Related papers: Manifold learning techniques and model reduction a…
This work explores optimization methods on hyperbolic manifolds. Building on Riemannian optimization principles, we extend the Hyperbolic Stochastic Gradient Descent (a specialization of Riemannian SGD) to a Hyperbolic Adam optimizer. While…
Many reduced order models are neither robust with respect to the parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In this study, we put forth a robust machine learning framework…
The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…
The quantification of multivariate uncertainties in partial differential equations can easily exceed any computing capacity unless proper measures are taken to reduce the complexity of the model. In this work, we propose a multidimensional…
Diffusion-based methods represented as stochastic differential equations on a continuous-time domain have recently proven successful as a non-adversarial generative model. Training such models relies on denoising score matching, which can…
The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a…
Manifold-learning techniques are routinely used in mining complex spatiotemporal data to extract useful, parsimonious data representations/parametrizations; these are, in turn, useful in nonlinear model identification tasks. We focus here…
The properties of nonlinear PDEs that generate filtered solutions are explored with particular attention given to the constraints on the residual term. The analysis is carried out for nonlinear PDEs with an emphasis on evolution problems…
We propose an online learning framework for forecasting nonlinear spatio-temporal signals (fields). The method integrates (i) dimensionality reduction, here, a simple proper orthogonal decomposition (POD) projection; (ii) a generalized…
In this work, we develop a method for learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) based on the Conservation-dissipation Formalism of irreversible thermodynamics. As governing…
A data-driven closure modeling based on proper orthogonal decomposition (POD) temporal modes is used to obtain stable and accurate reduced order models (ROMs) of unsteady compressible flows. Model reduction is obtained via Galerkin and…
We introduce a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The proposed method is based on a coarse grid and iteratively improves the solution's accuracy by solving local elliptic problems in…
We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the…
Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A…
We consider evolutionary systems, i.e. systems of linear partial differential equations arising from the mathematical physics. For these systems there exists a general solution theory in exponentially weighted spaces which can be exploited…
Real-time accurate solutions of large-scale complex dynamical systems are critically needed for control, optimization, uncertainty quantification, and decision-making in practical engineering and science applications, particularly in…
This paper discusses a Python interface for the recently published DUNE-FEM-DG module which provides highly efficient implementations of the Discontinuous Galerkin (DG) method for solving a wide range of non linear partial differential…
An hp-adaptive Discontinuous Galerkin Method for electromagnetic wave propagation phenomena in the time-domain is proposed. The method is highly efficient and allows for the first time the adaptive full-wave simulation of transient problems…
This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many…
Traditional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) suffer from severe limitations when dealing with nonlinear time-dependent parametrized PDEs,…