Abstract

Abstract

Numerical solutions of Partial Differential Equations with Finite Element Method have multiple applications in science and engineering. Several challenging problems require special stabilization methods to deliver accurate results of the numerical simulations. The advection-dominated diffusion problem is an example of such problems. They are employed to model pollution propagation in the atmosphere. Unstable numerical methods generate unphysical oscillations, and they make no physical sense. Obtaining accurate and stable numerical simulations is difficult, and the method of stabilization depends on the parameters of the partial differential equations. They require a deep knowledge of an expert in the field of numerical analysis. We propose a method to construct and train an artificial expert in stabilizing numerical simulations based on partial differential equations. We create a neural network-driven artificial intelligence that makes decisions about the method of stabilizing computer simulations. It will automatically stabilize difficult numerical simulations in a linear computational cost by generating the optimal test functions. These test functions can be utilized for building an unconditionally stable system of linear equations. The optimal test functions proposed by artificial intelligence will not depend on the right-hand side, and thus they may be utilized in a large class of PDE-based simulations with different forcing and boundary conditions. We test our method on the model one-dimensional advection-dominated diffusion problem.

Abstract

We investigate scaling and efficiency of the deep neural network multigrid method (DNN-MG), a novel neural network-based technique for the simulation of the Navier-Stokes equations that combines an adaptive geometric multigrid solver with a recurrent neural network with memory. The neural network replaces in DNN-MG one or multiple finest multigrid layers and provides a correction for the classical solve in the next time step. This leads to little degradation in the solution quality while substantially reducing the overall computational costs. At the same time, the use of the multigrid solver at the coarse scales allows for a compact network that is easy to train, generalizes well, and allows for the incorporation of physical constraints. In this work, we investigate how the network size affects training and solution quality and the overall runtime of the computations.

Abstract

Abstract

Abstract

This contribution presents a combined framework to perform parametric surrogate modeling of vibroacoustic problems that enables efficient training of large-scale problems. The proposed framework combines the active subspace method to perform dimensionality reduction of high-dimensional problems and thereafter a clustering-based approach within the identified active subspace region to yield smaller training clusters. Finally, a trained neural network assists the cluster classification task for any desired parameter point so as to query the parametric system response during the online phase.