arXiv subjects

Yuri M. Laevsky

Publications and source records attributed to Yuri M. Laevsky.

3 recordsLinked to original sources

arXiv2025

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

Large linear systems are ubiquitous in modern computational science and engineering. The main recipe for solving them is the use of Krylov subspace iterative methods with well-designed preconditioners. Recently, GNNs have been shown to be a promising tool for designing preconditioners to reduce the overall computational cost of iterative methods by constructing them more efficiently than with classical linear algebra techniques. Preconditioners designed with these approaches cannot outperform those designed with classical methods in terms of the number of iterations in CG. In our work, we recall well-established preconditioners from linear algebra and use them as a starting point for training the GNN to obtain preconditioners that reduce the condition number of the system more significantly than classical preconditioners. Numerical experiments show that our approach outperforms both classical and neural network-based methods for an important class of parametric partial differential equations. We also provide a heuristic justification for the loss function used and show that preconditioners obtained by learning with this loss function reduce the condition number in a more desirable way for CG.

cs.LG↗

arXiv2025

ConDiff: A Challenging Dataset for Neural Solvers of Partial Differential Equations

We present ConDiff, a novel dataset for scientific machine learning. ConDiff focuses on the parametric diffusion equation with space dependent coefficients, a fundamental problem in many applications of partial differential equations (PDEs). The main novelty of the proposed dataset is that we consider discontinuous coefficients with high contrast. These coefficient functions are sampled from a selected set of distributions. This class of problems is not only of great academic interest, but is also the basis for describing various environmental and industrial problems. In this way, ConDiff shortens the gap with real-world problems while remaining fully synthetic and easy to use. ConDiff consists of a diverse set of diffusion equations with coefficients covering a wide range of contrast levels and heterogeneity with a measurable complexity metric for clearer comparison between different coefficient functions. We baseline ConDiff on standard deep learning models in the field of scientific machine learning. By providing a large number of problem instances, each with its own coefficient function and right-hand side, we hope to encourage the development of novel physics-based deep learning approaches, such as neural operators, ultimately driving progress towards more accurate and efficient solutions of complex PDE problems.

cs.LG↗

arXiv2024

Neural operators meet conjugate gradients: The FCG-NO method for efficient PDE solving

Deep learning solvers for partial differential equations typically have limited accuracy. We propose to overcome this problem by using them as preconditioners. More specifically, we apply discretization-invariant neural operators to learn preconditioners for the flexible conjugate gradient method (FCG). Architecture paired with novel loss function and training scheme allows for learning efficient preconditioners that can be used across different resolutions. On the theoretical side, FCG theory allows us to safely use nonlinear preconditioners that can be applied in $O(N)$ operations without constraining the form of the preconditioners matrix. To justify learning scheme components (the loss function and the way training data is collected) we perform several ablation studies. Numerical results indicate that our approach favorably compares with classical preconditioners and allows to reuse of preconditioners learned for lower resolution to the higher resolution data.

math.NA↗