Method and device for operating technical system

Abstract

An apparatus and computer-implemented method for machine learning with time series data representing observations associated with a technical system (102), the method comprises providing (200) time series data, and model parameters of distributions over the time series data and over the first potential variable and over the second potential variable, and variational parameters of an approximate distribution over the second potential variable; sampling (204-1) the value of the second potential variable from the approximate distribution over the second potential variable; finding (204-2) the value of the first potential variable, in particular maximizing the distribution density over the time series data and over the values of the first potential variable and of the second potential variable, depending on the distribution density over the time series data and over the values of the first potential variable and of the second potential variable; determining (204-3) a Hessian matrix depending on the time series data and the distribution over the first potential variable and a second order Taylor approximation of a value of a second potential variable evaluated at a value of the first potential variable; determining (204-4) a determinant of the Hesse matrix; determining (204-5) a Laplace approximation of the distribution on the time series data conditioned on the value of the second potential variable, depending on the determinant of the Hesse matrix; determining (204-6) an inverse of the Hesse matrix; determining (204-7) a Jacobian matrix of distribution over values of the time series data and the first potential variable and the second potential variable; evaluating (202-1) an approximation lower limit that depends on a Laplacian approximation determined for the plurality of values of the second potential variable; determining (202-2) a gradient of the Laplacian approximation depending on the inverse Hessian matrix and the Jacobian matrix; depending on the gradient, the model parameters and the variational parameters are updated (202-3).

Description

Technical Field

[0001] The invention relates to a method and a device for operating a technical system. Background Art

[0002] Gaussian process state-space models use Gaussian processes as the transfer function in the state-space model in order to describe time series data in a fully probabilistic manner. These models have two types of latent variables, the temporal states required to model the noisy continuous observations, and the so-called induced outputs required to handle the Gaussian process part of the model in an efficient manner.

[0003] “Overcoming Mean-Field Approximations in Recurrent Gaussian Process Models” by Ialongo, Alessandro Davide, Mark Van Der Wilk, James Hensman, and Carl Edward Rasmussen (International Conference on Machine Learning, 2019) discloses the use of variational inference to approximate the true posterior of a Gaussian process model. The conditional dependence of the temporal state on the induced output is taken into account, and a Markov Gaussian model over the temporal state is assumed. The Markov Gaussian model is parameterized and allows for nonlinear transformations.

[0004] Skaug, Hans Julius and David A. Fournier, "Automatic approximation of the marginal likelihood in non-Gaussian hierarchical models" (Computer Statistical Data Analysis 51, pp. 699-709, 2006) discloses applying the Laplace approximation to general, i.e., state-space models of non-Gaussian processes in an efficient manner. This is possible by using the implicit function theorem and exploiting the sparsity and structure of the Hessian matrix, which is the matrix required to apply the Laplace approximation. Summary of the invention

[0005] The computer-implemented method and apparatus according to the independent claims provide a model and a combination of these inference methods by distinguishing between two different types of latent variables in a Gaussian process state space model and applying variational inference to the Gaussian process portion of the model and applying Laplace approximation to the temporal states of the model. The distinction between the two types of latent variables makes it possible to efficiently process the model. The method does not require sequential sampling of the temporal states during inference, and instead performs a Laplace approximation that involves a joint optimization of those temporal states. This helps optimize the model. The approximate posteriors used in the model further assume that the dynamic characteristics can be locally linearly approximated. The optimization improvements provided by the model also lead to better calibrated uncertainties for different time series prediction tasks.

[0006] A computer-implemented method for machine learning using time series data representing observations associated with a technical system comprises: providing the time series data, and model parameters of distributions on the time series data and on a first latent variable and a second latent variable, and variational parameters of an approximate distribution on the second latent variable; sampling values ​​of the second latent variable from the approximate distribution on the second latent variable; finding a value of the first latent variable depending on the density of the distributions on the time series data and on the values ​​of the first latent variable and the second latent variable, in particular maximizing the density of the distributions on the time series data and on the values ​​of the first latent variable and the second latent variable; maximizing the density of the distributions on the time series data and on the values ​​of the first latent variable and the second latent variable depending on the time series data and on the first latent variable; Determine a Hessian matrix by using a second-order Taylor approximation of the distribution over the variable and the value of the second latent variable evaluated at the value of the first latent variable; determine the determinant of the Hessian matrix; determine a Laplace approximation of the distribution over the time series data conditioned on the value of the second latent variable depending on the determinant of the Hessian matrix; determine the inverse of the Hessian matrix; determine the Jacobian matrix of the distribution over the time series data and the values ​​of the first latent variable and the second latent variable; evaluate an approximate lower bound that depends on the Laplace approximation determined for multiple values ​​of the second latent variable; determine the gradient of the Laplace approximation depending on the inverse Hessian matrix and the Jacobian matrix; update model parameters and variational parameters depending on the gradient. The method uses a distinction between two types of latent variables, uses an approximate posteriori, and assumes that the dynamic characteristics can be locally linearly approximated. The method provides an improved method for inference in a Gaussian process state space model. The method has the following advantages: the method does not require sequential sampling of time states during inference, and instead performs a Laplace approximation that involves joint optimization of those time states.

[0007] Preferably, providing the time series data comprises receiving the time series data or receiving a sensor signal comprising information about the technical system and determining the time series data depending on the sensor signal.

[0008] The method preferably comprises determining instructions for actuating the technical system in dependence on the time series data, the model parameters and the variational parameters, and outputting the instructions to cause the technical system to act.

[0009] Preferably, the technical system is a computer-controlled machine, such as a robot, in particular a vehicle, a domestic appliance, a power tool, a manufacturing machine, a personal assistant or an access control system.

[0010] The technical system may comprise an engine or a part thereof, wherein the time series data comprises speed and / or load as input to the technical system and emissions, temperature of the engine or oxygen content in the engine as output of the technical system.

[0011] The technical system may include a fuel cell stack or a portion thereof, wherein the time series data includes as inputs to the technical system the current in the fuel cell stack, the hydrogen concentration in the fuel cell stack, the stoichiometry of the anode or cathode of the fuel cell stack, the volume flow of the coolant for the fuel cell stack, the anode pressure for the anode of the fuel cell stack, the cathode pressure for the cathode of the fuel cell stack, the inlet temperature of the coolant for the fuel cell stack, the outlet temperature of the coolant for the fuel cell stack, the anode dew point temperature of the anode of the fuel cell stack, the cathode dew point temperature of the cathode of the fuel cell stack, and as outputs of the technical system (102), the average value of the cell tension between cells of the fuel cell stack, the anode pressure drop at the anode of the fuel cell stack, the cathode pressure drop at the cathode of the fuel cell stack, the coolant pressure drop between the coolant inlet and outlet of the fuel cell stack, or the coolant temperature rise between the coolant inlet and outlet of the fuel cell stack.

[0012] The instruction preferably comprises a target operating mode for the technical system.

[0013] The method may include determining the determinant of a Hessian matrix depending on a factorization comprising a strictly upper triangular portion of a portion of the Hessian matrix, a strictly lower triangular portion of the portion of the Hessian matrix, and a block diagonal matrix of a recursively defined block of the matrix. This is an efficient way of determining a Hessian matrix that is very computationally efficient.

[0014] The method may include determining the inverse of a Hessian matrix depending on a factorization comprising a strictly upper triangular portion of a portion of the Hessian matrix, a strictly lower triangular portion of the portion of the Hessian matrix, and a block diagonal matrix of a recursively defined block of the matrix. This is an efficient way of determining the inverse of a Hessian matrix that is very computationally efficient.

[0015] Evaluating the approximate lower bound may include sampling the second latent variable with a sample drawn from an approximate distribution over the second latent variable.

[0016] An apparatus for machine learning using time series data representing observations related to a technical system comprises at least one processor and at least one memory, wherein the at least one processor is adapted to execute instructions which, when executed by the at least one processor, cause the apparatus to perform the steps of a method for operating a technical system. The advantages provided by the apparatus correspond to those provided by the method.

[0017] The device may comprise an interface adapted to receive information about the technical system and / or adapted to output instructions for causing the technical system to act.The device is capable of interacting with the technical system.

[0018] The computer program may comprise computer readable instructions which, when executed by a computer, cause the computer to perform the steps of the method. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Further advantageous embodiments can be derived from the following description and the accompanying drawings. In the drawings:

[0020] Figure 1 schematically depicts an apparatus for operating a technical system,

[0021] Figure 2 Steps in a method for operating a technical system are schematically depicted. DETAILED DESCRIPTION

[0022] Figure 1 An apparatus 100 for operating a technical system 102 is schematically depicted.

[0023] The apparatus 100 comprises at least one processor 104 and at least one memory 106. The at least one processor 104 is adapted to execute instructions which, when executed by the at least one processor 104, cause the apparatus 100 to perform the steps of a method for operating the technical system 102.

[0024] The device 100 in this example comprises an interface 108. The interface 108 is adapted, for example, to receive information about the technical system 102. The interface 108 is adapted, for example, to output instructions for causing the technical system 102 to act. The technical system 102 may comprise an actuator 110. The actuator 110 may be connected to the interface 108 via a signal line 112 at least temporarily.

[0025] Figure 2 The steps of the method are described. The method includes analyzing data, such as given a dimension d y Given time series data in And then operate the technical system 102 accordingly.

[0026] Time series data Y TIncluding, for example, noisy observations from the technical system 102 .

[0027] In addition to the time series data Y T In addition, the method can consider the additional dimension d u Time Series Data T ,in

[0028] The technical system 102 may include an engine or a part thereof. The time series data may include speed and / or load as inputs to the technical system 102 and emissions, temperature of the engine or oxygen content in the engine as outputs of the technical system 102 .

[0029] The technical system 102 may include a fuel cell stack or a portion thereof. The time series data may include as inputs to the technical system 102 the current in the fuel cell stack, the hydrogen concentration in the fuel cell stack, the stoichiometry of the anode or cathode of the fuel cell stack, the volume flow of the coolant for the fuel cell stack, the anode pressure for the anode of the fuel cell stack, the cathode pressure for the cathode of the fuel cell stack, the inlet temperature of the coolant for the fuel cell stack, the outlet temperature of the coolant for the fuel cell stack, the anode dew point temperature of the anode of the fuel cell stack, the cathode dew point temperature of the cathode of the fuel cell stack, and as outputs of the technical system 102, the average value of the cell tension between the cells of the fuel cell stack, the anode pressure drop at the anode of the fuel cell stack, the cathode pressure drop at the cathode of the fuel cell stack, the coolant pressure drop between the inlet and outlet of the coolant for the fuel cell stack, or the coolant temperature rise between the inlet and outlet of the coolant for the fuel cell stack.

[0030] This method is used for a given time series data Y with a given number of iterations i and a given number of samples n. T to perform the operation.

[0031] The method is based on a probabilistic model and an approximate model. The approximate model is based on a completely independent training condition, the FITC assumption. The details of this assumption are described, for example, in Edward Snelson and Zoubin Ghahramani's "Sparse Gaussian Processes using Pseudo-inputs" (Advances in Neural Information Processing Systems, 2005).

[0032] The probabilistic model is based on the Gaussian process state space model, where the Gaussian process prior is used to learn the probability distribution of a latent state x t-1 To the next potential state x t The mean of the transfer model of the mapping is:

[0033]

[0034] Among them, the initial distribution p Θ (x0) and emission model p Θ (y t |x t ) is not specified, and the transfer model is given by

[0035] p Θ (x t |x t-1 ,f t-1 )=N(x t |x t-1 +f t-1 ,Q)

[0036] where N is independent similarly distributed Gaussian noise with variance Q, where And in this example, f~GP(0,k(·,·)) is a zero-mean Gaussian process, i.e., a process consisting entirely of the positive definite symmetric kernel The distribution over the specified function.

[0037] If the method considers adding d u Dimensional time series data U T in, Then the acceptance dimension of the kernel of the Gaussian process is The input pair.

[0038] The probability model includes a first latent variable. In this example, the first latent variable is a given dimension d x Time status in

[0039] The Gaussian process posterior can be summarized by a sparse Gaussian process, where the information of the posterior is contained in the pseudo dataset (X M , F M ), where X M Induced input, and F M is the induced output.

[0040] Induced output F M and F T Shared joint Gaussian distribution p Θ (F T ,F M ). The model uses a fully independent training conditional approximation that assumes independence of the underlying GP evaluations given the induced output:

[0041]

[0042] This results in:

[0043]

[0044] in

[0045]

[0046] Induced output F M is the second latent variable.

[0047] If the method considers adding d u Time series data of dimension U T ,in Then the induction point X M Existing in this higher dimensional space middle.

[0048] The approximate model includes the time series data and the first latent variable (e.g., time state ) and the second latent variable (e.g., the induced output F M ) distribution on .

[0049] The marginal log-likelihood log p is obtained by variational inference Θ (Y T ) to find the lower bound that allows finding the induced output p Θ (F M |Y T ) is an approximation of the true posterior probability on :

[0050] log p θ (Y T )≥∫q Ψ (F M )log p θ (Y T |F M )dF M -KL(q Ψ (F M )||p θ (F M )),

[0051] in

[0052] The approximate model includes an approximate distribution over the second latent variable, such as the induced output F with mean m and variance S M The variable distribution q Ψ (F M )=N(F M |m, S). These are for example given the initial variational parameters Ψ = {m, S}.

[0053] For each induced input in Induced output F MAccording to the given induction output F M The approximate distribution q on Ψ (F M ) and distributed.

[0054] Induced input X M and the induced output F M is called a pseudo data point. The prior on the induced output is given by the Gaussian process prior

[0055] p(F M )=N(F M |0,K MM ),in,

[0056] The approximate model consists of predicting the distribution p(x t |x t-1 , F M )=N(x t |x t-1 +μ(x t-1 , F M ),∑(x t-1 )+Q), where the mean is given by is given by , and the covariance is given by Given, where k tt = k(x t , x t )and

[0057] The method comprises step 200 .

[0058] Step 200 includes providing time series data Y T . Time series data U can additionally be provided and used T .

[0059] The method may include receiving time series data Y at interface 108 T .

[0060] Step 200 may include receiving a sensor signal including information about the technical system 102 and determining the time series data Y depending on the sensor signal. T Additionally, time series data U may be received or determined from received sensor signals. T . Time series datau t For example, with the latent state x t concatenated and used as input to the transformation model and kernel function.

[0061] Step 200 includes providing time series data Y T and potential state Given the true distribution on The induced output F M This means that, providing the time series data Y T and the first latent variable and the second latent variable F M The distribution on .

[0062] The method operates with given initial model parameters Θ and given initial variational parameters Ψ = {m, S}.

[0063] The method includes an outer loop 202 and an inner loop 204 .

[0064] The outer loop 202 is processed for iterations i=1, ..., i. In the iterations, the model and the variational parameters are optimized.

[0065] The inner loop 204 is processed for samples n=1, ..., N. The samples are used to obtain a stochastic approximation of the log-likelihood.

[0066] The inner loop 204 includes step 204 - 1 .

[0067] Step 204 - 1 includes sampling values ​​of the second latent variable from an approximate distribution over the second latent variable.

[0068] In this example, the output from the induction M The distribution on q Ψ (F M ) Determine the induced output sample

[0069]

[0070] The inner loop 204 includes step 204 - 2 .

[0071] Step 204-2 includes finding a value of the first latent variable depending on the time series data and the distribution density on the first latent variable and on the values ​​of the second latent variable. In this example, step 204-2 includes finding a value of the first latent variable so that the distribution density on the time series data and the first latent variable and on the values ​​of the second latent variable is maximized.

[0072] In this example, find the mode that maximizes the log density

[0073]

[0074] Regarding potential status,

[0075]

[0076] This means that the method consists in finding the pattern that maximizes the log density

[0077] The inner loop 204 includes step 204 - 3 .

[0078] Step 204-3 includes depending on the first latent variable The model parameters Θ and the second latent variable The value of log density is determined by The Hessian matrix.

[0079] In this example, the Hessian matrix is ​​obtained The non-zero elements of

[0080]

[0081] In this example, the non-zero elements include the number and The Hessian matrix is ​​used to provide the pattern The second-order Taylor approximation around Please note that Y T is a constant.

[0082] In this example, the non-zero elements of the Hessian matrix consist only of the 3d x The vector-Hessian matrix product is determined, thus reducing the memory and time requirements to

[0083] The inner loop 204 includes step 204 - 4 .

[0084] Step 204 - 4 includes determining the determinant of the Hessian matrix.

[0085] The determinant of the Hessian matrix is ​​determined, for example, depending on a factorization comprising a strictly upper triangular part of a portion of the Hessian matrix, a strictly lower triangular part of a portion of the Hessian matrix and a block diagonal matrix of a recursively defined block of the matrix.

[0086] In this example, the Hessian matrix H(A t ,B t ) is the determinant det H(A t ,B t ) is evaluated.

[0087] In this example, the Hessian matrix H(A t ,B t ) is the determinant det H(A t ,B t ) is determined by factoring

[0088] H(A t ,B t)=(Λ+B T )Λ -1 (Λ+B)

[0089] Where B is the Hessian matrix H(A t ,B t ) of the strictly upper triangular part, including different B t , and Λ is a block diagonal matrix of a block defined recursively:

[0090] Λ0=Λ0, t=1,…,T

[0091] as

[0092]

[0093] Among them, Λ t is a block diagonal matrix, and B t is a strictly upper triangular sum is strictly lower triangular. These operations can be Execute in step.

[0094] The inner loop 204 includes step 204 - 5 .

[0095] Step 204 - 5 includes determining, depending on the determinant of the Hessian matrix, a Laplace approximation of a distribution on the time series data conditioned on the value of the second latent variable.

[0096] In this example, the condition Laplace approximation of Evaluation:

[0097]

[0098] The inner loop 204 includes step 204 - 6 .

[0099] Step 204-6 includes determining the inverse of the Hessian matrix.

[0100] The inverse of the Hessian matrix is ​​determined, for example, depending on the factorization of a block diagonal matrix comprising a strictly upper triangular part of a portion of the Hessian matrix, a strictly lower triangular part of a portion of the Hessian matrix and a recursively defined block of the matrix.

[0101] In this example, the inverse H of the Hessian matrix H is determined -1 .

[0102] The inverse of the Hessian matrix is ​​determined by factorization.

[0103] H -1 =(Λ+B) -1 Λ(Λ+B T ) -1

[0104] The inner loop 204 includes step 204 - 7 .

[0105] Step 204 - 7 includes determining the Jacobian of the distribution over the time series data and the values ​​of the first latent variable and the second latent variable.

[0106] In this example, determine the function The Jacobian of h is:

[0107]

[0108] The outer loop 202 includes step 202 - 1 .

[0109] Step 202 - 1 includes evaluating an approximate lower bound that depends on a Laplace approximation determined for a plurality of values ​​of a second latent variable.

[0110] In this example, the approximate lower limit L(Θ,Ψ) is evaluated

[0111]

[0112] It includes the Kullback-Leibler term and the KL term, which are used to approximate the distribution q Ψ (F M ) and the true posterior distribution p Θ (F M ) for comparison. This distribution Depend on The Laplace approximation around is given.

[0113] In this example, the plurality of values ​​of the second latent variable are samples of the second latent variable determined in step 204-1 when the inner loop is repeatedly processed. This means that the approximate lower bound is evaluated depending on samples of the second latent variable drawn from the approximate distribution over the second latent variable.

[0114] In this example, to evaluate and optimize the optimization objective, we use the approximate distribution q Ψ (F M )Select the parameter family.

[0115] In this example, q Ψ (F M ) is a Gaussian distribution. This allows the KL term to be evaluated analytically. The other term of L(Θ,Ψ) is analytically intractable. In this example, the other term is optimized by sampling:

[0116]

[0117] Among them, from the induced output F M The approximate distribution q onΨ (F M ) is a sample of the induced output extracted from

[0118]

[0119] The outer loop 202 includes step 202 - 2 .

[0120] Step 202 - 2 includes determining the gradient of the Laplace approximation depending on the inverse Hessian matrix and the Jacobian matrix.

[0121] In this example, the gradient of L(Θ,Ψ) is obtained using the following formula and

[0122]

[0123] and

[0124]

[0125] where Ψ is the induced output at the sampling The variational parameters that implicitly enter into the formula are

[0126] This replaces the potentially expensive automatic differentiation computation with solving the Hessian matrix. This only requires the value This makes it unnecessary to have a complete computational graph of how the value was obtained.

[0127] The outer loop 202 includes step 202 - 3 .

[0128] Step 202-3 includes updating the model parameters and the variational parameters depending on the gradients.

[0129] In this example, the model parameters Θ and the variational parameters Ψ = {m, S} are updated.

[0130] Updating the model parameters Θ and the variational parameters Ψ={m,S} includes determining the model parameters Θ and the variational parameters Ψ={m,S} so that L(Θ,Ψ) is minimized. This means that the model parameters Θ and the variational parameters Ψ={m,S} are determined so that L(Θ,Ψ) is less than other model parameters Θ and the variational parameters Ψ={m,S}.

[0131] The above steps of the method describe the inference method for the model parameters Θ and the variational parameters Ψ of the Gaussian process state space model learned during training. These steps can be determined in an offline phase, for example, for a given time series data Y T and optionally additional time series data U given T .

[0132] The following steps of the method can be performed, for example, in an online phase for prediction. These steps can be performed with a trained model, i.e., with given model parameters θ and variational parameters Ψ. These steps can be performed independently of training, i.e., without training, or after training and in conjunction with training.

[0133] In this example, the model parameters Θ and variational parameters Ψ determined in the last iteration of updating the model parameters Θ and variational parameters Ψ are used for prediction.

[0134] The method may include step 206 .

[0135] In step 206, the method includes depending on the time series data Y T , model parameters Θ and variational parameters Ψ={m,S} to determine instructions for actuating the technical system 102 .

[0136] Optionally, additional time series data U can also be used T .

[0137] For example, the time series data include speed and / or load as input to the approximation model of the technical system 102. For example, the output of the approximation model of the technical system 102 is emissions, the temperature of the engine or the oxygen content in the engine.

[0138] For example, the time series data includes as inputs to the approximate model of the technical system 102 the current in the fuel cell stack, the hydrogen concentration in the fuel cell stack, the stoichiometry of the anode or cathode of the fuel cell stack, the volume flow of the coolant for the fuel cell stack, the anode pressure for the anode of the fuel cell stack, the cathode pressure for the cathode of the fuel cell stack, the inlet temperature of the coolant for the fuel cell stack, the outlet temperature of the coolant for the fuel cell stack, the anode dew point temperature of the anode of the fuel cell stack, the cathode dew point temperature of the cathode of the fuel cell stack. For example, the output of the approximate model of the technical system 102 is the average value of the cell tension between the cells of the fuel cell stack, the anode pressure drop at the anode of the fuel cell stack, the cathode pressure drop at the cathode of the fuel cell stack, the coolant pressure drop between the coolant inlet and outlet for the fuel cell stack, or the coolant temperature rise between the coolant inlet and outlet for the fuel cell stack.

[0139] The instructions comprise, for example, a target operating mode for the technical system 102. The target operating mode can be determined depending on an output of the approximation model, for example by a controller or a characteristic curve or a mapping which maps the output to a target operating mode.

[0140] The method may include step 208 .

[0141] In step 208 , the method includes outputting instructions to cause the technical system 102 to act.

[0142] The instruction comprises, for example, a target operating mode for the technical system 102 .

[0143] Time series data Y T It can be processed in mini-batch training. A mini-batch is a b The time series data Y starts at an arbitrary time index t0 T subsequence of The method can be applied to small batches. The method can include Ψ (F M ) and approximate this term:

[0144]

[0145] The method is also applicable to one-dimensional or multi-dimensional latent states x t For a multidimensional latent state x t , we can calculate the potential state x t Use a separate Gaussian process for each dimension of :

[0146]

[0147] in is the set of all induced outputs, is the dth dimension of the latent state, and μ (d) is the mean, and Σ (d) is the covariance of the d-th Gaussian process.

Claims

1. A computer-implemented method for performing machine learning using time series data representing observations associated with a technical system (102), characterized in that Providing (200) time series data, and model parameters of distributions on the time series data and on a first latent variable and a second latent variable, and variational parameters of an approximate distribution on the second latent variable; sampling (204-1) values ​​of the second latent variable from the approximate distribution on the second latent variable; finding (204-2) the value of the first latent variable depending on the density of distributions on the time series data and on the values ​​of the first latent variable and the second latent variable, in particular maximizing the density of distributions on the time series data and on the values ​​of the first latent variable and the second latent variable; Determine (204-3) a Hessian matrix based on a second-order Taylor approximation of the distribution over the time series data and a first latent variable and the value of a second latent variable evaluated at the value of the first latent variable; determine (204-4) the determinant of the Hessian matrix, and determine (204-5) a Laplace approximation of the distribution over the time series data conditioned on the value of the second latent variable based on the determinant of the Hessian matrix; determine (204-6) the inverse of the Hessian matrix; determine (204-7) a Jacobian matrix of the distribution over the time series data and the values ​​of the first latent variable and the second latent variable; evaluate (202-1) an approximate lower limit that depends on a Laplace approximation determined for multiple values ​​of the second latent variable; determine (202-2) the gradient of the Laplace approximation based on the inverse Hessian matrix and the Jacobian matrix; and update (202-3) model parameters and variational parameters based on the gradient.

2. The method according to claim 1, characterized in that Providing (200) the time series data comprises receiving the time series data or receiving a sensor signal comprising information about the technical system (102) and determining the time series data depending on the sensor signal.

3. The method according to claim 1, characterized in that Instructions for actuating the technical system (102) are determined (206) depending on the time series data, the model parameters and the variational parameters, and the instructions are output (208) to cause the technical system (102) to act.

4. The method according to any one of the preceding claims, characterized in that The technical system ( 102 ) is a computer-controlled machine, such as a robot, in particular a vehicle, a household appliance, a power tool, a manufacturing machine, a personal assistant or an access control system.

5. The method according to any one of the preceding claims, characterized in that The technical system (102) comprises an engine or a part thereof, wherein the time series data comprises speed and / or load as input to the technical system (102) and emissions, temperature of the engine or oxygen content in the engine as output of the technical system (102).

6. The method according to any one of the preceding claims, characterized in that The technical system (102) includes a fuel cell stack or a part thereof, wherein the time series data includes as inputs to the technical system (102) the current in the fuel cell stack, the hydrogen concentration in the fuel cell stack, the stoichiometry of the anode or cathode of the fuel cell stack, the volume flow of the coolant for the fuel cell stack, the anode pressure for the anode of the fuel cell stack, the cathode pressure for the cathode of the fuel cell stack, the inlet temperature of the coolant for the fuel cell stack, the outlet temperature of the coolant for the fuel cell stack, the anode dew point temperature of the anode of the fuel cell stack, the cathode dew point temperature of the cathode of the fuel cell stack, and as outputs of the technical system (102) the average value of the cell tension between cells of the fuel cell stack, the anode pressure drop at the anode of the fuel cell stack, the cathode pressure drop at the cathode of the fuel cell stack, the coolant pressure drop between the inlet and outlet of the coolant for the fuel cell stack, or the coolant temperature rise between the inlet and outlet of the coolant for the fuel cell stack.

7. The method according to any one of the preceding claims, characterized in that The instructions include a target operating mode for the technical system (102).

8. The method according to any one of the preceding claims, characterized in that The determinant of the (204-4) Hessian matrix is ​​determined depending on a factorization comprising a strictly upper triangular portion of a portion of the Hessian matrix, a strictly lower triangular portion of a portion of the Hessian matrix, and a block diagonal matrix of a recursively defined block of the matrix.

9. The method according to any one of the preceding claims, characterized in that The inverse of the (204-6) Hessian matrix is ​​determined depending on a factorization comprising a strictly upper triangular portion of a portion of the Hessian matrix, a strictly lower triangular portion of a portion of the Hessian matrix, and a block diagonal matrix of a recursively defined block of the matrix.

10. The method according to any one of the preceding claims, characterized in that Evaluating the approximate lower bound (202-1) includes sampling with samples of the second latent variable, the samples being drawn from an approximate distribution on the second latent variable.

11. An apparatus (100) for performing machine learning using time series data representing observations associated with a technical system (102), characterized in that The device (100) comprises at least one processor (104) and at least one memory (106), wherein the at least one processor (104) is adapted to execute instructions which, when executed by the at least one processor (104), cause the device (100) to perform steps in a method for operating the technical system (102).

12. The device (100) according to claim 11, characterized in that The device (100) comprises an interface (108) adapted to receive information about a technical system (102) and / or adapted to output instructions for causing the technical system (102) to act.

13. A computer program, characterized in that The computer program comprises computer readable instructions which, when executed by a computer, cause the computer to perform the steps of the method according to one of claims 1 to 12 .

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