Timo Welti

Arnulf Jentzen, Diyora Salimova, Timo Welti

In this paper we propose and analyze explicit space-time discrete numerical approximations for additive space-time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space-time white noise. The main result of this paper proves that the proposed explicit space-time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space-time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space-time discrete approximation method in the case of the stochastic Burgers equations with space-time white noise.

Ladislas Jacobe de Naurois, Arnulf Jentzen, Timo Welti

Although for a number of semilinear stochastic wave equations existence and uniqueness results for corresponding solution processes are known from the literature, these solution processes are typically not explicitly known and numerical approximation methods are needed in order for mathematical modelling with stochastic wave equations to become relevant for real world applications. This, in turn, requires the numerical analysis of convergence rates for such numerical approximation processes. A recent article by the authors proves upper bounds for weak errors for spatial spectral Galerkin approximations of a class of semilinear stochastic wave equations. The findings there are complemented by the main result of this work, that provides lower bounds for weak errors which show that in the general framework considered the established upper bounds can essentially not be improved.

Arnulf Jentzen, Timo Welti

In spite of the accomplishments of deep learning based algorithms in numerous applications and very broad corresponding research interest, at the moment there is still no rigorous understanding of the reasons why such algorithms produce useful results in certain situations. A thorough mathematical analysis of deep learning based algorithms seems to be crucial in order to improve our understanding and to make their implementation more effective and efficient. In this article we provide a mathematically rigorous full error analysis of deep learning based empirical risk minimisation with quadratic loss function in the probabilistically strong sense, where the underlying deep neural networks are trained using stochastic gradient descent with random initialisation. The convergence speed we obtain is presumably far from optimal and suffers under the curse of dimensionality. To the best of our knowledge, we establish, however, the first full error analysis in the scientific literature for a deep learning based algorithm in the probabilistically strong sense and, moreover, the first full error analysis in the scientific literature for a deep learning based algorithm where stochastic gradient descent with random initialisation is the employed optimisation method.