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DEEP KERNEL LEARNING OF DYNAMICAL MODELS FROM HIGH-DIMENSIONAL NOISY DATA

NICOLÒ BOTTEGHI*, MENGWU GUO, CHRISTOPH BRUNE

�MATHEMATICS OF IMAGING & AI GROUP�DEPARTMENT OF APPLIED MATHEMATICS�

SIAM CSE

27 FEBRUARY 2023

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Learning Dynamical Models from Data

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robotics

human-machine interaction

symmetries in nature

micro-macro

bio-inspired robots

forecasting

pattern-forming systems

In our everyday life we can see many example of systems evolving over time. You can think of robots, autonomous vehicles, birds flying, patterns appearing on certain animals, or the weather (especially in the Netherlands evolving very fast as you might have noticed from rainy to cloudy, sunny for a moment and back again to rainy). Understanding and predicting how these systems evolve over time is a crucial aspect of science and engineering.

While for simple systems we may rely on physical laws to describe their evolution, for complex systems this is not often possible.

In many modern systems, governing equations are often unknown, and even when partially known, the computational size of the derived model is simply too high and forces users to trade accuracy for efficient computations.

However, we can often gather an abundance of data due to sensors and measurement technologies from these systems (e.g., in robotics, biology and climate science).

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Challenges of Learning Dynamics from Data

HIGH DIMENSIONALITY AND NOISE

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Data are indirect, high-dimensional, and noisy measurements of the systems, making the learning of dynamics from data is extremely challenging.

Therefore, data-driven methods must:

1- learn the latent state variables

dimensionality reduction

2- learn the dynamics from the latent state variables

🡪 reduce-order modeling

3- quantify the uncertainties over the latent state variables

🡪 uncertainty quantification

Learning the evolution of the systems from indirect measurement data however has several challenges! The two critical aspects we tackled in our research are high-dimensionality of the measurements over space and time and they are corrupted by noise.

In the context of learning dynamics from data, in general we need to three steps to properly learns models:

-First reduce the dimensionality of the data and infer the low dimensional set of latent state variables governing the evolution of the systems,

-Then learn the dynamics given the latent state variable and if present control inputs,

-At the same time properly quantify the uncertainties deriving from the measurement noise and the dynamics of the system itself (e.g. double pendulum).

The key message is that without properly quantify uncertainties for example whether they come from noise on the data or the dynamics of the systems itself, we cannot properly learn latent state variables and latent dynamical models from data.

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Neural Networks and Gaussian Processes

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Neural Network

Gaussian Process

+ representing complex functions

+ handling high-dimensional data (parametric approach)

-- quantifying uncertainties

(no kernel)

+ representing complex functions

-- handling high-dimensional data (non-parametric approach)

+ quantifying uncertainties (kernel structure)

What tools do we have for learning dynamics from data? For the sake of time, I will just briefly introduce neural networks and gaussian processes since they are the core of our research. Of course, there are many other methods.

Let’s assume for a moment the classic supervised learning, where we want to learn an unknow function mapping x to y given some target values y*.

If for this problem we use a neural network, we know that it can represent complex functions (neural networks are universal function approximators), handle high-dimensional data (e.g. convolutional NN for images), but they do not necessarily quantify nicely the uncertainties.

On the other hand, a GP can as well represent complex functions, can nicely quantify uncertainties non-local view on the data given by the kernel structure, but because it’s a non-parametric approach, it doesn't handle well high-dimensional data. We need to invert a matrix that has the dimensions of our data.

For learning dynamics, we need a methods that scores well in uncertainty quantification as well as handling of high-dimensional data.

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Deep Kernel Learning

THE BEST OF BOTH WORLDS

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Neural

Network

Gaussian

Process

+ representing complex functions

+ handling high-dimensional data

+ quantifying uncertainties (kernel)

Wilson, Andrew Gordon, et al. "Deep kernel learning." Artificial intelligence and statistics. PMLR, 2016.

Wilson, Andrew G., et al. "Stochastic variational deep kernel learning." Advances in neural information processing systems 29 (2016).

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Deep Kernel Learning for Dynamical Systems

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How to use Deep Kernel Learning for learning dynamical models:

from high-dimensional noisy data

and without labelled data

Neural

Network

Gaussian

Process

time

Data

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Deep Kernel Learning of Dynamical Models

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Botteghi, Nicolò, Mengwu Guo, and Christoph Brune. "Deep kernel learning of dynamical models from high-dimensional noisy data." Scientific reports 12.1 (2022): 21530.

Underlying Dynamics

Deep Kernel Learning Encoder

Deep Kernel Learning (Forward) Dynamical Model

Decoder

We start with a dynamical systems that is unknown, but that is measurable using one or more sensors. For example a camera. The data are fed into the first DKL (orange) that reduces the dimensionality of the data and learns latent state variables z and their distribution.

This latent variables are decoded via a decoder neural network to reconstruct a denoised version of the original measurements. The latent state is fed together with a control input (if available) to the second DKL model learning the actual one-step ahead dynamical model. The latent next state variables indicated by z t+1 can also be decoded for checking the predictions of the model.

As a numerical example we use the pendulum. We apply a control law u and collect high-dimensional and noisy measurements (each measurement is a RGB image of size 84x84x3). We use two DKL models because we try to decouple and quantify the uncertainties on the dynamics and well as the uncertainties on the data.

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Joint Training of the Models

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Underlying Dynamics

Deep Kernel Learning Encoder

Decoder

Deep Kernel Learning (Forward) Dynamical Model

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Numerical Results

DENOISING – MEASUREMENTS RECONSTRUCTION

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Underlying Dynamics

Deep Kernel Learning Encoder

Decoder

Deep Kernel Learning (Forward) Dynamical Model

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Numerical Results

RECOVERY OF COHERENT LATENT STATE VARIABLE MEANS

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Underlying Dynamics

Deep Kernel Learning Encoder

Decoder

Deep Kernel Learning (Forward) Dynamical Model

Measurement corruption

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Numerical Results

UNCERTAINTY QUANTIFICATION

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Latent state variables mean with uncertainty bounds (+/-std)

True state variables

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Conclusion & Future Work

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The combination of Neural Networks (local) and Gaussian Processes (non-local) is the key ingredient for learning without supervision:

reduced-order latent dynamical models and
to quantify uncertainties

Our framework can be used for:

Reinforcement Learning and model-based control
Bifurcation analysis in pattern forming systems (e.g. Turing patterns)
In combination with other (gray box) approaches for learning dynamics (e.g. physic-informed neural networks, SINDy, Hamiltonian neural networks)

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DEEP KERNEL LEARNING OF DYNAMICAL MODELS FROM HIGH-DIMENSIONAL NOISY DATA

NICOLÒ BOTTEGHI*, MENGWU GUO, CHRISTOPH BRUNE

�MATHEMATICS OF IMAGING & AI GROUP�DEPARTMENT OF APPLIED MATHEMATICS�

SIAM CSE

27 FEBRUARY 2023

Thanks for your attention!

Botteghi, Nicolò, Mengwu Guo, and Christoph Brune. "Deep kernel learning of dynamical models from high-dimensional noisy data." Scientific reports 12.1 (2022): 21530.

An implementation of our framework is available at: https://github.com/nicob15/DeepKernelLearningOfDynamicalModels

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Numerical Results

UNCERTAINTY QUANTIFICATION

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Latent state variables mean with uncertainty bounds (+/-std)

True state variables

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Deep Kernel Learning of Dynamical Models

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