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

Numerical approximation of statistical solutions of scalar conservation laws

We propose efficient numerical algorithms for approximating statistical solutions of scalar conservation laws. The proposed algorithms combine finite volume spatio-temporal approximations with Monte Carlo and multi-level Monte Carlo discretizations of the probability space. Both sets of methods are proved to converge to the entropy statistical solution. We also prove that there is a considerable gain in efficiency resulting from the multi-level Monte Carlo method over the standard Monte Carlo method. Numerical experiments illustrating the ability of both methods to accurately compute multi-point statistical quantities of interest are also presented

Statistical solutions of hyperbolic conservation laws I: Foundations

We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on $p$-integrable functions. To do so, we prove the equivalence between probability measures on $L^p$ spaces and infinite families of \textit{correlation measures}. Each member of this family, termed a \textit{correlation marginal}, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions. We also prove that any probability measure on a $L^p$ space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure. We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal. These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions. Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions even for atomic initial data. For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting \textit{entropy statistical solutions} exist, are unique and are stable with respect to the $1$-Wasserstein metric on probability measures on $L^1$

Stable finite difference schemes for the magnetic induction equation with Hall effect

We consider a sub-model of the Hall-MHD equations: the so-called magnetic induction equations with Hall effect. These equations are non-linear and include third-order spatial and spatio-temporal mixed derivatives. We show that the energy of the solutions is bounded and design finite difference schemes that preserve the energy bounds for the continuous problem. We design both divergence preserving schemes and schemes with bounded divergence. We present a set of numerical experiments that demonstrate the robustness of the proposed scheme

Entropy Stable Numerical Schemes for Two-Fluid Plasma Equations

Two-fluid ideal plasma equations are a generalized form of the ideal MHD equations in which electrons and ions are considered as separate species. The design of efficient numerical schemes for the these equations is complicated on account of their non-linear nature and the presence of stiff source terms, especially for high charge to mass ratios and for low Larmor radii. In this article, we design entropy stable finite difference schemes for the two-fluid equations by combining entropy conservative fluxes and suitable numerical diffusion operators. Furthermore, to overcome the time step restrictions imposed by the stiff source terms, we devise time-stepping routines based on implicit-explicit (IMEX)-Runge Kutta (RK) schemes. The special structure of the two-fluid plasma equations is exploited by us to design IMEX schemes in which only local (in each cell) linear equations need to be solved at each time step. Benchmark numerical experiments are presented to illustrate the robustness and accuracy of these scheme

Physics Informed Neural Networks for Simulating Radiative Transfer

We propose a novel machine learning algorithm for simulating radiative transfer. Our algorithm is based on physics informed neural networks (PINNs), which are trained by minimizing the residual of the underlying radiative tranfer equations. We present extensive experiments and theoretical error estimates to demonstrate that PINNs provide a very easy to implement, fast, robust and accurate method for simulating radiative transfer. We also present a PINN based algorithm for simulating inverse problems for radiative transfer efficiently

Computation of measure-valued solutions for the incompressible Euler equations

We combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness of measure valued solutions as a solution framework for the Euler equations. Furthermore, we report an extensive computational study of the two dimensional vortex sheet, which indicates that the computed measure valued solution is non-atomic and implies possible non-uniqueness of weak solutions constructed by Delort

Error analysis for deep neural network approximations of parametric hyperbolic conservation laws

We derive rigorous bounds on the error resulting from the approximation of the solution of parametric hyperbolic scalar conservation laws with ReLU neural networks. We show that the approximation error can be made as small as desired with ReLU neural networks that overcome the curse of dimensionality. In addition, we provide an explicit upper bound on the generalization error in terms of the training error, number of training samples and the neural network size. The theoretical results are illustrated by numerical experiments