L3

Neural networks are undoubtedly successful in practical applications. However complete mathematical theory of why and when machine learning algorithms based on neural networks work has been elusive. Although various representation theorems ensures the existence of the ``perfect’’ parameters of the network, it has not been proved that these perfect parameters can be (efficiently) approximated by conventional algorithms, such as the stochastic gradient descent. This problem is well known, since the arising optimisation problem is non-convex. In this talk we show how the optimization problem becomes convex in the mean field limit for one-hidden layer networks and certain deep neural networks. Moreover we present optimality criteria for the distribution of the network parameters and show that the nonlinear Langevin dynamics converges to this optimal distribution. This is joint work with Kaitong Hu, Zhenjie Ren and Lukasz Szpruch. 

Many time-dependent PDEs -- KdV, Burgers, Gray-Scott, Allen-Cahn, Navier-Stokes and many others -- combine a higher-order linear term with a lower-order nonlinear term.  This talk will review the method of exponential integrators for solving such problems with better than 2nd-order accuracy in time.

Numerical simulations indicate that deep artificial neural networks (DNNs) seem to be able to overcome the curse of dimensionality in many computational  problems in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the dimension of the function which the DNN aims to approximate. However, there are only a few special situations where results in the literature can rigorously explain the success of DNNs when approximating high-dimensional functions.

In this talk it is revealed that DNNs do indeed overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. A crucial ingredient in our proof of this result is the fact that the artificial neural network used to approximate the PDE solution really is a deep artificial neural network with a large number of hidden layers.

Graph sampling with noise is a fundamental problem in graph signal processing (GSP). A popular biased scheme using graph Laplacian regularization (GLR) solves a system of linear equations for its reconstruction. Assuming this GLR-based reconstruction scheme, we propose a fast sampling strategy to maximize the numerical stability of the linear system--i.e., minimize the condition number of the coefficient matrix. Specifically, we maximize the eigenvalue lower bounds of the matrix that are left-ends of Gershgorin discs of the coefficient matrix, without eigen-decomposition. We propose an iterative algorithm to traverse the graph nodes via Breadth First Search (BFS) and align the left-ends of all corresponding Gershgorin discs at lower-bound threshold T using two basic operations: disc shifting and scaling. We then perform binary search to maximize T given a sample budget K. Experiments on real graph data show that the proposed algorithm can effectively promote large eigenvalue lower bounds, and the reconstruction MSE is the same or smaller than existing sampling methods for different budget K at much lower complexity.

A defect-based a posteriori error estimate for Krylov subspace approximations to the matrix exponential is introduced. This error estimate constitutes an upper norm bound on the error and can be computed during the construction of the Krylov subspace with nearly no computational effort. The matrix exponential function itself can be understood as a time propagation with restarts. In practice, we are interested in finding time steps for which the error of the Krylov subspace approximation is smaller than a given tolerance. Finding correct time steps is a simple task with our error estimate. Apart from step size control, the upper error bound can be used on the fly to test if the dimension of the Krylov subspace is already sufficiently large to solve the problem in a single time step with the required accuracy.

During this seminar, we will present a new mathematical model describing the transport of nitric oxide (NO) in a realistic geometrical representation of the lungs. Nitric oxide (NO) is naturally produced in the bronchial region of the lungs. It is a physiological molecule that has antimicrobial properties and allows the relaxation of muscles. It is well known that the measurement of the molar fraction of NO in the exhaled air, the so-called FeNO, allows a monitoring of asthmatic patients, since the production of this molecule in the lungs is increased in case of inflammation. However, recent clinical studies have shown that the amount of NO in the exhaled air can also be affected by « non-inflammatory » processes, such as the action of a bronchodilator or a respiratory physiotherapy session for a patient with cystic fibrosis. Using our new model, we will highlight the complex interplay between different transport phenomena in the lungs. More specifically, we will show why changes taking place in the deepest part of the lungs are expected to impact the FeNO. This gives a new light on the clinical studies mentioned below, allowing to confer a new role to the NO for the management of various pulmonary pathologies.

In Formula 1 engineers strive to produce the fastest car possible for their drivers. A lap simulation provides an objective evaluation of the performance of the car and the subsequent lap time achieved. Using this information, engineers aim to test new car concepts, determine performance limitations or compromises, and identify the sensitivity of performance to car setup parameters.

The latest state of the art lap simulation techniques use optimal control approaches. Optimisation methods are employed to derive the optimal control inputs of the car that achieve the fastest lap time within the constraints of the system. The resulting state trajectories define the complete behaviour of the car. Such approaches aim to create more robust, realistic and powerful simulation output compared to traditional methods.

In this talk we discuss our latest work in this area. A dynamic vehicle model is used within a free-trajectory solver based on direct optimal control methods. We discuss the reasons behind our design choices, our progress to date, and the issues we have faced during development. Further, we look at the short and long term aims of our project and how we wish to develop our mathematical methods in the future.

Simple mathematical models have had remarkable successes in biology, framing how we understand a host of mechanisms and processes. However, with the advent of a host of new experimental technologies, the last ten years has seen an explosion in the amount and types of data now being generated. Increasingly larger and more complicated processes are now being explored, including large signalling or gene regulatory networks, and the development, dynamics and disease of entire cells and tissues. As such, the mechanistic, mathematical models developed to interrogate these processes are also necessarily growing in size and complexity. These detailed models have the potential to provide vital insights where data alone cannot, but to achieve this goal requires meeting significant mathematical challenges. In this talk, I will outline some of these challenges, and recent steps we have taken in addressing them.

It is well known that a rough path is uniquely determined by its signature (the collection of global iterated path integrals) up to tree-like pieces. However, the proof the uniqueness theorem is non-constructive and does not give us information about how quantitative properties of the path can be explicitly recovered from its signature. In this talk, we examine the quantitative relationship between the local p-variation of a rough path and the tail asymptotics of its signature for the simplest type of rough paths ("line segments"). What lies at the core of the work a novel technique based on the representation theory of complex semisimple Lie algebras. 

This talk is based on joint work with Horatio Boedihardjo and Nikolaos Souris