Large linear systems are ubiquitous in modern computational science. The main
recipe for solving them is iterative solvers with well-designed
preconditioners. Deep learning models may be used to precondition residuals
during iteration of such linear solvers as the conjugate gradient (CG) method.
Neural network models require an enormous number of parameters to approximate
well in this setup. Another approach is to take advantage of small graph neural
networks (GNNs) to construct preconditioners of the predefined sparsity
pattern. In our work, we recall well-established preconditioners from linear
algebra and use them as a starting point for training the GNN. Numerical
experiments demonstrate that our approach outperforms both classical methods
and neural network-based preconditioning. We also provide a heuristic
justification for the loss function used and validate our approach on complex
datasets.

大规模线性系统中使用迭代求解器和预处理器，我们使用小型图神经网络作为预处理器并与传统方法和神经网络预处理进行对比实验证明我们的方法更优。

从线性代数中学习：一种用于共轭梯度解算器的图神经网络预处理器设计方法

Learning from Linear Algebra: A Graph Neural Network Approach to  Preconditioner Design for Conjugate Gradient Solvers

Applying iterative solvers on sparsity-constrained optimization (SCO)
requires tedious mathematical deduction and careful programming/debugging that
hinders these solvers' broad impact. In the paper, the library skscope is
introduced to overcome such an obstacle. With skscope, users can solve the SCO
by just programming the objective function. The convenience of skscope is
demonstrated through two examples in the paper, where sparse linear regression
and trend filtering are addressed with just four lines of code. More
importantly, skscope's efficient implementation allows state-of-the-art solvers
to quickly attain the sparse solution regardless of the high dimensionality of
parameter space. Numerical experiments reveal the available solvers in skscope
can achieve up to 80x speedup on the competing relaxation solutions obtained
via the benchmarked convex solver. skscope is published on the Python Package
Index (PyPI) and Conda, and its source code is available at:
this https URL

用 skscope 可以通过编程目标函数来解决稀疏约束优化问题，该库的高效实现使得先进的求解器可以快速获得稀疏解。

skscope：Python 中的快速稀疏约束优化

skscope: Fast Sparsity-Constrained Optimization in Python

Recent advances in the field of machine learning open a new era in high
performance computing. Applications of machine learning algorithms for the
development of accurate and cost-efficient surrogates of complex problems have
already attracted major attention from scientists. Despite their powerful
approximation capabilities, however, surrogates cannot produce the `exact'
solution to the problem. To address this issue, this paper exploits up-to-date
ML tools and delivers customized iterative solvers of linear equation systems,
capable of solving large-scale parametrized problems at any desired level of
accuracy. Specifically, the proposed approach consists of the following two
steps. At first, a reduced set of model evaluations is performed and the
corresponding solutions are used to establish an approximate mapping from the
problem's parametric space to its solution space using deep feedforward neural
networks and convolutional autoencoders. This mapping serves a means to obtain
very accurate initial predictions of the system's response to new query points
at negligible computational cost. Subsequently, an iterative solver inspired by
the Algebraic Multigrid method in combination with Proper Orthogonal
Decomposition, termed POD-2G, is developed that successively refines the
initial predictions towards the exact system solutions. The application of
POD-2G as a standalone solver or as preconditioner in the context of
preconditioned conjugate gradient methods is demonstrated on several numerical
examples of large scale systems, with the results indicating its superiority
over conventional iterative solution schemes.

该论文基于最新的机器学习工具，提供了一种定制的迭代求解器，结合代数多重网格方法和 proper orthogonal decomposition（POD-2G），能够在任意所需精度下，求解大规模参数化问题中的线性方程系统。