s movements are fast in some phases and slow in others.
  3. Energy or Constraint Preservation:
    • By carefully choosing specialized ODE solvers (e.g., symplectic integrators) within a Neural ODE framework, you can preserve known physical constraints such as conservation of energy.
    • This ensures more stable and realistic robotic control.

In practice: You can embed a Neural ODE block inside a larger control architecture, letting it serve as an adaptive and data-driven model of your system’s continuous motion.

7. Applications in Physics

Many physical processes—like planetary motion, fluid flows, chemical kinetics—are naturally described by differential equations:

  1. Data-Driven Discovery of Dynamics:
    • Traditional PDE or ODE derivations can be very complex. Neural ODEs let you approximate the right-hand side of physical laws directly from data.
  2. Hamiltonian/Lagrangian Neural Networks:
    • Extensions of Neural ODEs, such as Hamiltonian Neural Networks or Lagrangian Neural Networks, incorporate energy-conserving properties or variational principles into the architecture.
    • This can be helpful in modeling and simulating physical systems long-term without numerical drift in energy.
  3. Irregular Sampling:
    • Physical measurements are often taken at uneven intervals (due to sensor scheduling, experimental constraints, etc.). Neural ODEs are particularly convenient in this setting, as the solver adaptively handles time steps.
  4. Predictive Modeling and Forecasting:
    • Once trained, a Neural ODE can predict future states (e.g., pressure, temperature, or mechanical stress) by simply integrating forward in time.
    • Similarly, you can model long-term behaviors more stably than with discrete updates, if you choose appropriate integrators.

8. Concluding Remarks

Neural ODEs mark a conceptual shift from thinking of neural networks as layered, discrete transformations to viewing them as continuously evolving dynamical systems. This perspective:

  • Bridges machine learning with classical ODE-based modeling.
  • Improves memory usage and allows for adaptive evaluation.
  • Offers new avenues in generative modeling, robotics, and physics.

When you need to handle irregularly sampled data, embed domain-specific constraints, or require memory-efficient training of very deep models, Neural ODEs can be a compelling solution.

Whether you’re building a manipulator control system or studying planetary orbits, Neural ODEs provide a flexible, elegant approach to learning continuous-time dynamics from data. It is an exciting technique that continues to inspire new research and innovative applications in the broader landscape of physics-informed machine learning.

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