Regularized dynamical parametric approximation of stiff evolution problems

Evolutionary deep neural networks have emerged as a rapidly growing field of research. This paper studies numerical integrators for such and other classes of nonlinear parametrizations $ u(t) = \Phi(\theta(t)) $, where the evolving parameters $\theta(t)$ are to be computed. The primary focus is on tackling the challenges posed by the combination of stiff evolution problems and irregular parametrizations, which typically arise with neural networks, tensor networks, flocks of evolving Gaussians, and in further cases of overparametrization. We propose and analyse regularized parametric versions of the implicit Euler method and higher-order implicit Runge--Kutta methods for the time integration of the parameters in nonlinear approximations to evolutionary partial differential equations and large systems of stiff ordinary differential equations. At each time step, an ill-conditioned nonlinear optimization problem is solved approximately with a few regularized Gauss--Newton iterations. Error bounds for the resulting parametric integrator are derived by relating the computationally accessible Gauss--Newton iteration for the parameters to the computationally inaccessible Newton iteration for the underlying non-parametric time integration scheme. The theoretical findings are supported by numerical experiments that are designed to show key properties of the proposed parametric integrators.

Precise localization within the GI tract by combining classification of CNNs and time-series analysis of HMMs

This paper presents a method to efficiently classify the gastroenterologic section of images derived from Video Capsule Endoscopy (VCE) studies by exploring the combination of a Convolutional Neural Network (CNN) for classification with the time-series analysis properties of a Hidden Markov Model (HMM). It is demonstrated that successive time-series analysis identifies and corrects errors in the CNN output. Our approach achieves an accuracy of $98.04\%$ on the Rhode Island (RI) Gastroenterology dataset. This allows for precise localization within the gastrointestinal (GI) tract while requiring only approximately 1M parameters and thus, provides a method suitable for low power devices

The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions

Inference and simulation in the context of high-dimensional dynamical systems remain computationally challenging problems. Some form of dimensionality reduction is required to make the problem tractable in general. In this paper, we propose a novel approximate Gaussian filtering and smoothing method which propagates low-rank approximations of the covariance matrices. This is accomplished by projecting the Lyapunov equations associated with the prediction step to a manifold of low-rank matrices, which are then solved by a recently developed, numerically stable, dynamical low-rank integrator. Meanwhile, the update steps are made tractable by noting that the covariance update only transforms the column space of the covariance matrix, which is low-rank by construction. The algorithm differentiates itself from existing ensemble-based approaches in that the low-rank approximations of the covariance matrices are deterministic, rather than stochastic. Crucially, this enables the method to reproduce the exact Kalman filter as the low-rank dimension approaches the true dimensionality of the problem. Our method reduces computational complexity from cubic (for the Kalman filter) to \emph{quadratic} in the state-space size in the worst-case, and can achieve \emph{linear} complexity if the state-space model satisfies certain criteria. Through a set of experiments in classical data-assimilation and spatio-temporal regression, we show that the proposed method consistently outperforms the ensemble-based methods in terms of error in the mean and covariance with respect to the exact Kalman filter. This comes at no additional cost in terms of asymptotic computational complexity.