Original: Learning flows of control systems
Revised: Learning flows of control systems

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: A recurrent neural network architecture is presented to learn the flow of a causaland time-invariant control system from data.
Revised: A recurrent neural network architecture is presented to learn the flow of a causal and time-invariant control system from data.

Original: For piecewise constant control inputs, we show that the proposed architecture is able to approximate the flow function by exploiting the system’s causality and time-invariance.
Revised: For piecewise constant control inputs, we show that the proposed architecture is able to approximate the flow function by exploiting the system’s causality and time-invariance.

Original: The output of thelearned flow function can be queried at any time instant.
Revised: The output of the learned flow function can be queried at any time instant.

Original: We demonstrate the generalisation capabilities of the trained model with respect to the simulation timehorizon and the class of input signals.
Revised: We demonstrate the generalisation capabilities of the trained model with respect to the simulation time horizon and the class of input signals.

Original:  Introduction
Revised:  Introduction

Original: Models play a vital role in designing control systems.
Revised: Models play a vital role in designing control systems.

Original: For instance, in receding horizon control(Maciejowski [2002]), the model predicts the future evolution of the state variables and acts as aconstraint in formulating the optimal control problem.
Revised: For instance, in receding horizon control (Maciejowski [2002]), the model predicts the future evolution of the state variables and acts as a constraint in formulating the optimal control problem.

Original: With increasing complexity, the curse of dimensionality limits the usefulness of standard first-principle models.
Revised: With increasing complexity, the curse of dimensionality limits the usefulness of standard first-principle models.

Original: This limitation has motivatedresearch onthe data-driven approximation of sucha physical model.
Revised: This limitation has motivated research on data-driven approximation of such physical models.

Original: Besides fast predictions forarbitrary initial conditions, in designing control systems, another advantage of many data-drivenmethods is the efficient computation of the model’s gradient with respect to initial conditions, parameters or input signals.
Revised: Besides fast predictions for arbitrary initial conditions, in designing control systems, another advantage of many data-driven methods is the efficient computation of the model’s gradient with respect to initial conditions, parameters or input signals.

Original: To leverage these advantages, in [Li et al., 2021, Geneva and Zabaras,2022, Lu et al., 2021, Kissas et al., 2022], the focus is on learning the map from initial conditions,parameters and inputs to the solution of a differential equation.
Revised: To leverage these advantages, in [Li et al., 2021, Geneva and Zabaras, 2022, Lu et al., 2021, Kissas et al., 2022], the focus is on learning the map from initial conditions, parameters and inputs to the solution of a differential equation.

Original: The problem of approximating physical models from data is also tackled in system identification,see Schoukens and Ljung [2019] for an overview.
Revised: The problem of approximating physical models from data is also tackled in system identification, see Schoukens and Ljung [2019] for an overview.

Original: Forgione and Piga [2021] have proposed methodsfor identifying dynamics of continuous-time control systems using neural ordinary differential equations (ODEs).
Revised: Forgione and Piga [2021] have proposed methods for identifying dynamics of continuous-time control systems using neural ordinary differential equations (ODEs).

Original: However, as the dynamics correspond to the time derivative of the flow, the neural
Revised: However, as the dynamics correspond to the time derivative of the flow, the neural

Original: ODEs must be integrated through an ODE solver to obtain the system trajectories, representing anextra computational burden both for prediction and for computing gradients.
Revised: ODEs must be integrated through an ODE solver to obtain the system trajectories, representing an extra computational burden both for prediction and for computing gradients.

Original: Furthermore, errors inthe learned dynamics will accumulate over time when the dynamics are integrated, and the error inthe simulated trajectory can become unbounded.
Revised: Furthermore, errors in the learned dynamics will accumulate over time when the dynamics are integrated, and the error in the simulated trajectory can become unbounded.

Original: To directly learn the flow of an autonomous dynamical system, Biloš et al.
Revised: To directly learn the flow of an autonomous dynamical system, Biloš et al.

Original: [2021] has proposedan alternative to neural ODEs that avoids the step of using an ODE solver and allows for fasterprediction.
Revised: [2021] has proposed an alternative to neural ODEs that avoids the step of using an ODE solver and allows for faster prediction.

Original: This motivates the search for a corresponding learning scheme for controlled dynamicalsystems where inputs are present.
Revised: This motivates the search for a corresponding learning scheme for controlled dynamical systems where inputs are present.

Original: However, this problem is more difficult since the domain of theflow of a continuous-time control system is infinite-dimensional.
Revised: However, this problem is more difficult since the domain of the flow of a continuous-time control system is infinite-dimensional.

Original: Hanson and Raginsky [2020] have shown that continuous-time recurrent neural networks are universal approximators for flow functions of stable continuous-time dynamical systems, where theapproximation quality is uniform over time.
Revised: Hanson and Raginsky [2020] have shown that continuous-time recurrent neural networks are universal approximators for flow functions of stable continuous-time dynamical systems, where the approximation quality is uniform over time.

Original: But the question of whether learning such a modelfrom data is suitable in practice is to the best of our knowledge open.
Revised: But the question of whether learning such a model from data is suitable in practice is to the best of our knowledge open.

Original: In this paper, we propose a neural network-based architecture for learning flows of controlled dynamical systems.
Revised: In this paper, we propose a neural network-based architecture for learning flows of controlled dynamical systems.

Original: We consider the class of piece-wise constant inputs which are practically relevantsince digital controllers typically produce such control inputs.
Revised: We consider the class of piece-wise constant inputs which are practically relevant since digital controllers typically produce such control inputs.

Original: Taking advantage of the causality andtime-invariance properties of the considered class of systems, we show that the flow function canbe efficiently approximated by a recurrent neural network-based architecture.
Revised: Taking advantage of the causality and time-invariance properties of the considered class of systems, we show that the flow function can be efficiently approximated by a recurrent neural network-based architecture.

Original: We illustrate usingexamples of nonlinear oscillators such as the Van der Pol oscillator, highlighting the generalisationcapabilities of the proposed methodology.
Revised: We illustrate using examples of nonlinear oscillators such as the Van der Pol oscillator, highlighting the generalisation capabilities of the proposed methodology.

Original:  Problem formulation2.1 Considered class of control systems and flow functions
Revised:  Problem formulation 2.1 Considered class of control systems and flow functions

Original: A control system Σ consists of the following quadruple (see Sontag [1998], pp. 26)
Revised: A control system Σ consists of the following quadruple (see Sontag [1998], pp. 26)

Original: [Equation]
Revised: [Equation]

Original:  describing the evolution of state-variables of the dynamical system over a time interval T dependingon its initial condition x ∈ X and input u ∈ U , where U is a set of functions u : T → U .
Revised: ) describing the evolution of state-variables of the dynamical system over a time interval T depending on its initial condition x ∈ X and input u ∈ U , where U is a set of functions u : T → U .

Original: The flow ,dictating this evolution, is defined as a mapping φ : T ×
Revised: The flow , dictating this evolution, is defined as a mapping φ : T ×

Original: X × U → X .
Revised: X × U → X .

Original: We assume that Σ time-invariant and finite dimensional.
Revised: We assume that Σ time-invariant and finite dimensional.

Original: In particular, T ⊆ R ≥ 0 , X ⊂ R n and
Revised: In particular, T ⊆ R ≥ 0 , X ⊂ R n and

Original: U ⊂ R m .
Revised: U ⊂ R m .

Original: We also assume that U is the set of piecewise constant controls 1 of period ∆ .
Revised: We also assume that U is the set of piecewise constant controls 2 of period ∆ .

Original: In otherwords, given a sequence { u k } ∞ k =1 with u k ∈ U , the control input u is defined by
Revised: In other words, given a sequence { u k } ∞ k =1 with u k ∈ U , the control input u is defined by

Original: [Equation]
Revised: [Equation]

Original: We will exploit two properties of the flow: causality which implies that the flow at time T ≥ 0 ,φ ( T, x, u ) depends only on the values of u ( t ) for 0 ≤ t < T , and continuity in the sense thatt (cid:55)→ φ ( t, x, u ) is continuous for each x, u .
Revised: We will exploit two properties of the flow: causality which implies that the flow at time T ≥ 0 , φ ( T, x, u ) depends only on the values of u ( t ) for 0 ≤ t < T , and continuity in the sense that t (cid:55)→ φ ( t, x, u ) is continuous for each x, u .

Original: As an example, we can consider the flow function generated by a system of ordinary differentialequations ˙ ξ ( t ) =
Revised: As an example, we can consider the flow function generated by a system of ordinary differential equations ˙ ξ ( t ) =

Original: f ( ξ ( t ) , u ( t )) , t ≥ 0 with initial condition x where u is generated by a digitalcontroller.
Revised: f ( ξ ( t ) , u ( t )) , t ≥ 0 with initial condition x where u is generated by a digital controller.

Original: 2.2 Mathematical formulation of the learning problem
Revised: 2.2 Mathematical formulation of the learning problem

Original: We are interested in learning the flow from data on a time interval [0 , T ] with T > ∆ .
Revised: We are interested in learning the flow from data on a time interval [0 , T ] with T > ∆ .

Original: The inputsignals u and initial conditions x of interest are assumed to be drawn from probability distributions
Revised: The input signals u and initial conditions x of interest are assumed to be drawn from probability distributions

Original: P u on U and P x on X , respectively.
Revised: P u on U and P x on X , respectively.

Original: Finding an approximated flow ˆ φ over a hypothesis class Hamounts to minimising the following loss function
Revised: Finding an approximated flow ˆ φ over a hypothesis class H amounts to minimising the following loss function

Original: [Equation]
Revised: [Equation]

Original: where X ∼ P x and U ∼ P u are independent.
Revised: where X ∼ P x and U ∼ P u are independent.

Original: In practice, the data from a control system is typicallyavailable in the following form
Revised: In practice, the data from a control system is typically available in the following form

Original: [Equation]
Revised: [Equation]

Original: for a given K, N ∈ N where ξ ik = φ ( t ik , x i , u i ) , t ik ∈ [0 , T ] is an increasing sequence of timesamples, and x i ,
Revised: for a given K, N ∈ N where ξ ik = φ ( t ik , x i , u i ) , t ik ∈ [0 , T ] is an increasing sequence of time samples, and x i ,

Original: u i are i.i.d samples from P x and P u .
Revised: u i are i.i.d samples from P x and P u .

Original: Thus we define the following empirical lossfunction ˆ ℓ T and search for a minimiser in H :
Revised: Thus we define the following empirical loss function ˆ ℓ T and search for a minimiser in H :

Original: [Equation]
Revised: [Equation]

Original: [Equation] The objective of this paper is to define an hypothesis space H which renders the above problemtractable and provides an approximation ˆ φ of the true flow function φ while preserving causalityand continuity.
Revised: The objective of this paper is to define an hypothesis space H which renders the above problem tractable and provides an approximation ˆ φ of the true flow function φ while preserving causality and continuity.

Original: In the next section, we propose a neural network-based architecture to solve thisproblem.
Revised: In the next section, we propose a neural network-based architecture to solve this problem.

Original:  Proposed model architecture
Revised:  Proposed model architecture

Original: Due to causality and the considered class of inputs (2), the flow φ ( s, x, u ) at a time instant s ≥ 0depends only on a finite number of the input values { u k } .
Revised: Due to causality and the considered class of inputs (2), the flow φ ( s, x, u ) at a time instant s ≥depends only on a finite number of the input values { u k } .

Original: Thus, at any time during the first controlperiod [0 , ∆] , only the value of u 1 ∈ U and the initial condition x ∈ X are required to define φ .
Revised: Thus, at any time during the first control period [0 , ∆] , only the value of u 1 ∈ U and the initial condition x ∈ X are required to define φ .

Original: Therefore, we define Φ : [0 , 1] ×
Revised: Therefore, we define Φ : [0 , 1] ×

Original: X × U → X as
Revised: X × U → X as

Original: [Equation]
Revised: [Equation]

Original: such that a finite-dimensional vector of parameters (as opposed to functions) directly maps to theflow.
Revised: such that a finite-dimensional vector of parameters (as opposed to functions) directly maps to the flow.

Original: For an arbitrary time instant s > 0 , the flow φ ( s, x, u ) can be computed as follows:
Revised: For an arbitrary time instant s > 0 , the flow φ ( s, x, u ) can be computed as follows:

Original: Construct a map d ∆ : ( s, u ) (cid:55)→ { τ k , u k } k s +1 k
Revised: Construct a map d ∆ : ( s, u ) (cid:55)→ { τ k , u k } k s +1 k

Original: =
Revised: =

Original: such that u k is given according to (2) and
Revised: such that u k is given according to (2) and

Original: [Equation]
Revised: [Equation]

Original: Define the sequence x k ∈ X for all k = 1 , . . .
Revised: Define the sequence x k ∈ X for all k = 1 , . . .

Original: , k s + 1 as
Revised: , k s + 1 as

Original: [Equation]
Revised: [Equation]

Original: As a consequence of these two steps, we obtain x k s +

Revised: As a consequence of these two steps, we obtain x k s +

Original: = φ ( s, x, u ) .
Revised: = φ ( s, x, u ) .

Original: Thus , trajectories of φ can beequivalently represented by the trajectories of a discrete dynamical system with inputs ( τ k , u k ) .
Revised: Thus, trajectories of φ can be equivalently represented by the trajectories of a discrete dynamical system with inputs ( τ k , u k ) .

Original: To approximate (5), we define the hypothesis class H as a subset of the set of causal and continuousfunctions ˆ φ : R ≥ 

Revised: To approximate (5), we define the hypothesis class H as a subset of the set of causal and continuous functions ˆ φ : R ≥

Original: ×
Revised: ×

Original: X × U → X , defined by the composition of three neural networks as illustratedin Figure 1b.
Revised: X × U → X , defined by the composition of three neural networks as illustrated in Figure 1b.

Original: [Equation]
Revised: [Equation]

Original: τ 2 ∆ τ 3 ∆ τ 4 ∆ (a)
Revised: a)

Original: Encoder x
Revised: Encoder x

Original: RNN
Revised: RNN

Original: RNNcell z( u 1 ,τ 1 )
Revised: RNNcell z( u 1 ,τ 1 )

Original: RNNcell z( u 2 ,τ 2 )
Revised: RNNcell z( u 2 ,τ 2 )

Original: RNNcell z( u 3 ,τ 3 )
Revised: RNNcell z( u 3 ,τ 3 )

Original: RNNcell z( u 4 ,τ 4 )
Revised: RNNcell z( u 4 ,τ 4 )

Original: Decoder τ 4 z 4 +(1 − τ 4 )
Revised: Decoder τ 4 z 4 +(1 − τ 4 )

Original: zˆ φ ( s,x 0 ,u ) (b)
Revised: zˆ φ ( s,x 0 ,u ) (

Revised: [Figure]

Original: More precisely, the hypothesis space is defined as
Revised: More precisely, the hypothesis space is defined as

Original: [Equation]
Revised: [Equation]

Original: where h enc , h dec and h RNN are the maps corresponding to the encoder, decoder and the recurrentnetwork, respectively.
Revised: where h enc , h dec and h RNN are the maps corresponding to the encoder, decoder and the recurrent network, respectively.

Original: The map g z ensures that the approximated mapping is continuous in time byinterpolating between the two last RNN states.
Revised: The map g z ensures that the approximated mapping is continuous in time by interpolating between the two last RNN states.

Original:  Experimental evaluation
Revised:  Experimental evaluation

Original: [Figure] We illustrate the proposed method on a Van der Pol oscillator model, see Appendix A for details.
Revised: We illustrate the proposed method on a Van der Pol oscillator model, see Appendix A for details.

Original: Themodel is trained on N = 300 trajectories on [0 , 15] using the stochastic gradient descent algorithm
Revised: The model is trained on N = 300 trajectories on [0 , 15] using the stochastic gradient descent algorithm

Original: Adam with early stopping and a gradient descent scheduler monitoring the validation loss.
Revised: Adam with early stopping and a gradient descent scheduler monitoring the validation loss.

Original: The RNN is a single-layer LSTM network with 24 hidden states.
Revised: The RNN is a single-layer LSTM network with 24 hidden states.

Original: The encoder network maps theinitial state x to the initial LSTM hidden and cell states, and is a 3-layer feedforward net with48 nodes in the hidden layers.
Revised: The encoder network maps the initial state x to the initial LSTM hidden and cell states, and is a 3-layer feedforward net with 48 nodes in the hidden layers.

Original: The decoder network maps the hidden LSTM state to the flowvalue and is a 3-layer feedforward network with 24 nodes in the hidden layers.
Revised: The decoder network maps the hidden LSTM state to the flow value and is a 3-layer feedforward network with 24 nodes in the hidden layers.

Original: All networks usetanh activations.
Revised: All networks use tanh activations.

Original: For each trajectory, K = 200 time points t ik are sampled using Latin hypercubesampling.
Revised: For each trajectory, K = 200 time points t ik are sampled using Latin hypercube sampling.

Original: [Figure]
Revised: [Figure]

Original: Figure 2a shows a predicted trajectory on [0 , 15] for two new pairs of initial conditions and inputsdrawn from P x and P u , that are unseen during training.
Revised: Figure 2a shows a predicted trajectory on [0 , 15] for two new pairs of initial conditions and inputs drawn from P x and P u , that are unseen during training.

Original: Further examples of trajectory predictionare shown in Appendix B.1.
Revised: Further examples of trajectory prediction are shown in Appendix B.1.

Original: Note that the actual trajectory and the predicted trajectory are nearlyindistinguishable.
Revised: Note that the actual trajectory and the predicted trajectory are nearly indistinguishable.

Original: The trajectories considered for training have a time horizon of 15 seconds.
Revised: The trajectories considered for training have a time horizon of 15 seconds.

Original: Due to the recurrentstructure of the model and the stability of the system under consideration, we expect that the prediction performance is maintained for longer prediction horizons.
Revised: Due to the recurrent structure of the model and the stability of the system under consideration, we expect that the prediction performance is maintained for longer prediction horizons.

Original: To check this, we compute anestimate of the mean and variance of ℓ t ( ˆ φ ) as a function of t , where ˆ φ is the model trained on[0 , 15] .
Revised: To check this, we compute an estimate of the mean and variance of ℓ t ( ˆ φ ) as a function of t , where ˆ φ is the model trained on [0 , 15] .

Original: To this end, we compute the test loss on a different set of 300 trajectories on [0 , t ] for a set ofgridded values 15 ≤ t ≤ 100 .
Revised: To this end, we compute the test loss on a different set of 300 trajectories on [0 , t ] for a set of gridded values 15 ≤ t ≤ 100 .

Original: The result is shown in Figure 2b, where the coloured area representsthe 95% confidence interval approximated using the empirical variance estimate.
Revised: The result is shown in Figure 2b, where the coloured area represents the 95% confidence interval approximated using the empirical variance estimate.

Original: We observe that ℓ tremains approximately constant as t increases, indicating that the model gives reliable predictionsfor t much larger than the value of T used for the training trajectories.
Revised: We observe that ℓ t remains approximately constant as t increases, indicating that the model gives reliable predictions for t much larger than the value of T used for the training trajectories.

Original: In Appendix B.2 we additionally investigate the generalisation with respect to the input distribution.
Revised: In Appendix B.2 we additionally investigate the generalisation with respect to the input distribution.

Original:  Concluding remarks
Revised:  Concluding remarks

Original: We presented a recurrent neural network architecture to learn the flow of a continuous causal andtime-invariant control system in continuous time from trajectory data.
Revised: We presented a recurrent neural network architecture to learn the flow of a continuous causal and time-invariant control system in continuous time from trajectory data.

Original: Exploiting causality andtime-invariance, we show that the problem of learning the flow function can be cast as the problemof learning a discrete dynamical system, motivating the use of an RNN-based architecture.
Revised: Exploiting causality and time-invariance, we show that the problem of learning the flow function can be cast as the problem of learning a discrete dynamical system, motivating the use of an RNN-based architecture.

Original: Ourexperimental results on the Van der Pol oscillator show that the learned model has good predictionperformance, and furthermore demonstrate that the model is able to generalise to longer predictiontime horizons and new classes of input signals.
Revised: Our experimental results on the Van der Pol oscillator show that the learned model has good prediction performance, and furthermore demonstrate that the model is able to generalise to longer prediction time horizons and new classes of input signals.

Original: We envision that our approach can provide an alternative to traditional models in control problems, bypassing the need of solving complex dynamicsequations by directly predicting trajectories using our model.
Revised: We envision that our approach can provide an alternative to traditional models in control problems, bypassing the need of solving complex dynamics equations by directly predicting trajectories using our model.

Original: A Van der Pol oscillator
Revised: A Van der Pol oscillator

Original: The Van der Pol oscillator is described by the system of ordinary differential equations
Revised: The Van der Pol oscillator is described by the system of ordinary differential equations

Original: [Equation]
Revised: [Equation]

Original: (6) with µ = 1 .
Revised: with µ = 1 .

Original: We take x (0) ∼ N (0 , I ) , i.e. a standard normal distribution.
Revised: We take x (0) ∼ N (0 , I ) , i.e. a standard normal distribution.

Original: The control input samplingtime is ∆ = 0 .
Revised: The control input sampling time is ∆ = 0 .

Original: 1 and the inputs considered are square wave inputs with period 5∆ and amplitudessampled i.i.d.
Revised: 1 and the inputs considered are square wave inputs with period 5∆ and amplitudes sampled i.i.d.

Original: from N (0 , σ = 5) , i.e.u 1+5 k ∼ N (0 , 5) ,u
Revised: from N (0 , σ = 5) , i.e. u 1+5 k ∼ N (0 , 5) ,

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: = u [Equation] [Equation] , j = 2 , 3 , 4 ,holds for all k ∈ N .
Revised: ,holds for all k ∈ N .

Original: To generate the data used to train the model in Section 4, we integrate (6) withan RK45 solver.
Revised: To generate the data used to train the model in Section 4, we integrate (6) with an RK45 solver.

Original: B Additional experimental results 
Revised: B Additional experimental results B.1 Van der Pol prediction results

Original: [Figure] B.1 Van der Pol prediction results

Original: [Figure] Figure 3 shows four additional examples of trajectories predicted by the model described in Section 4, with x ∼ P x , u ∼ P u .
Revised: Figure 3 shows four additional examples of trajectories predicted by the model described in Section 4, with x ∼ P x , u ∼ P u .

Original: As in Figure 2a, the predicted trajectory (in black) is nearly indistinguishable from the actual trajectory (in blue, dashed).
Revised: As in Figure 2a, the predicted trajectory (in black) is nearly indistinguishable from the actual trajectory (in blue, dashed).

Original: Figure 4 shows four additional trajectories on the larger prediction horizon T = 100 .
Revised: Figure 4 shows four additional trajectories on the larger prediction horizon T = 100 .

Original: Recall thatthe model used to predict these was trained with trajectories on the time interval [0 , 15] , confirmingthe observation in Figure 2b that the loss function ℓ t remains approximately constant even as theprediction horizon increases well beyond the length of the trajectories considered during training. 
Revised: Recall that the model used to predict these was trained with trajectories on the time interval [0 , 15] , confirming the observation in Figure 2b that the loss function ℓ t remains approximately constant even as the prediction horizon increases well beyond the length of the trajectories considered during training. [Figure] [Table] [Figure]

Revised: [Table]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: B.2 Generalisation to different input distributions
Revised: B.2 Generalisation to different input distributions

Original: We additionally investigate the performance of the model trained in section 4 to an input distributiondifferent from the training distribution P u .
Revised: We additionally investigate the performance of the model trained in section 4 to an input distribution different from the training distribution P u .

Original: In particular, we consider here a distribution Q u on Uconsisting of sinusoidal sequences with random amplitude and frequency, that is,
Revised: In particular, we consider here a distribution Q u on U consisting of sinusoidal sequences with random amplitude and frequency, that is,

Original: [Equation]
Revised: [Equation]

Original: where A ∼ LogNormal(0 , 1) and Ω ∼ Uniform(0 , 2 π ) .
Revised: where A ∼ LogNormal(0 , 1) and Ω ∼ Uniform(0 , 2 π ) .

Original: This corresponds to sinusoidal signals withfrequency in the band [0 , 1 / 2∆] , which is chosen to prevent aliasing.
Revised: This corresponds to sinusoidal signals with a maximum frequency of 0 . 5 Hz.

Original: To verify the performance on this class of inputs, we compute an estimate of ℓ T ( ˆ φ ; Q u ) , defined asin equation (3) with the expectation taken with U ∼ Q u .
Revised: To verify the performance on this class of inputs, we compute an estimate of ℓ T ( ˆ φ ; Q u ) , defined as in equation (3) with the expectation taken with U ∼ Q u .

Original: Figure 5 shows four trajectories with initialconditions and input signals drawn from P x and Q u , respectively.
Revised: Figure 5 shows four trajectories with initial conditions and input signals drawn from P x and Q u , respectively.

Original: Figure 6, shows a box plot of thedistribution of the estimate of ℓ t computed on 500 trajectories for each of the two input distributions
Revised: Figure 6, shows a box plot of the distribution of the estimate of ℓ t computed on 500 trajectories for each of the two input distributions

Original: P u and Q u .
Revised: P u and Q u .

Original: As expected, the mean and variance of the prediction loss for the distribution Q uare slightly higher than for P u , but remain reasonably closer to that of P u .
Revised: As expected, the mean and variance of the prediction loss for the distribution Q u are slightly higher than for P u , but remain reasonably closer to that of P u .

Original: This is confirmed by
Revised: This is confirmed by

Original: Figure 5, where we observe that there is little degradation in prediction performance compared tothe prediction on inputs drawn from P u .
Revised: Figure 5, where we observe that there is little degradation in prediction performance compared to the prediction on inputs drawn from P u .

Original: [Figure] [Figure]
Revised: [Figure] [Figure]