Danny Lathouwers

Alessandro Longhi, Danny Lathouwers, Zoltán Perkó

Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently increasingly Deep Learning (DL) techniques. A conclusive methodology capable of accounting both for (continuous) time and parameter dependency in such DL models however is still lacking. In this paper, we propose an autoregressive and data-driven method using the analogy with classical numerical solvers for time-dependent, parametric and (typically) nonlinear PDEs. We present how Dimensionality Reduction (DR) can be coupled with Neural Ordinary Differential Equations (NODEs) in order to learn the solution operator of arbitrary PDEs. The idea of our work is that it is possible to map the high-fidelity (i.e., high-dimensional) PDE solution space into a reduced (low-dimensional) space, which subsequently exhibits dynamics governed by a (latent) Ordinary Differential Equation (ODE). Solving this (easier) ODE in the reduced space allows avoiding solving the PDE in the high-dimensional solution space, thus decreasing the computational burden for repeated calculations for e.g., uncertainty quantification or design optimization purposes. The main outcome of this work is the importance of exploiting DR as opposed to the recent trend of building large and complex architectures: we show that by leveraging DR we can deliver not only more accurate predictions, but also a considerably lighter and faster DL model compared to existing methodologies.

Oscar Pastor-Serrano, Danny Lathouwers, Zoltán Perkó

One of the main problems with biomedical signals is the limited amount of patient-specific data and the significant amount of time needed to record the sufficient number of samples needed for diagnostic and treatment purposes. In this study, we present a framework to simultaneously generate and classify biomedical time series based on a modified Adversarial Autoencoder (AAE) algorithm and one-dimensional convolutions. Our work is based on breathing time series, with specific motivation to capture breathing motion during radiotherapy lung cancer treatments. First, we explore the potential in using the Variational Autoencoder (VAE) and AAE algorithms to model breathing from individual patients. We extend the AAE algorithm to allow joint semi-supervised classification and generation of different types of signals. To simplify the modeling task, we introduce a pre-processing and post-processing compressing algorithm that transforms the multi-dimensional time series into vectors containing time and position values, which are transformed back into time series through an additional neural network. By incorporating few labeled samples during training, our model outperforms other purely discriminative networks in classifying breathing baseline shift irregularities from a dataset completely different from the training set. To our knowledge, the presented framework is the first approach that unifies generation and classification within a single model for this type of biomedical data, enabling both computer aided diagnosis and augmentation of labeled samples within a single framework.