SynopsisThe aim of this paper is to showcase a handful of mathematical challenges found in classical literature and to offer possible ways of integrating classical literature in mathematics lessons. We analyze works from a range of authors such as Jules Verne, Anton Chekhov, and others. We also propose ideas for further tasks. Most of the problems can be restated in terms of simple mathematical equations, and they can often be solved without a computer. Nevertheless, we use the computer program Mathcad to solve the problems and to illustrate the solutions to enhance the reader's mathematical experience.
We propose a novel scheme for fitting heavily parameterized non-linear stochastic differential equations (SDEs). We assign a prior on the parameters of the SDE drift and diffusion functions to achieve a Bayesian model. We then infer this model using the well-known local reparameterized trick for the first time for empirical Bayes, i.e. to integrate out the SDE parameters. The model is then fit by maximizing the likelihood of the resultant marginal with respect to a potentially large number of hyperparameters, which prohibits stable training. As the prior parameters are marginalized, the model also no longer provides a principled means to incorporate prior knowledge. We overcome both of these drawbacks by deriving a training loss that comprises the marginal likelihood of the predictor and a PAC-Bayesian complexity penalty. We observe on synthetic as well as real-world time series prediction tasks that our method provides an improved model fit accompanied with favorable extrapolation properties when provided a partial description of the environment dynamics. Hence, we view the outcome as a promising attempt for building cutting-edge hybrid learning systems that effectively combine firstprinciple physics and data-driven approaches. * equal contribution Preprint. Under review.
Graph neural networks are often used to model interacting dynamical systems since they gracefully scale to systems with a varying and high number of agents. While there has been much progress made for deterministic interacting systems, modeling is much more challenging for stochastic systems in which one is interested in obtaining a predictive distribution over future trajectories. Existing methods are either computationally slow since they rely on Monte Carlo sampling or make simplifying assumptions such that the predictive distribution is unimodal. In this work, we present a deep state-space model which employs graph neural networks in order to model the underlying interacting dynamical system. The predictive distribution is multimodal and has the form of a Gaussian mixture model, where the moments of the Gaussian components can be computed via deterministic moment matching rules. Our moment matching scheme can be exploited for sample-free inference, leading to more efficient and stable training compared to Monte Carlo alternatives. Furthermore, we propose structured approximations to the covariance matrices of the Gaussian components in order to scale up to systems with many agents. We benchmark our novel framework on two challenging autonomous driving datasets. Both confirm the benefits of our method compared to state-of-the-art methods. We further demonstrate the usefulness of our individual contributions in a carefully designed ablation study and provide a detailed runtime analysis of our proposed covariance approximations. Finally, we empirically demonstrate the generalization ability of our method by evaluating its performance on unseen scenarios.
In this paper a method for detecting and furthermore estimating the intensity of cavitation occurrences in hydraulic turbines is presented. The method relies on analyzing high frequency signals with a convolutional neural network (CNN). The CNN is trained in an adversarial manner in order to get more robust results. After successful training the obtained network is modified in such a way, that it is possible to obtain estimations of the intensity. For evaluation purposes a separate dataset is investigated.
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