Original: Approximate Uncertainty Propagation for Continuous Gaussian Process Dynamical Systems
Revised: Approximate Uncertainty Propagation for Continuous Gaussian Process Dynamical Systems

Original: Anonymous Author(s)

Original: Affiliation

Original: Address

Original: Abstract
Revised: Abstract

Original: One of the benefits of using Gaussian Process models is the availability of uncertainty quantification. However, when learning continuous dynamical systemsobtaining trajectories requires repeatedly mapping  uncertain inputs through thelearned nonlinear function, which is generally non-tractable.
Revised: When learning continuous dynamical systems with Gaussian Processes, computing trajectories requires repeatedly mapping the distributions of uncertain states through the distribution of learned nonlinear functions, which is generally intractable.

Original: As sampling-basedapproaches are computationally expensive, we consider approximations of theoutput and trajectory distribution.
Revised: Since sampling-based approaches are computationally expensive, we consider approximations of the output and trajectory distributions.

Original: We show that existing approaches make an incorrect implicit independence assumption and underestimate the model-induceduncertainty.
Revised: We show that existing methods make an incorrect implicit independence assumption and underestimate the model-induced uncertainty.

Original: We propose a piecewise linear approximation of the GP model anda numerical solver for efficient uncertainty estimates matching sampling-basedmethods at a lower computational cost
Revised: We propose a piecewise linear approximation of the GP model yielding a class of numerical solvers for efficient uncertainty estimates matching sampling-based methods

Original:  Introduction
Revised:  Introduction

Original: Gaussian Processes (GPs) represent a distribution over a function space, assumed to contain the trueunderlying function, conditioned on available pairs of input-output data.
Revised: Gaussian Processes (GPs) represent a distribution over a function space, assumed to contain the true underlying function, conditioned on available pairs of input-output data.

Original: This means, that GP modelscapture epistemic uncertainty [1], as we explicitly assume that with additional data the uncertaintyabout the true model decreases.
Revised: This means, that GP models capture epistemic uncertainty [1], as we explicitly assume that with additional data the uncertainty about the true model decreases.

Original: In this work, we consider learning the vector field of a dynamical system using GPs [2].
Revised: The context for this work is learning the vector field of a dynamical system using GPs [2].

Original: Specifically,we consider an ordinary differential equation (ODE)
Revised: Specifically, we consider an ordinary differential equation (ODE)

Original: [Equation]
Revised: [Equation]

Original: ) where f is a GP.
Revised: where f is a GP.

Original: This is conceptually similar to using NeuralODEs, which have a Neural Networkin place of the GP [3].
Revised: This is conceptually similar to using NeuralODEs, which use a Neural Network in place of the GP [3].

Original: We note explicitly, that the resulting model is not a stochastic differential equation or random dynamical system with uncertainty arising from inherent stochasticity [1].
Revised: We note explicitly that the resulting model is not a stochastic differential equation or random dynamical system with uncertainty arising from inherent stochasticity [1].

Original: While there is some noisein themeasured data, the GP model aims to identify the true underlying function.
Revised: While we assume the presence of measurement noise in the data, the GP model aims to identify the true underlying function.

Revised: The context of this work is also different from the contributions of Schober et al.

Revised: [4], who assume that the solution of the ODE is described by a GP, and the discrete steps of the solver represent observations affected by noise through computational error.

Original: This vector-field approach to learning the  continuous dynamics is in contrast to learning the flowmap via a state-space GP , which was done by several authors in recent years [4, 5, 6, 7].
Revised: A more popular alternative to learning the vector-field representing the continuous dynamics is directly learning the flow map via a state-space GP [5, 6, 7, 8]

Original: Thesemodels generally have the form
Revised: These models generally have the form

Original: [Equation]
Revised: [Equation]

Original: ) and from a given state directly predict the next state of the trajectory after some fixed time step.
Revised: directly predicting the next state of the trajectory after some fixed time step.

Original: Some of these models use Gaussian auto-regression, taking into account l past states [7, 8, 9].
Revised: Some of these models use Gaussian auto-regression, taking into account l past states [8, 9, 10].

Original: While these approaches  it easy  collected data which is often available as a time series they  a flow map with a fixed step size and are therefore unable to adapt based on the speed of the dynamics in a particular location
Revised: While these approaches make it easy to use collected data, which is often available as a time series, they are implicitly learning a flow map with a fixed step size and are therefore unable to adapt to speed of the dynamics in a particular location.

Original: In addition for many  systems the flow map is a highly complex function that cannot be expressed analytically, even for simple vectorfields.
Revised: In addition, for many dynamical systems the flow map is a highly complex function that cannot be expressed analytically, even for simple vector fields.

Original: Learning the continuous dynamics from time series data is more difficult, but also allows the inclusion of additional data, such as known terms from first principle models [3], symmetries [2], andsymplectic structure [10].
Revised: Learning the continuous dynamics from time series data is more difficult, but also allows the inclusion of additional data, such as known terms from first principle models [3], symmetries [2], and symplectic structure [11].

Original: In most cases, we obtain a trajectory from the continuous vector field via numerical integrationmethods.
Revised: In most cases, we obtain a trajectory from the continuous vector field via numerical integration methods.

Original: The simplest such method is the explicit Euler
Revised: The simplest such method is the explicit Euler

Original: [Equation]
Revised: [Equation]

Original: where h is the step size, which can be varied over subsequent steps.
Revised: where h is the step size, which can be varied over subsequent steps.

Original: There exists a wide varietyof more complex numerical solvers, differing in properties like the order of convergence, stability,or computational cost that make them well suited for specific classes of systems.
Revised: There exists a wide range of more complex numerical solvers, differing in their order of convergence, stability, or computational cost.

Original: One of the keymotivations for this work is applying those methods to GP-based ODEs.
Revised: One of the key motivations for this work is applying those methods to GP-based ODEs and including uncertainty propagation.

Original: The challenge with uncertainty propagation is that mapping a random variable with a tractable,parametrizable distribution, like a normal distribution, through a nonlinear function is generallyintractable.
Revised: The challenge with uncertainty propagation is that mapping a random variable with a tractable, parametrizable distribution, like a normal distribution, through a nonlinear function is generally intractable.

Original: In the context of dynamical system models, this problem is compounded.
Revised: In the context of dynamical system models, this problem is compounded.

Original: For state-space GPs, weneed to repeatedly map a random variable X n through the distribution of nonlinear functions represented by the GP, to obtain the distribution of X n +1 , X n +2 and so forth.
Revised: For state-space GPs, one repeatedly maps a random variable X n through the distribution of nonlinear functions represented by the GP to obtain the distribution of X n +1 , X n +2 and so forth.

Original: In the context of this work,the vector field is non-linear, which means that we obtain a distribution of gradients f ( X ) for state X , which in the context of numerical solvers needs to be combined with the distribution of currentstate X n and, for higher order methods, the gradients at other states.
Revised: In the context of this work, the vector field is non-linear, which means that we obtain a distribution of gradients f ( X ) for state X , which for numerical solvers needs to be combined with the distribution of current state X n and, for higher order methods, the gradients at other states.

Original: To our knowledge, this work is the first to consider approximate uncertainty propagation throughnumerical integrators for ODEs.
Revised: To our knowledge, this work is the first to consider approximate uncertainty propagation through numerical integrators for ODEs.

Original: However, there has been some work onpropagation uncertainty instate space GPs.
Revised: However, there has been some work on uncertainty propagation in state-space GPs.

Original: Girard et al. published a series of results on approximating the output distribution when mappinga normal distribution through a GP.
Revised: Girard et al. published a series of results on approximating the output distribution when mapping a normal distribution through a GP.

Original: In [11] the authors show an approximation for the mean andvariance of the output based on Taylor series approximations of the predicted mean and variance,using derivatives for the GP kernel function.
Revised: In [12] the authors show an approximation for the mean and variance of the output based on Taylor series approximations of the predicted mean and variance, using derivatives for the GP kernel function.

Original: A subsequent result [6] shows that when using thesquared exponential kernel specifically, the mean and the variance of the output distribution canbe obtained analytically.
Revised: A subsequent result [7] showed that when using the squared exponential kernel specifically, the mean and the variance of the output distribution can be obtained analytically.

Original: These results have been called moment matching and have been used forstate-space GPs [4].
Revised: These results are called moment matching and have been used for state-space GPs [5].

Original: Other work has applied sampling based approaches tostate-space GPs to estimate uncertainty [12,13].
Revised: Other work has applied sampling-based approaches to estimate uncertainty in state-space GPs [13] and for continuous GP dynamics [14].

Original: To construct a similarapproach for this work, one option is to sample a representative numberof vector field functions from the GP distribution and integrate each function over the desired timeframe to get an estimate for the state distributions.
Revised: For comparison, we use a similar option in this work: We sample a representative number of vector field functions from the GP distribution and integrate each function over the desired time frame to get an estimate of the state distributions.

Original: Weconsider briefly heuristically the expected outcome.
Revised: We briefly discuss the expected outcome.

Original: We assume that the data-generating systemhas an unknown, but fixed model.
Revised: We assume that the data-generating system has an unknown, but fixed model.

Original: While the observations are noisy, the implicit parameters of thesystem are constant, and the finite GP conditioned on the collected data represents a distributionover different possible models and their likelihood.
Revised: While the observations are noisy, the implicit parameters of the system are constant, and the finite GP conditioned on the collected data represents a distribution over different possible models and their likelihood.

Original: An ideal experimental setup would allow us toadd additional data and narrow down to the true underlying model for the system in the Reproducing Kernel Hilbert Space.
Revised: An ideal experimental setup would allow us to add additional data and narrow down to the true underlying model for the system in the Reproducing Kernel Hilbert Space.

Original: When considering the resulting trajectories, a fixed point of the system shouldbecome a fixed point distribution which a numerical integrator converges to through repeated steps,assuming that the step size is chosen from within the stability region.
Revised: When considering the resulting trajectories, a fixed point of the system should become a fixed point distribution which a numerical integrator converges to through repeated steps, assuming that the step size is chosen from within the stability region.

Revised: Our main contributions are the following: We demonstrate how existing approaches to uncertainty propagation based on approximating the output distribution of the exact model make an incorrect implicit independence assumption.

Revised: We further propose an alternative approach based on a piecewise linear model approximation that can be solved exactly, resulting in what we call the PULL (Propagating Uncertainty through Local Linearization) class of solvers.

Original:  Review2.1 Gaussian Processes
Revised:  Review 2.1 Gaussian Processes

Original: We first review briefly the basics of Gaussian Processes [14].
Revised: We first review briefly the basics of Gaussian Processes [15].

Original: We assume we have N output observations
Revised: We assume we have N output observations

Original: y i =
Revised: y i =

Original: f ( x i )
Revised: f ( x i )

Original: + ϵ
Revised: + ϵ

Original: i with Gaussian noise ϵ i ∼ N (0 , σ 2 n ) for known inputs x i to an unknownfunction f : R m → R d .
Revised: i with Gaussian noise ϵ i ∼ N (0 , σ 2 n ) for known inputs x i to an unknown function f : R m → R d .

Original: We now express f as a GP, that is as a distribution over functions.
Revised: We now express f as a GP, that is as a distribution over functions.

Original: We specify a prior P ( f ) in the form of a mean function m ( x ) , generally assumed to be zero, and a kernelfunction k ( x, x ′ ) , which uniquely determines a Reproducing Kernel Hilbert Space containing allpossible realizations of f .
Revised: We specify a prior P ( f ) in the form of a mean function m ( x ) , generally assumed to be zero, and a kernel function k ( x, x ′ ) , which uniquely determines a Reproducing Kernel Hilbert Space containing all possible realizations of f .

Original: For any finite subset of random variables f i corresponding to known inputs x i the GP determines ajoint Gaussian distribution, and by conditioning on the initial observations
Revised: For any finite subset of random variables f i corresponding to known inputs x i the GP determines a joint Gaussian distribution, and by conditioning on the initial observations

Original: D =
Revised: D =

Original: ( x i , y i )
Revised: ( x i , y i )

Original: we obtainthe posterior distribution P ( f |D ) , which allows us to predict the output f
Revised: we obtain the posterior distribution P ( f |D ) , which allows us to predict the output f

Original: (ˆ x ) at a new input ˆ x .
Revised: (ˆ x ) at a new input ˆ x .

Original: Aswe are considering a distribution of functions, we will obtain a distribution of outputs with the meanµ f (ˆ x ) and variance σ 2 f (ˆ x ) determined by
Revised: As we are considering a distribution of functions, we will obtain a distribution of outputs with the mean µ f (ˆ x ) and variance σ 2 f (ˆ x ) determined by

Original: [Equation]
Revised: [Equation]

Original: where x = [ x 1 , . .
Revised: where x = [ x 1 , . .

Original: , x N ] , y =
Revised: , x N ] , y =

Original: [ y 1 , . . .
Revised: [ y 1 , . . .

Original: , y N ] , and K x,x ′ =
Revised: , y N ] , and K x,x ′ =

Original: [ k ( x i , x ′ j )]
Revised: [ k ( x i , x ′ j )]

Original: i,j is the kernel or covariancematrix.
Revised: i,j is the kernel or covariance matrix.

Original: The most commonly used Kernel is the squared exponential kernel, which we also use in theexamples shown in this work.
Revised: The most commonly used Kernel is the squared exponential kernel, which we also use in the examples shown in this work.

Original: It is defined as
Revised: It is defined as

Original: [Equation]
Revised: [Equation]

Original: (5) where W = diag ( w 1 , . . . , w M ) is the diagonal matrix of length scales.
Revised: where W = diag ( w 1 , . . . , w M ) is the diagonal matrix of length scales.

Original: In the context of dynamical systems the dimension of the input and the output are the same, hencem =
Revised: In the context of dynamical systems the dimension of the input and the output are the same, hence m =

Original: d While a number of vector-valued kernels exist [15], it is often assumed that each outputcan be treated independently with d GPs f : R m → R .
Revised: d While a number of vector-valued kernels exist [16], it is often assumed that each output can be treated independently with d GPs f : R m → R .

Original: In this work, we further restrict to onedimensional dynamical systems, allowing us to consider only one-dimensional GPs.
Revised: In this work, we further restrict to onedimensional dynamical systems, allowing us to consider only one-dimensional GPs.

Original: 2.2 Sampling GPs
Revised: 2.2 Sampling GPs

Original: GPs are distributions over, generally infinite dimensional, function spaces and therefore inherentlydifficult to sample numerically.
Revised: GPs are distributions over, generally infinite dimensional, function spaces and therefore inherently difficult to sample numerically.

Original: To obtain some approximate function realizations f ∗ from the posterior distribution at input locations x ∗ , the standard option [16] is to generate normally distributedrandom variables ζ ∼ N (0 , I ) , and transform them according to the GP posteriorf
Revised: To obtain some approximate function realizations f ∗ from the posterior distribution at input locations x ∗ , the standard option [17] is to generate normally distributed random variables ζ ∼ N (0 , I ) , and transform them according to the GP posterior

Original: [Equation]
Revised: [Equation]

Original: Here, ( · ) 1 / 2 indicates a matrix square root like the Cholesky factor, m ∗ is the GP mean µ f ( x ∗ ) from(4a) and K ∗ , ∗ is the covariance matrix for the sample input x ∗ via (4b).
Revised: Here, ( · ) 1 / 2 indicates a matrix square root like the Cholesky factor, m ∗ is the GP mean µ f ( x ∗ ) from (4a) and K ∗ , ∗ is the covariance matrix for the sample input x ∗ via (4b).

Original: In essence, this is a gridbased approach, and while it is numerically exact, it is computationally costly due to cubic scalingwith the number of function values sampled.
Revised: In essence, this is a gridbased approach, and while it is numerically exact, it is computationally costly due to cubic scaling with the number of function values sampled.

Original: To evaluate the function at arbitrary locations betweengrid points, one can use standard interpolation methods.
Revised: To evaluate the function at arbitrary locations between grid points, one can use standard interpolation methods.

Original: To mitigate the cubic scaling, a number of alternate algorithms have been developed.
Revised: To mitigate the cubic scaling, a number of alternate algorithms have been developed.

Original: Wilson et al.[16] propose a combination of decoupled bases, specifically Fourier basis functions [17], and kernelbases via a sparse GP approach.
Revised: Wilson et al. [17] propose a combination of decoupled bases, specifically Fourier basis functions [18], and kernel bases via a sparse GP approach.

Original: This has also been used in other recent work [10, 13] in the contextof vector field models.
Revised: This has also been used in other recent work [11, 14] in the context of vector field models.

Original: An approach of this kind was also mentioned in [12], along with a memory based approach, whichgenerates samples subsequently while conditioning each sample on previous ones.
Revised: An approach of this kind was also mentioned in [13], along with a memory based approach, which generates samples subsequently while conditioning each sample on previous ones.

Original: It has also beenused in a similar context as in this work [2].
Revised: It has also been used in a similar context as in this work [2].

Original: For this work, we will use samples via (6) as a source of ground truth for comparison, as it is themost accurate option to compare against. 
Revised: For this work, we will use samples via (6) as a source of ground truth for comparison, as it is the most accurate option to compare against. [Figure]

Original:  Linear Approximations
Revised:  Linear Approximations

Original: Using a linear approximation is attractive in many applications, as it avoids many of the complexbehaviours that appear in nonlinear functions.
Revised: Using a linear approximation is attractive in many applications, as it avoids many of the complex behaviours that appear in nonlinear functions.

Original: However, the question is what part of the problemrequires, or is best suited for, linearization.
Revised: However, the question is what part of the problem requires, or is best suited for, linearization.

Original: 3.1 Approximating the Output Distribution
Revised: 3.1 Approximating the Output Distribution

Original: Even though the distribution of each subsequent state random variable X n is generally not normallydistributed, we can compute its mean and variance and instead of the unknown general distributionuse a normal distribution with the same moments.[Figure]
Revised: Even though the distribution of each subsequent state random variable X n is generally not normally distributed, we can compute its mean and variance and instead of the unknown general distribution use a normal distribution with the same moments.

Original: Applying the approach of Girard et al.
Revised: Applying the approach of Girard et al.

Original: [6] in the context of the explicit Euler method (3), an initialidea might use to apply it to the GP f
Revised: [7] in the context of the explicit Euler method (3), an initial idea might use to apply it to the GP f

Original: and then treat the sum of X n and f ( X n ) as independent.
Revised: and then treat the sum of X n and f ( X n ) as independent.

Original: However, this leads to an inaccurate growth in the uncertainty, as they are clearly not independent.
Revised: However, this leads to an inaccurate growth in the uncertainty, as they are clearly not independent.

Original: Instead, we can define a new function g ( x ) =
Revised: Instead, we can define a new function g ( x ) =

Original: x + hf ( x ) and derive the equivalent expressions.
Revised: x + hf ( x ) and derive the equivalent expressions.

Original: Letthe input X n be distributed as
Revised: Let the input X n be distributed as

Original: [Equation]
Revised: [Equation]

Original: and the output of the GP for a particular sample x ∗ of X n is distributed as
Revised: and the output of the GP for a particular sample x ∗ of X n is distributed as

Original: [Equation]
Revised: [Equation]

Original: The distribution of X n +1 =
Revised: The distribution of X n +1 =

Original: g ( X n ) will generally not be normal, but we can compute its meanν n [-+
-]
Revised: g ( X n ) will generally not be normal, but we can compute its mean ν n [-+
-]+1 =

Original: = E [ x n +1 ] and variance Σ n +1 = var ( x n +1 ) .
Revised: E [ x n +1 ] and variance Σ n +1 = var ( x n +1 ) .

Original: From the law of iterated expectations follows that the mean of the output distribution of one Eulerstep is given by
Revised: From the law of iterated expectations follows that the mean of the output distribution of one Euler step is given by

Original: [Equation]
Revised: [Equation]

Original: The corresponding variance is given by
Revised: The corresponding variance is given by

Original: [Equation]
Revised: [Equation]

Original: 10) and from the law of total variance follows
Revised: and from the law of total variance follows

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: This leaves the covariance between X n and f ( X n ) .
Revised: This leaves the covariance between X n and f ( X n ) .

Original: In [6], the authors include a linearized expression in the footnote.
Revised: In [7], the authors include a linearized expression in the footnote.

Original: In [7], the authors suggest
Revised: In [8], the authors suggest

Original: [Equation]
Revised: [Equation]

Original: 12) which results in
Revised: which results in

Original: [Equation]
Revised: [Equation]

Original: [Equation] To compute values for the mean m ( ν n +1 , Σ n +1 ) and the variance v ( ν n +1 , Σ n +1 ) , we need expressions for the expected values E [ µ ( x n )] , E [ σ 2 ( x n )] , E [ µ ( x n ) 2 ] and E [ x n µ ( x n )] .
Revised: To compute values for the mean m ( ν n +1 , Σ n +1 ) and the variance v ( ν n +1 , Σ n +1 ) , we need expressions for the expected values E [ µ ( x n )] , E [ σ 2 ( x n )] , E [ µ ( x n ) 2 ] and E [ x n µ ( x n )] .

Original: For the first threeterms, we can find analytical expressions [11] for the squared exponential kernel, and approximateexpressions using linearization for all other kernels in [6].
Revised: For the first three terms, we can find analytical expressions [12] for the squared exponential kernel, and approximate expressions using linearization for all other kernels in [7].

Original: In [7] we also find the previously listedexpressions for the squared exponential kernel and in addition an expression for the term E [ µ ( x n )]for the covariance.
Revised: In [8] we also find the previously listed expressions for the squared exponential kernel and in addition an expression for the term E [ µ ( x n )] for the covariance.

Original: However, we will demonstrate in the next section that this approach makes an incorrect implicitassumption about the independence of the states X n and the model f .
Revised: However, we will demonstrate in the next section that this approach makes an incorrect implicit assumption about the independence of the states X n and the model f .

Original: 3.2 A linear model
Revised: 3.2 A linear model

Original: Consider a very simple model, a distribution of stable linear functions of the formf
Revised: Consider a very simple model, a distribution of stable linear functions of the form

Original: [Equation]
Revised: [Equation]

Original: with a ∈ R + , B ∼ N (0 , β ) , (14) as shown in Fig.
Revised: 14) as shown in Fig.

Original: 1, which means that
Revised: 1, which means that

Original: [Equation]
Revised: [Equation]

Original: The integrals from the previous sections resolve to
Revised: The integrals from the previous sections resolve to

Original: [Equation]
Revised: [Equation]

Original: 16b) [Equation] 16d
Revised: E

Original: [Equation]
Revised: [Equation]

Original: which results in the iterations
Revised: which results in the iterations

Original: [Equation]
Revised: [Equation]

Original: 17a

Original: [Equation]

Original: 17b)

Original: In addition, the analytical flow for (14) is
Revised: In addition, the analytical flow for (14) is

Original: [Equation]
Revised: [Equation]

Original: (18) which has the time-dependent mean and varianceν
Revised: which has the time-dependent mean and variance

Original: [Equation]

Original: 19b

Original: [Equation]
Revised: [Equation]

Original: Here, we assume that the initial value and the model parameter B are independent, so cov ( X 0 , B ) =0 .
Revised: Here, we assume that the initial value and the model parameter B are independent, so cov ( X 0 , B ) = 0 .

Original: Note that the variance from the Euler step (17b) and the exact solution (19b) do not have the samefixed points:
Revised: Note that the variance from the Euler step (17b) and the exact solution (19b) do not have the same fixed points:

Original: [Equation]
Revised: [Equation]

Original: The result for ˆΣ exact matches the variance of the fixed point distribution of the model (14).
Revised: The result for ˆΣ exact matches the variance of the fixed point distribution of the model (14).

Original: The Euler steps converge to a fixed point that depends on the step size h , as we demonstrate in Fig.1.
Revised: The Euler steps converge to a fixed point that depends on the step size h , as we demonstrate in Fig.

Original: This is problematic, as the dynamics should not depend on solver parameters.
Revised: This is problematic, as the dynamics should not depend on solver parameters.

Original: Remarkably, weobtain a similar behaviour if we consider (19b) as an iterable function of X i and step size h , whichhas the fixed point
Revised: Remarkably, we obtain a similar behaviour if we consider (19b) as an iterable function of X i and step size h , which has the fixed point

Original: [Equation]
Revised: [Equation]

Original: This is surprising, as the flow map should have the semi-group property, such that φ t ◦ φ s X 0 =
Revised: This is surprising, as the flow map should have the semi-group property, such that φ t ◦ φ s X 0 =

Original: φ t + s X 0 .
Revised: φ t + s X 0 .

Original: Therefore, there should be no difference between applying the flow over multiple smallerintervals compared to one large interval.
Revised: Therefore, there should be no difference between applying the flow over multiple smaller intervals compared to one large interval.

Original: Taking a closer look at (18), we note that it is made up of two terms.
Revised: Taking a closer look at (18), we note that it is made up of two terms.

Original: For stable systems, the first termcaptures the transient effect of the initial value, whereas the second term represents the asymptoticbehaviour determined by the model parameter B .
Revised: For stable systems, the first term captures the transient effect of the initial value, whereas the second term represents the asymptotic behaviour determined by the model parameter B .

Original: If we apply the flow to some state X n instead ofthe initial value X 0 , the state has already been affected by the model and some decay of the initialvalue has occurred.
Revised: If we apply the flow to some state X n instead of the initial value X 0 , the state has already been affected by the model and some decay of the initial value has occurred.

Original: Thus, the iterative schemes mentioned above are equivalent to starting over withan independent initial value after each time step.
Revised: Thus, the iterative schemes mentioned above are equivalent to starting over with an independent initial value after each time step.

Original: This can be seen in more detail in Fig.
Revised: This can be seen in more detail in Fig.

Original: The repeated transient explains the additional factor in ˆΣ iter.
Revised: The repeated transient explains the additional factor in ˆΣ iter.

Original: flow .
Revised: flow .

Original: As ah → ∞ , we see either increasingly large steps or increasingly fast dynamics, which means that the transient behaviour fullyconcludes in each step.
Revised: As ah → ∞ , we see either increasingly large steps or increasingly fast dynamics, which means that the transient behaviour fully concludes in each step.

Original: The additional factor in ˆΣ euler also depends on ah .
Revised: The additional factor in ˆΣ euler also depends on ah .

Original: However, the expressionbreaks down as ah → 2 , which matches the stability region of the explicit Euler method.
Revised: However, the expression breaks down as ah → 2 , which matches the stability region of the explicit Euler method.

Original: Therefore, the issue lies in the covariance between the states and the model parameter.
Revised: Therefore, the issue lies in the covariance between the states and the model parameter.

Original: While it isvalid to assume that cov ( X 0 , B ) = 0 , all subsequent states
Revised: While it is valid to assume that cov ( X 0 , B ) = 0 , all subsequent states

Original: X
Revised: X

Original: i depend on the model, specificallythis case on the random model parameter B , which means that they are not independent.
Revised: i depend on the model, specifically this case on the random model parameter B , which means that they are not independent.

Original: Instead, wefind
Revised: Instead, we find

Original: [Equation]
Revised: [Equation]

Original: Appending this to (19b) yields the correct result, as demonstrated on the right side of Fig.
Revised: Appending this to (19b) yields the correct result, as demonstrated on the right side of Fig.

Original: If X n is the result of subsequent Euler steps,
Revised: If X n is the result of subsequent Euler steps,

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: we find a telescope sum, such that
Revised: we find a telescope sum, such that

Original: [Equation]
Revised: [Equation]

Original: which results in
Revised: which results in

Original: [Equation]
Revised: [Equation]

Original: Using this expression, we find that the variance from the iterative solutions matches the analyticsolution and the variance of the sampled solutions (see Fig. 1).
Revised: Using this expression, we find that the variance from the iterative solutions matches the analytic solution and the variance of the sampled solutions (see Fig. 1).

Original: This result is noteworthy beyond the continuous dynamics discussed in this work.
Revised: This result is noteworthy beyond the continuous dynamics discussed in this work.

Original: Applying the flowmap iteratively without the additional covariance term for some fixed step size h is equivalent torepeatedly applying the learned flow map in the state-space GP context.
Revised: Applying the flow map iteratively without the additional covariance term for some fixed step size h is equivalent to repeatedly applying the learned flow map in the state-space GP context.

Original: As this phenomenon is thusfar unaddressed in previous work using moment matching for multiple step ahead predictions, theuncertainty has likely been underestimated.
Revised: As this phenomenon is thus far unaddressed in previous work using moment matching for multiple step ahead predictions, the uncertainty has likely been underestimated.

Original: However, as the effect depends on the implicit step sizeh in each step of the flow and might be affected by the inclusion of multiple previous points in theauto-regression context, further study is needed to understand the scope.
Revised: However, as the effect depends on the implicit step size h in each step of the flow and might be affected by the inclusion of multiple previous points in the auto-regression context, further study is needed to understand the scope.

Original: Local Linearization

Original: We demonstrated in the previous section that we have to take the history of each state into accountto take another iteration step.

Original: However, doing so for a general nonlinear GP model is substantiallymore complicated.

Original: To address this, instead of approximating the output distribution of the exact GP, we propose a piecewise linear approximation of the GP by defining a series of linear ODEs.

Original: Therefore, approximatethe model as a whole with a model for which we can compute the exact state distributions.

Original: We define

Original: [Equation]

Original: [Equation]

Original: −−   
Revised: −−

Original: −−−    x f ( x )
Revised: −−−    x f ( x )

Original: GP Samplesdata 0 .2 .

Revised: GP Samplesdata 0 .2 .

Original: .time
Revised: .time

Original: t tr a j ec t
Revised: t tr a j ec t

Original: o r y x (t) density samplemean samplestandarddeviationapprox.meanapprox.standarddeviation 1 . 0 1 . 2 1 .
Revised: o r y x (t) density samplemean samplestandarddeviationapprox.meanapprox.standarddeviation 1 . 0 1 . 2 1 .

Original: 8    state x(2.5) p r o b a b ili t y  wherea i
Revised: 8    state x(2.5) p r o b a b ili t y  Local Linearization

Revised: We demonstrated in the previous section that we have to take the history of each state into account to take another iteration step.

Revised: However, doing so for a general nonlinear GP model is substantially more complicated.

Revised: To address this, instead of approximating the output distribution of the exact GP, we propose a piecewise linear approximation of the GP by defining a series of linear ODEs.

Revised: Therefore, approximate the model as a whole with a model for which we can compute the exact state distributions.

Original:  = µ f ( ν i ) , and B i =f
Revised: We define [Equation] (27) where a i = µ f ( ν i ) , and B i =

Original:  ( ν i ) − ai
Revised: f ( ν i ) − a

Original:  X i , (28) which match (14) as we have
Revised: i X i , (28) which match (14) as we have

Original: [Equation]
Revised: [Equation]

Original: While it would be more accurate to have a i as a random variable, similar to Nghiem’s linGP [9],this would introduce the product of two random variables and additional covariances.
Revised: While it would be more accurate to have a i as a random variable, similar to Nghiem’s linGP [10], this would introduce the product of two random variables and additional covariances.

Original: With this approximation we can take Euler steps on the linear system, resulting in the iterativescheme
Revised: With this approximation we can take Euler steps on the linear system, resulting in the iterative scheme

Original: [Equation]
Revised: [Equation]

Original: 30a) [Equation] Our approximation allows us to find an expression for the covariance in (30b) using a similar telescope sum as in (25).
Revised: Our approximation allows us to find an expression for the covariance in (30b) using a similar telescope sum as in (25).

Original: This results in
Revised: This results in

Original: [Equation]
Revised: [Equation]

Original: This expression requires storing all previous a i , but since the trajectory states X i already need tobe stored, the solver still has only linearly scaling storage requirements.
Revised: This expression requires storing all previous a i , but since the trajectory states X i already need to be stored, the solver still has only linearly scaling storage requirements.

Original: The bigger computationalchallenge is computing the covariances cov ( ν i , ν n ) between the current state and all previous ones.
Revised: The bigger computational challenge is computing the covariances cov ( ν i , ν n ) between the current state and all previous ones.

Original: This is done via (4b) which causes quadratic scaling with the number of steps already taken.
Revised: This is done via (4b) which causes quadratic scaling with the number of steps already taken.

Original: Fortunately, there are two possible justifications to truncate the series in (31).
Revised: Fortunately, there are two possible justifications to truncate the series in (31).

Original: Firstly, in practice, weoften operate within the basin of attraction of a fixed point.
Revised: Firstly, in practice, we often operate within the basin of attraction of a fixed point.

Original: Then it will generally be the case thata i < 0 and (cid:81) n − 1 j = i +

Revised: Then it will generally be the case that a i < 0 and (cid:81) n − 1 j = i +

Original: (1 + a j h ) → 0 for n → ∞ .
Revised: (1 + a j h ) → 0 for n → ∞ .

Original: Secondly, when using a stationary kernel for the
Revised: Secondly, when using a stationary kernel for the

Original: GP, the covariance is determined by the distance of the two points and a length scale, where distantpoints are assumed to be nearly uncorrelated.
Revised: GP, the covariance is determined by the distance of the two points and a length scale, where distant points are assumed to be nearly uncorrelated.

Original: Hence, we will generally find cov ( ν i , ν n ) ≈ 0 fori ≪ n .
Revised: Hence, we will generally find cov ( ν i , ν n ) ≈ 0 for i ≪ n .

Original: Both terms have complementary behaviour.
Revised: Both terms have complementary behaviour.

Original: Under stable dynamics, successive x n will be close andtherefore correlated, but since we have a i < 0 we can truncate based on the first term.
Revised: Under stable dynamics, successive x n will be close and therefore correlated, but since we have a i < 0 we can truncate based on the first term.

Original: Under unstabledynamics, we will have a i > 0 , but successive points will be further apart and the covariance termdecreases faster.
Revised: Under unstable dynamics, we will have a i > 0 , but successive points will be further apart and the covariance term decreases faster.

Original: As the terms of the series are simple to compute, it is possible to implement adaptive truncationbased on the smallest term in the sum.
Revised: As the terms of the series are simple to compute, it is possible to implement adaptive truncation based on the smallest term in the sum.

Original: We also note that we have made no assumption about theunderlying kernel beyond stationarity.
Revised: We also note that we have made no assumption about the underlying kernel beyond stationarity.

Original: .10 .−−−−−time

Original: [-t l og m e a n e rr o r 0 .2 .
-]
Revised: [-t l og m e a n e rr o r 0 .2 .
-]

Original: .time t s t a nd a r d d e v i a t i o n σ

Original: [Figure]

Original: samplingh=0.02h=0.01h=0.005h=0.

Original: [-bufferfull 0 .2 .
-]
Revised: [-bufferfull 0 .2 .
-]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: d d e v i a t

Original: i

Original: [Figure]

Original:  Numerical Experiments
Revised:  Numerical Experiments

Original: To demonstrate the effectiveness of the local linearization, we consider the ODE˙ x = g ( x )
Revised: To demonstrate the effectiveness of the local linearization, we consider the ODE

Original: = x cos ( x ) , (32) and sample 10 data points in the interval [ − 4 , 4] and condition a GP with zero mean and the squaredexponential kernel (5).
Revised: [Equation] and sample 10 data points in the interval [ − 4 , 4] and condition a GP with zero mean and the squared exponential kernel (5).

Original: Via (6), we generate 3000 samples (as seen on the left in Fig. 2) and integrateeach one with initial values from 150 samples of the input distribution N (0 . 6 , 0 . 005) , resulting inthe distribution of trajectories shown on the right of Fig. 2.
Revised: Via (6), we generate 3000 samples (as seen on the left in Fig. 2) and integrate each one with initial values from 150 samples of the input distribution N (0 . 6 , 0 . 005) , resulting in the distribution of trajectories shown on the right of Fig. 2.

Original: We then apply the local-linearization Euler’s method to the same GP, starting from the same initialdistribution, and see that the results agree very well.
Revised: We then apply the local-linearization Euler’s method to the same GP, starting from the same initial distribution, and see that the results agree very well.

Original: However, using the full history of past states makes this method very expensive.
Revised: However, using the full history of past states makes this method very expensive.

Original: Therefore, weconsider using only a truncated version of the sum in (31) with only a limited buffer of past slopesa i , as discussed in the previous section, and also highlight the effect of different step sizes.
Revised: Therefore, we consider using only a truncated version of the sum in (31) with only a limited buffer of past slopes a i , as discussed in the previous section, and also highlight the effect of different step sizes.

Original: Theresults in Fig.
Revised: The results in Fig.

Original: 2 show that the behaviour of the mean is as expected for Euler’s method, the errordecreases with a smaller step size.
Revised: 2 show that the behaviour of the mean is as expected for Euler’s method, the error decreases with a smaller step size.

Original: We also see that if the buffer is insufficient, the variance is again step size dependent, but increasingthe buffer results in convergence towards the correct value.
Revised: We also see that if the buffer is insufficient, the variance is again step size dependent, but increasing the buffer results in convergence towards the correct value.

Original: While a smaller step size does decreasethe error in both the mean and the variance, the quadratic scaling of the computational cost for eachstep with the buffer size, as a result of (4b), provides an incentive to maximize the step size.
Revised: While a smaller step size does decrease the error in both the mean and the variance, the quadratic scaling of the computational cost for each step with the buffer size, as a result of (4b), provides an incentive to maximize the step size.

Original: Our solver returns accurate results as the estimated trajectory distribution converges towards a deterministic, model-dependent fixed point distribution.
Revised: Our solver returns accurate results as the estimated trajectory distribution converges towards a deterministic, model-dependent fixed point distribution.

Original: While the results can still be affected by solverparameters, their effect is understood and can be mitigated by adjusting the parameters.
Revised: While the results can still be affected by solver parameters, their effect is understood and can be mitigated by adjusting the parameters.

Original: Our slightly underestimates variance compared to the ground truth sampling, likely due to linearization error.
Revised: Our slightly underestimates variance compared to the ground truth sampling, likely due to linearization error.

Original: One option to improve the piecewise linear approximation might be incorporating thework of Girard et al.
Revised: One option to improve the piecewise linear approximation might be incorporating the work of Girard et al.

Original: [6] in a way that correctly incorporates the correlation of subsequent states.
Revised: [7] in a way that correctly incorporates the correlation of subsequent states.

Original: Inthis context, we note that using (13) is conceptually equivalent to using our solver with zero buffer.
Revised: In this context, we note that using (13) is conceptually equivalent to using our solver with zero buffer.

Original: 5.1 Limitations
Revised: 5.1 Limitations

Original: The linearization-based approach detailed in this work has substantially lower computational costthan a sampling-based approach, but has a fundamental limitation.
Revised: The linearization-based approach detailed in this work has substantially lower computational cost than a sampling-based approach, but has a fundamental limitation.

Original: While linear models only have asingle fixed point, nonlinear models introduce a wide array of additional behaviours, like additionalfixed points and complex basins of attraction.
Revised: While linear models only have a single fixed point, nonlinear models introduce a wide array of additional behaviours, like additional fixed points and complex basins of attraction.

Original: In the context of this section’s model (32), we note that there is an unstable fixed point in 0 , andtrajectories starting to the right of it will converge towards π/ 4 , as shown in Fig.
Revised: In the context of this section’s model (32), we note that there is an unstable fixed point in 0 , and trajectories starting to the right of it will converge towards π/ 4 , as shown in Fig.

Original: However, trajec- tories starting on the left of 0 will converge towards − π/ 4 .
Revised: However, trajectories starting on the left of 0 will converge towards − π/ 4 .

Original: This means that starting with a distribution of initial values near 0 , the distribution of trajectories will be bi-modal, as shown in Fig.4.
Revised: This means that starting with a distribution of initial values near 0 , the distribution of trajectories will be bi-modal, as shown in Fig.

Original: Thisbehaviour fundamentally cannot be captured by a linear approximation that assumes a uni-modaldistribution for all states.
Revised: This behaviour fundamentally cannot be captured by a linear approximation that assumes a uni-modal distribution for all states.

Original: However, Fig.
Revised: However, Fig.

Original: 4 highlights the usefulness of a cheap approximate solver.
Revised: 4 highlights the usefulness of a cheap approximate solver.

Original: Only solving the ODEdefined by the GP mean would completely miss the presence and effects of an added fixed point.
Revised: Only solving the ODE defined by the GP mean would completely miss the presence and effects of an added fixed point.

Revised: .10 .−−−−−time

Original: {+t l og m e a n e rr o r 0 .2 .
+}
Revised: {+t l og m e a n e rr o r 0 .2 .
+}

Revised: .time t s t a nd a r d d e v i a t i o n σ buffer =samplingh=0.02h=0.01h=0.005h=0.

Original: {+bufferfull 0 .2 .
+}
Revised: {+bufferfull 0 .2 .
+}

Revised: .time t s t a nd a r d d e v i a t i o n σ h = 0.

Original:  Our solver converges to the sensible fixed point, as the initial value distribution has its mean to theright of 0 , but the variance indicates increased initial uncertainty compared to the previous resultfrom Fig.
Revised: samplingbuffer [Figure] Our solver converges to the sensible fixed point, as the initial value distribution has its mean to the right of 0 , but the variance indicates increased initial uncertainty compared to the previous result from Fig.

Original:  Conclusion
Revised:  Conclusion

Original: Accurately evolving dynamical systems expressed by GPs, either by numerically integrating a continuous vector field or subsequently applying a learned flow map, introduces a correlation betweenthe distribution of functions expressed by the GP and subsequent states.
Revised: Accurately evolving dynamical systems expressed by GPs, either by numerically integrating a continuous vector field or subsequently applying a learned flow map, introduces a correlation between the distribution of functions expressed by the GP and subsequent states.

Original: We show that current approaches incorrectly assume independence between the distribution of the states and the GP distribution, resulting in an incorrect step size dependence of the trajectory variance.
Revised: We show that current approaches incorrectly assume independence between the distribution of the states and the GP distribution, resulting in an incorrect step size dependence of the trajectory variance.

Original: We derive and demonstrate the correct correlation for a simple linear model, and leverage thosefindings to combine local linearization and the explicit Euler method, resulting in a computationallyefficient and accurate solver.
Revised: We derive and demonstrate the correct correlation for a simple linear model, and leverage those findings to combine local linearization and the explicit Euler method, resulting in a computationally efficient and accurate solver.

Original: A possible application for this solver lies in creating a likelihood-based cost function for training
Revised: A possible application for this solver lies in creating a likelihood-based cost function for training

Original: GP-based models, as an alternative to multiple shooting methods [13].
Revised: GP-based models, as an alternative to multiple shooting methods [14].

Original: Another option is using it tomake ahead of time predictions for model predictive control and take the prediction uncertainty intoaccount.
Revised: Another option is using it to make ahead of time predictions for model predictive control and take the prediction uncertainty into account.

Original: Our results highlight a fundamental consideration for solving dynamical system and creates scopefor several extensions.
Revised: Our results highlight a fundamental consideration for solving dynamical system and creates scope for several extensions.

Original: This includes improving the linear approximation, combining local linearization with higher order methods than the explicit Euler, and studying the effect of incorporating thecorrect correlation in the state-space GP context.
Revised: This includes improving the linear approximation, combining local lineariza- tion with higher order methods than the explicit Euler, and studying the effect of incorporating the correct correlation in the state-space GP context.