A machine learning framework for data driven acceleration of computations of differential equations

We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an offline training process by (approximately) minimizing suitable (possibly non-convex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying differential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods

Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem

In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical H\"older smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method

Thermo-Mechanical Wave Propagation In Shape Memory Alloy Rod With Phase Transformations

Many new applications of ferroelastic materials require a better understanding of their dynamics that often involve phase transformations. In such cases, an important prerequisite is the understanding of wave propagation caused by pulse-like loadings. In the present study, a mathematical model is developed to analyze the wave propagation process in shape memory alloy rods. The first order martensite transformations and associated thermo-mechanical coupling effects are accounted for by employing the modified Ginzburg-Landau-Devonshire theory. The Landau-type free energy function is employed to characterize different phases, while a Ginzburg term is introduced to account for energy contributions from phase boundaries. The effect of internal friction is represented by a Rayleigh dissipation term. The resulted nonlinear system of PDEs is reduced to a differential-algebraic system, and Chebyshev's collocation method is employed together with the backward differentiation method. A series of numerical experiments are performed. Wave propagations caused by impact loadings are analyzed for different initial temperatures. It is demonstrated that coupled waves will be induced in the material. Such waves will be dissipated and dispersed during the propagation process, and phase transformations in the material will complicate their propagation patterns. Finally, the influence of internal friction and capillary effects on the process of wave propagation is analyzed numerically.Comment: Keywords: nonlinear waves, thermo-mechanical coupling, martensite transformations, Ginzburg-Landau theory, Chebyshev collocation metho

Second-order Stable Finite Difference Schemes for the Time-fractional Diffusion-wave Equation

We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and the central difference in space. We use the generalized Newton-Gregory formula in time for the second scheme and its modification for the third scheme. While the second scheme is conditionally stable, the first and the third schemes are stable. We apply the methodology to the considered equation with also linear advection-reaction terms and also obtain second-order schemes both in time and space. Numerical examples with comparisons among the proposed schemes and the existing ones verify the theoretical analysis and show that the present schemes exhibit better performances than the known ones