M.A. Botchev

Artem Chashchin, Mikhail Botchev, Ivan Oseledets et al.

Forecasting time series and time-dependent data is a common problem in many applications. One typical example is solving ordinary differential equation (ODE) systems $\dot{x}=F(x)$. Oftentimes the right hand side function $F(x)$ is not known explicitly and the ODE system is described by solution samples taken at some time points. Hence, ODE solvers cannot be used. In this paper, a data-driven approach to learning the evolution of dynamical systems is considered. We show how by training neural networks with ResNet-like architecture on the solution samples, models can be developed to predict the ODE system solution further in time. By evaluating the proposed approaches on three test ODE systems, we demonstrate that the neural network models are able to reproduce the main dynamics of the systems qualitatively well. Moreover, the predicted solution remains stable for much longer times than for other currently known models.

G. L. Kooij, M. A. Botchev, B. J. Geurts

A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as a source term, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection-diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection-diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.