In this paper, we train turbulence models based on convolutional neural networks. These learned improve under-resolved low resolution solutions to the incompressible Navier-Stokes equations at simulation time. Our study involves development of a differentiable numerical solver that supports propagation optimisation gradients through multiple steps. The significance property is demonstrated by s...

We present a novel, systematic procedure for smoothing non-smooth dynamical systems. In particular we introduce the Moving Average Transformation which can be thought of as a change of variables which transforms discontinuous systems into dynamically equivalent continuous systems and continuous but non-differentiable systems into dynamically equivalent differentiable systems. This smoothing giv...

By combining the explicit formula of the Shannon informational entropy ) , ( Y X H for two random variables X and Y , with the entropy )) ( ( X f H of ) (X f where (.) f is a real-valued differentiable function, we have shown that the density of the amount of information in Shannon sense involved in a non-random differentiable function is defined by the logarithm of the absolute value of its de...

We make a connection between famous analytical object introduced in the 1860s by Riemann, as well some variants of it, and nonlinear geometric PDE, binormal curvature flow. As consequence, this has non-obvious interpretation. recall that flow is standard model for evolution vortex filaments. prove existence solutions with smooth trajectories are close desired to curves multifractal behavior. Fi...

We extend the DuBois-Reymond necessary optimality condition and Noether’s symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether’s theorem are proved, covering problems of the calculus of variations with functionals defined on sets of non-differentiable functions, as well as more general non-differentiable problems of optimal control. As an ...