Diyora Salimova

TL;DR: It is revealed that DNNs do overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients.

TL;DR: The main result of this paper shows that if a function can be approximated by means of some suitable discrete approximation scheme without the curse of dimensionality and if there exist DNNs which satisfy certain regularity properties and which approximate this discrete approximation Scheme, then the function itself can also be approximating with DNN's without the cursed dimensionality.

TL;DR: In this article, a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations (SPDEs) with possibly non-globally monotone nonlinearities, such as Stochastic Kuramoto-Sivashinsky equations, is presented.

TL;DR: In this article, the authors consider the stochastic Kuramoto-Sivashinsky equations and show that the nonlinearity of convolutional neural networks can be characterized by Fernique's theorem.

TL;DR: In this article, it was shown that the number of parameters used to describe the employed DNN grows at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy.