arXiv subjects

Ekaterina Muravleva

Publications and source records attributed to Ekaterina Muravleva.

17 recordsLinked to original sources

arXiv2026

Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations

Graph Neural Networks (GNNs) have been proposed as a tool for learning sparse matrix preconditioners, which are key components in accelerating linear solvers. We present theoretical and empirical evidence that message-passing GNNs are fundamentally incapable of approximating sparse triangular factorizations for classes of matrices for which high-quality preconditioners exist but require non-local dependencies. To illustrate this, we construct a set of baselines using both synthetic matrices and real-world examples from the SuiteSparse collection. Across a range of GNN architectures, including Graph Attention Networks and Graph Transformers, we observe low cosine similarity ($\leq0.7$ in key cases) between predicted and reference factors. Our theoretical and empirical results suggest that architectural innovations beyond message-passing are necessary for applying GNNs to scientific computing tasks such as matrix factorization. Moreover, experiments demonstrate that overcoming non-locality alone is insufficient. Tailored architectures are necessary to capture the required dependencies since even a completely non-local Global Graph Transformer fails to match the proposed baselines.

cs.LG↗

arXiv2026

Deep Learning for Subspace Regression

It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of the problem. A practical way to apply such a scheme is to compute subspaces for a selected set of parameters in the computationally demanding offline stage and in the online stage approximate subspace for unknown parameters by interpolation. For realistic problems the space of parameters is high dimensional, which renders classical interpolation strategies infeasible or unreliable. We propose to relax the interpolation problem to regression, introduce several loss functions suitable for subspace data, and use a neural network as an approximation to high-dimensional target function. To further simplify a learning problem we introduce redundancy: in place of predicting subspace of a given dimension we predict larger subspace. We show theoretically that this strategy decreases the complexity of the mapping for elliptic eigenproblems with constant coefficients and makes the mapping smoother for general smooth function on the Grassmann manifold. Empirical results also show that accuracy significantly improves when larger-than-required subspaces are predicted. With the set of numerical illustrations we demonstrate that subspace regression can be useful for a range of tasks including parametric eigenproblems, deflation techniques, relaxation methods, optimal control and solution of parametric partial differential equations.

cs.LG↗

arXiv2026

Geological Field Restoration through the Lens of Image Inpainting

We study an ill-posed problem of geological field reconstruction under limited observations. Engineers often have to deal with the problem of reconstructing the subsurface geological field from sparse measurements such as exploration well data. Inspired by image inpainting, we model this partially observed spatial field as a multidimensional tensor and recover missing values by enforcing a global low rank structure together with spatial smoothness. We solve the resulting optimization via the Alternating Direction Method of Multipliers. On the SPE10 model 2 benchmark, this deterministic approach yields consistently lower relative squared error than ordinary kriging across various sampling densities and produces visually coherent reconstructions.

cs.CV↗

arXiv2026

Locally Subspace-Informed Neural Operators for Efficient Multiscale PDE Solving

Neural operators (NOs) struggle with high-contrast multiscale partial differential equations (PDEs), where fine-scale heterogeneities cause large errors. To address this, we use the Generalized Multiscale Finite Element Method (GMsFEM) that constructs localized spectral basis functions on coarse grids. This approach efficiently captures dominant multiscale features while solving heterogeneous PDEs accurately at reduced computational cost. However, computing these basis functions is computationally expensive. This gap motivates our core idea: to use a NO to learn the subspace itself - rather than individual basis functions - by employing a subspace-informed loss. On standard multiscale benchmarks - namely a linear elliptic diffusion problem and the nonlinear, steady-state Richards equation - our hybrid method cuts solution error by approximately $60\%$ compared with standalone NOs and reduces basis-construction time by about $60$ times relative to classical GMsFEM, while remaining independent of forcing terms and boundary conditions. The result fuses multiscale finite-element robustness with NO speed, yielding a practical solver for heterogeneous PDEs.

math.NA↗

arXiv2025

PERELMAN: Pipeline for scientific literature meta-analysis. Technical report

We present PERELMAN (PipEline foR sciEntific Literature Meta-ANalysis), an agentic framework designed to extract specific information from a large corpus of scientific articles to support large-scale literature reviews and meta-analyses. Our central goal is to reliably transform heterogeneous article content into a unified, machine-readable representation. PERELMAN first elicits domain knowledge-including target variables, inclusion criteria, units, and normalization rules-through a structured dialogue with a subject-matter expert. This domain knowledge is then reused across multiple stages of the pipeline and guides coordinated agents in extracting evidence from narrative text, tables, and figures, enabling consistent aggregation across studies. In order to assess reproducibility and validate our implementation, we evaluate the system on the task of reproducing the meta-analysis of layered Li-ion cathode properties (NMC811 material). We describe our solution, which has the potential to reduce the time required to prepare meta-analyses from months to minutes.

cs.MA↗

arXiv2025

Spectral Analysis of the Weighted Frobenius Objective

We analyze a weighted Frobenius loss for approximating symmetric positive definite matrices in the context of preconditioning iterative solvers. Unlike the standard Frobenius norm, the weighted loss penalizes error components associated with small eigenvalues of the system matrix more strongly. Our analysis reveals that each eigenmode is scaled by the corresponding square of its eigenvalue, and that, under a fixed error budget, the loss is minimized only when the error is confined to the direction of the largest eigenvalue. This provides a rigorous explanation of why minimizing the weighted loss naturally suppresses low-frequency components, which can be a desirable strategy for the conjugate gradient method. The analysis is independent of the specific approximation scheme or sparsity pattern, and applies equally to incomplete factorizations, algebraic updates, and learning-based constructions. Numerical experiments confirm the predictions of the theory, including an illustration where sparse factors are trained by a direct gradient updates to IC(0) factor entries, i.e., no trained neural network model is used.

math.NA↗

arXiv2025

Bayesian Inverse Problems Meet Flow Matching: Efficient and Flexible Inference via Transformers

The efficient resolution of Bayesian inverse problems remains challenging due to the high computational cost of traditional sampling methods. In this paper, we propose a novel framework that integrates Conditional Flow Matching (CFM) with a transformer-based architecture to enable fast and flexible sampling from complex posterior distributions. The proposed methodology involves the direct learning of conditional probability trajectories from the data, leveraging CFM's ability to bypass iterative simulation and transformers' capacity to process arbitrary numbers of observations. The efficacy of the proposed framework is demonstrated through its application to three problems: a simple nonlinear model, a disease dynamics framework, and a two-dimensional Darcy flow Partial Differential Equation. The primary outcomes demonstrate that the relative errors in parameters recovery are as low as 1.5%, and that the inference time is reduced by up to 2000 times on CPU in comparison with the Monte Carlo Markov Chain. This framework facilitates the expeditious resolution of Bayesian problems through the utilisation of sampling from the learned conditional distribution.

cs.LG↗

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

Quantization of Large Language Models with an Overdetermined Basis

In this paper, we introduce an algorithm for data quantization based on the principles of Kashin representation. This approach hinges on decomposing any given vector, matrix, or tensor into two factors. The first factor maintains a small infinity norm, while the second exhibits a similarly constrained norm when multiplied by an orthogonal matrix. Surprisingly, the entries of factors after decomposition are well-concentrated around several peaks, which allows us to efficiently replace them with corresponding centroids for quantization purposes. We study the theoretical properties of the proposed approach and rigorously evaluate our compression algorithm in the context of next-word prediction tasks and on a set of downstream tasks for text classification. Our findings demonstrate that Kashin Quantization achieves competitive or superior quality in model performance while ensuring data compression, marking a significant advancement in the field of data quantization.

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↗

arXiv2023

On the efficient preconditioning of the Stokes equations in tight geometries

If the Stokes equations are properly discretized, it is known that the Schur complement matrix is spectrally equivalent to the identity matrix. Moreover, in the case of simple geometries, it is often observed that most of its eigenvalues are equal to one. These facts form the basis for the famous Uzawa algorithm. Despite recent progress in developing efficient iterative methods for solving the Stokes problem, the Uzawa algorithm remains popular in science and engineering, especially when accelerated by Krylov subspace methods. However, in complex geometries, the Schur complement matrix can become severely ill-conditioned, having a significant portion of non-unit eigenvalues. This makes the established Uzawa preconditioner inefficient. To explain this behaviour, we examine the Pressure Schur Complement formulation for the staggered finite-difference discretization of the Stokes equations. Firstly, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement matrix. Secondly, we demonstrate that its condition number increases with increasing the surface-to-volume ratio of the flow domain. As an alternative to the Uzawa preconditioner, we propose using the diffusive SIMPLE preconditioner for geometries with a large surface-to-volume ratio. We show that the latter is much more fast and robust for such geometries. Furthermore, we show that the usage of the SIMPLE preconditioner leads to more accurate practical computation of the permeability of tight porous media. Keywords: Stokes problem, tight geometries, computing permeability, preconditioned Krylov subspace methods

math.NA↗

arXiv2023

On the structure of the Schur complement matrix for the Stokes equation

In this paper, we investigate the structure of the Schur complement matrix for the fully-staggered finite-difference discretization of the stationary Stokes equation. Specifically, we demonstrate that the structure of the Schur complement matrix depends qualitatively on a particular characteristic, namely the number of non-unit eigenvalues, and the two limiting cases are of special interest.

math.NA↗

arXiv2022

On a workflow for efficient computation of the permeability of tight sandstones

The paper presents a workflow for fast pore-scale simulation of single-phase flow in tight reservoirs typically characterized by low, multiscale porosity. Multiscale porosity implies that the computational domain contains porous voxels (unresolved porosity) in addition to pure fluid voxels. In this case, the Stokes-Brinkman equations govern the flow, with the Darcy term needed to account for the flow in the porous voxels. As the central part of our workflow, robust and efficient solvers for Stokes and Stokes-Brinkman equations are presented. The solvers are customized for low-porosity binary and multiclass images, respectively. Another essential component of the workflow is a preprocessing module for classifying images with respect to the connectivity of the multiscale pore space. Particularly, an approximation of the Stokes-Brinkman problem, namely, the Darcy problem, is investigated for the images that do not have pure fluid percolation paths. Thorough computational experiments demonstrate efficiency and robustness of the workflow for simulations on images from tight reservoirs. Raw files describing the used CT images are provided as supplementary materials to enable other researchers to use them.

math.NA↗

arXiv2021

Reconstruction of 3D Porous Media From 2D Slices

In many branches of earth sciences, the problem of rock study on the micro-level arises. However, a significant number of representative samples is not always feasible. Thus the problem of the generation of samples with similar properties becomes actual. In this paper, we propose a novel deep learning architecture for three-dimensional porous media reconstruction from two-dimensional slices. We fit a distribution on all possible three-dimensional structures of a specific type based on the given dataset of samples. Then, given partial information (central slices), we recover the three-dimensional structure around such slices as the most probable one according to that constructed distribution. Technically, we implement this in the form of a deep neural network with encoder, generator and discriminator modules. Numerical experiments show that this method provides a good reconstruction in terms of Minkowski functionals.

cs.CV↗

arXiv2019

Microstructure synthesis using style-based generative adversarial network

Work considers the usage of StyleGAN architecture for the task of microstructure synthesis. The task is the following: given number of samples of structure we try to generate similar samples at the same time preserving its properties. Since the considered architecture is not able to produce samples of sizes larger than the training images, we propose to use image quilting to merge fixed-sized samples. One of the key features of the considered architecture is that it uses multiple image resolutions. We also investigate the necessity of such an approach.

eess.IV↗

arXiv2019

Data-driven model for the identification of the rock type at a drilling bit

Directional oil well drilling requires high precision of the wellbore positioning inside the productive area. However, due to specifics of engineering design, sensors that explicitly determine the type of the drilled rock are located farther than 15m from the drilling bit. As a result, the target area runaways can be detected only after this distance, which in turn, leads to a loss in well productivity and the risk of the need for an expensive re-boring operation. We present a novel approach for identifying rock type at the drilling bit based on machine learning classification methods and data mining on sensors readings. We compare various machine-learning algorithms, examine extra features coming from mathematical modeling of drilling mechanics, and show that the real-time rock type classification error can be reduced from 13.5 % to 9 %. The approach is applicable for precise directional drilling in relatively thin target intervals of complex shapes and generalizes appropriately to new wells that are different from the ones used for training the machine learning model.

cs.LG↗