Related papers: Exa-Dune -- Flexible PDE Solvers, Numerical Method…
Many complex systems can be accurately modeled as a set of coupled time-dependent partial differential equations (PDEs). However, solving such equations can be prohibitively expensive, easily taxing the world's largest supercomputers. One…
The Multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for Uncertainty Quantification (UQ) in Partial Differential Equation (PDE) models, combining model computations at different levels…
The demand for substantial increases in the spatial resolution of global weather- and climate- prediction models makes it necessary to use numerically efficient and highly scalable algorithms to solve the equations of large scale…
We present a highly scalable Monte Carlo (MC) three-dimensional photon transport simulation platform designed for heterogeneous computing systems. Through the development of a massively parallel MC algorithm using the Open Computing…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…
We investigate the potential of Graphics Processing Units (GPUs) to solve large-scale nonlinear programs with a dynamic structure. Using ExaModels, a GPU-accelerated automatic differentiation tool, and the interior-point solver MadNLP, we…
Fractional PDEs involving the fractional Laplacian on bounded domains are challenging because of hypersingular nonlocal kernels, exterior Dirichlet constraints, reduced boundary regularity, and the high computational cost in high…
Eulerian nonlinear uncertainty propagation methods often suffer from finite domain limitations and computational inefficiencies. A recent approach to this class of algorithm, Grid-based Bayesian Estimation Exploiting Sparsity, addresses the…
Discrete variational methods show excellent performance in numerical simulations of different mechanical systems. In this paper, we introduce an iterative procedure for the solution of discrete variational equations for boundary value…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial…
A new flow solver scalable on multiple Graphics Processing Units (GPUs) for direct numerical simulation of wall-bounded incompressible flow is presented. This solver utilizes a previously reported work (J. Comp. Physics, vol. 352 (2018),…
Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free…
Simulation is a powerful tool to better understand physical systems, but generally requires computationally expensive numerical methods. Downstream applications of such simulations can become computationally infeasible if they require many…
Communication overhead is a major obstacle to scaling distributed training systems. Gradient sparsification is a potential optimization approach to reduce the communication volume without significant loss of model fidelity. However,…
Neural networks have promise as surrogate partial differential equation (PDE) solvers, but it remains a challenge to use these concepts to solve problems with high accuracy and scalability. In this work, we show that neural network…
The combination of machine learning and physical laws has shown immense potential for solving scientific problems driven by partial differential equations (PDEs) with the promise of fast inference, zero-shot generalisation, and the ability…
Among the algorithms that are likely to play a major role in future exascale computing, the fast multipole method (FMM) appears as a rising star. Our previous recent work showed scaling of an FMM on GPU clusters, with problem sizes in the…
Machine learning (ML) models have emerged as a promising approach for solving partial differential equations (PDEs) in science and engineering. Previous ML models typically cannot generalize outside the training data; for example, a trained…