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Neural Ordinary Differential Equations (NODEs) are a new class of models that transform data continuously through infinite-depth architectures. The continuous nature of NODEs has made them particularly suitable for learning the dynamics of complex physical systems. While previous work has mostly been focused on first order ODEs, the dynamics of many systems, especially in classical physics, are governed by second order laws. In this work, we consider Second Order Neural ODEs (SONODEs). We show how the adjoint sensitivity method can be extended to SONODEs and prove that the optimisation of a first order coupled ODE is equivalent and computationally more efficient. Furthermore, we extend the theoretical understanding of the broader class of Augmented NODEs (ANODEs) by showing they can also learn higher order dynamics with a minimal number of augmented dimensions, but at the cost of interpretability. This indicates that the advantages of ANODEs go beyond the extra space offered by the augmented dimensions, as originally thought. Finally, we compare SONODEs and ANODEs on synthetic and real dynamical systems and demonstrate that the inductive biases of the former generally result in faster training and better performance., Comment: Camera-ready version for NeurIPS 2020. Contains 28 pages, 15 figures

Deep learning methods hold great promise for the automatic analysis of large-scale remote sensing data in archaeological research. Here, we present a robust approach to locating ancient Maya architectures (buildings, aguadas, and platforms) based on integrated segmentation of satellite imagery and aerial laser scanning data. Deep learning models with different architectures and loss functions were trained and combined to form an ensemble for pixel-wise classification. We applied both training data augmentation as well as test-time augmentation and performed morphological cleaning in the postprocessing phase. Our approach was evaluated in the context of the “Discover the mysteries of the Maya: An Integrated Image Segmentation Challenge” at ECML PKDD 2021 and achieved one of the best results with an average IoU of 0.8183.

International initiatives such as the Molecular Taxonomy of Breast Cancer International Consortium (METABRIC), Cancer Genome Atlas (TCGA), and the International Cancer Genome Consortium (ICGC) are collecting multiple data sets at different genome-scales with the aim to identify novel cancer bio-markers and predict patient survival. To analyse such data, several machine learning, bioinformatics and statistical methods have been applied, among them neural networks such as autoencoders. Although these models provide a good statistical learning framework to analyse multi-omic and/or clinical data, there is a distinct lack of work on how to integrate diverse patient data and identify the optimal design best suited to the available data. In this paper, we investigate several autoencoder architectures that integrate a variety of cancer patient data types (e.g., multi-omics and clinical data). We perform extensive analyses of these approaches and provide a clear methodological and computational framework for designing systems that enable clinicians to investigate cancer traits and translate the results into clinical applications. We demonstrate how these networks can be designed, built and, in particular, applied to tasks of integrative analyses of heterogeneous breast cancer data. The results show that these approaches yield relevant data representations that, in turn, lead to accurate and stable diagnosis.