Abstract: A computer-implemented method of training a model for making time-series predictions of a computer-controlled system. The model uses a stochastic differential equation (SDE) comprising a drift component and a diffusion component. The drift component has a predefined part representing domain knowledge, that is received as an input to the training; and a trainable part. When training the model, values of the set of SDE variables at a current time point are predicted based on their values at a previous time point, and based on this, the model is refined. In order to predict the values of the set of SDE variables, the predefined part of the drift component is evaluated to get a first drift, and the first drift is combined with a second drift obtained by evaluating the trainable part of the drift component.

Abstract: A computer-implemented method for predicting a behavior of agents in a dynamic system with a multiplicity of interacting agents depending on the latent state thereof. For a plurality of components and for a plurality of time points up to a prediction time point, a value of a first moment of a first distribution, which models the latent state of the agents, is determined for each component. A value of a second moment of the first distribution is determined. An expected value for a first moment of a second distribution at the prediction time point is determined for each component depending on the value of the first moment of the first distribution at the prediction time point and depending on the value of the second moment of the first distribution at the prediction time point. The second distribution models the behavior of the agents depending on the latent state thereof.

Abstract: A method for training the neural drift network and the neural diffusion network of a neural stochastic differential equation. The method includes drawing a training trajectory from training sensor data, and, starting from the training data point which the training trajectory includes for a starting instant, determining the data-point mean and the data-point covariance at the prediction instant for each prediction instant of the sequence of prediction instants using the neural networks. The method also includes determining a dependency of the probability that the data-point distributions of the prediction instants—which are given by the ascertained data-point means and the ascertained data-point covariances—will supply the training data points at the prediction instants, on the weights of the neural drift network and of the neural diffusion network, and adapting the neural drift network and the neural diffusion network to increase the probability.

Abstract: A computer-implemented method of training a model for making time-series predictions of a computer-controlled system. The model uses a stochastic differential equation (SDE) comprising a drift component and a diffusion component. The drift component has a predefined part representing domain knowledge, that is received as an input to the training; and a trainable part. When training the model, values of the set of SDE variables at a current time point are predicted based on their values at a previous time point, and based on this, the model is refined. In order to predict the values of the set of SDE variables, the predefined part of the drift component is evaluated to get a first drift, and the first drift is combined with a second drift obtained by evaluating the trainable part of the drift component.

Abstract: A computer-implemented method for enabling control or monitoring of a computer-controlled entity operating in an environment by predicting a future state of the computer-controlled entity and/or its environment using sensor data which is indicative of a current state of the computer-controlled entity and/or its environment. The method includes using a first neural network for approximating a drift component of a stochastic differential equation and a second neural network for approximating a diffusion component of the stochastic differential equation, and discretizing the stochastic differential equation into time steps, and obtaining time-evolving mean and covariance functions based on the discretization and determining a probability distribution of a second state of the computer-controlled entity and/or its environment therefrom.

Abstract: A method for processing sensor data. The method includes receiving input sensor data, determining, starting from the input sensor data as initial state, a plurality of end states, including determining, for each end state, a sequence of states, wherein determining the sequence of states comprises, for each state of the sequence beginning with the initial state until the end state, a first Bayesian neural network determining a sample of a drift term in response to inputting the respective state, a second Bayesian neural network determining a sample of a diffusion term in response to inputting the respective state and determining a subsequent state by sampling a stochastic differential equation including the sample of the drift term as drift term and the sample of the diffusion term as diffusion term. An end state probability distribution is determined, and a processing result is determined from the end state probability distribution.