9 research outputs found
A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs
Algebraic multigrid (AMG) methods are among the most efficient solvers for
linear systems of equations and they are widely used for the solution of
problems stemming from the discretization of Partial Differential Equations
(PDEs). The most severe limitation of AMG methods is the dependence on
parameters that require to be fine-tuned. In particular, the strong threshold
parameter is the most relevant since it stands at the basis of the construction
of successively coarser grids needed by the AMG methods. We introduce a novel
Deep Learning algorithm that minimizes the computational cost of the AMG method
when used as a finite element solver. We show that our algorithm requires
minimal changes to any existing code. The proposed Artificial Neural Network
(ANN) tunes the value of the strong threshold parameter by interpreting the
sparse matrix of the linear system as a black-and-white image and exploiting a
pooling operator to transform it into a small multi-channel image. We
experimentally prove that the pooling successfully reduces the computational
cost of processing a large sparse matrix and preserves the features needed for
the regression task at hand. We train the proposed algorithm on a large dataset
containing problems with a highly heterogeneous diffusion coefficient defined
in different three-dimensional geometries and discretized with unstructured
grids and linear elasticity problems with a highly heterogeneous Young's
modulus. When tested on problems with coefficients or geometries not present in
the training dataset, our approach reduces the computational time by up to 30%
Reinforcement Learning for Adaptive Mesh Refinement
Large-scale finite element simulations of complex physical systems governed
by partial differential equations crucially depend on adaptive mesh refinement
(AMR) to allocate computational budget to regions where higher resolution is
required. Existing scalable AMR methods make heuristic refinement decisions
based on instantaneous error estimation and thus do not aim for long-term
optimality over an entire simulation. We propose a novel formulation of AMR as
a Markov decision process and apply deep reinforcement learning (RL) to train
refinement policies directly from simulation. AMR poses a new problem for RL in
that both the state dimension and available action set changes at every step,
which we solve by proposing new policy architectures with differing generality
and inductive bias. The model sizes of these policy architectures are
independent of the mesh size and hence scale to arbitrarily large and complex
simulations. We demonstrate in comprehensive experiments on static function
estimation and the advection of different fields that RL policies can be
competitive with a widely-used error estimator and generalize to larger, more
complex, and unseen test problems.Comment: 14 pages, 13 figure
Graph Neural Networks and Applied Linear Algebra
Sparse matrix computations are ubiquitous in scientific computing. With the
recent interest in scientific machine learning, it is natural to ask how sparse
matrix computations can leverage neural networks (NN). Unfortunately,
multi-layer perceptron (MLP) neural networks are typically not natural for
either graph or sparse matrix computations. The issue lies with the fact that
MLPs require fixed-sized inputs while scientific applications generally
generate sparse matrices with arbitrary dimensions and a wide range of nonzero
patterns (or matrix graph vertex interconnections). While convolutional NNs
could possibly address matrix graphs where all vertices have the same number of
nearest neighbors, a more general approach is needed for arbitrary sparse
matrices, e.g. arising from discretized partial differential equations on
unstructured meshes. Graph neural networks (GNNs) are one approach suitable to
sparse matrices. GNNs define aggregation functions (e.g., summations) that
operate on variable size input data to produce data of a fixed output size so
that MLPs can be applied. The goal of this paper is to provide an introduction
to GNNs for a numerical linear algebra audience. Concrete examples are provided
to illustrate how many common linear algebra tasks can be accomplished using
GNNs. We focus on iterative methods that employ computational kernels such as
matrix-vector products, interpolation, relaxation methods, and
strength-of-connection measures. Our GNN examples include cases where
parameters are determined a-priori as well as cases where parameters must be
learned. The intent with this article is to help computational scientists
understand how GNNs can be used to adapt machine learning concepts to
computational tasks associated with sparse matrices. It is hoped that this
understanding will stimulate data-driven extensions of classical sparse linear
algebra tasks