arXiv subjects

Anna Paszynska

Publications and source records attributed to Anna Paszynska.

3 recordsLinked to original sources

arXiv2023

Physics Informed Neural Networks with strong and weak residuals for advection-dominated diffusion problems

This paper deals with the following important research questions. Is it possible to solve challenging advection-dominated diffusion problems in one and two dimensions using Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN)? How does it compare to the higher-order and continuity Finite Element Method (FEM)? How to define the loss functions for PINN and VPINN so they converge to the correct solutions? How to select points or test functions for training of PINN and VPINN? We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson-Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov-Galerkin (PG) formulation and present the FEM solution obtained with the PG method. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods.

math.NA↗

arXiv2022

Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices

Petrov-Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approximation in terms of a problem-dependent energy norm. This ideal approach has two shortcomings: first, we need to explicitly know the set of optimal test functions; and second, the optimal test functions may have large supports inducing expensive dense linear systems. Nevertheless, parametric families of PDEs are an example where it is worth investing some (offline) computational effort to obtain stabilized linear systems that can be solved efficiently, for a given set of parameters, in an online stage. Therefore, as a remedy for the first shortcoming, we explicitly compute (offline) a function mapping any PDE-parameter, to the matrix of coefficients of optimal test functions (in a basis expansion) associated with that PDE-parameter. Next, as a remedy for the second shortcoming, we use the low-rank approximation to hierarchically compress the (non-square) matrix of coefficients of optimal test functions. In order to accelerate this process, we train a neural network to learn a critical bottleneck of the compression algorithm (for a given set of PDE-parameters). When solving online the resulting (compressed) Petrov-Galerkin formulation, we employ a GMRES iterative solver with inexpensive matrix-vector multiplications thanks to the low-rank features of the compressed matrix. We perform experiments showing that the full online procedure as fast as the original (unstable) Galerkin approach. In other words, we get the stabilization with hierarchical matrices and neural networks practically for free. We illustrate our findings by means of 2D Eriksson-Johnson and Hemholtz model problems.

math.NA↗

arXiv2022

Complexity of direct and iterative solvers on space-time formulations versus time--marching schemes for h-refined grids towards singularities

We study computational complexity aspects for Finite Element formulations considering hypercubic space--time full and time--marching discretization schemes for $h$--refined grids towards singularities. We perform a relatively comprehensive study of comparing the computational time via time complexities of direct and iterative solvers. We focus on the space-time formulation with refined computational grids and on the corresponding time slabs, namely, computational grids obtained by taking the "cross-sections" of the refined space-time mesh. We compare the computational complexity of the space-time formulation and the corresponding time--marching scheme. Our consideration concerns the computational complexity of the multi-frontal solvers, the iterative solvers, as well as the static condensation. Numerical experiments with Octave confirm our theoretical findings.

math.NA↗