NeuraLSP: An Efficient and Rigorous Neural Left Singular Subspace Preconditioner for Conjugate Gradient Methods

• URL: http://arxiv.org/abs/2601.20174v2
• Date: Thu, 29 Jan 2026 01:54:05 GMT
• Title: NeuraLSP: An Efficient and Rigorous Neural Left Singular Subspace Preconditioner for Conjugate Gradient Methods
• Authors: Alexander Benanti, Xi Han, Hong Qin,
• Abstract summary: NeuraLSP is a novel neural preconditioner combined with a novel loss metric.<n>Our method exhibits both theoretical guarantees and empirical robustness to rank inflation, up to a 53% speedup.
• Score: 49.84495044725856
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Numerical techniques for solving partial differential equations (PDEs) are integral for many fields across science and engineering. Such techniques usually involve solving large, sparse linear systems, where preconditioning methods are critical. In recent years, neural methods, particularly graph neural networks (GNNs), have demonstrated their potential through accelerated convergence. Nonetheless, to extract connective structures, existing techniques aggregate discretized system matrices into graphs, and suffer from rank inflation and a suboptimal convergence rate. In this paper, we articulate NeuraLSP, a novel neural preconditioner combined with a novel loss metric that leverages the left singular subspace of the system matrix's near-nullspace vectors. By compressing spectral information into a fixed low-rank operator, our method exhibits both theoretical guarantees and empirical robustness to rank inflation, affording up to a 53% speedup. Besides the theoretical guarantees for our newly-formulated loss function, our comprehensive experimental results across diverse families of PDEs also substantiate the aforementioned theoretical advances.

Related papers

• Global Convergence of Adjoint-Optimized Neural PDEs [0.0]
  We study the convergence of the adjoint gradient descent optimization method for training neural-network PDE models in the limit where both the number of hidden units and the training time tend to infinity.<n>Specifically, for a general class of nonlinear parabolic PDEs with a neural network embedded in the source term, we prove the trained neural-network PDE solution to the target data (i.e., a global minimizer)<n>The global convergence proof poses a unique mathematical challenge that is not encountered in finite-dimensional convergence analyses.
  arXiv  Detail & Related papers  (2025-06-16T16:00:00Z)
• Spectral operator learning for parametric PDEs without data reliance [6.7083321695379885]
  We introduce a novel operator learning-based approach for solving parametric partial differential equations (PDEs) without the need for data harnessing. The proposed framework demonstrates superior performance compared to existing scientific machine learning techniques.
  arXiv  Detail & Related papers  (2023-10-03T12:37:15Z)
• Neural incomplete factorization: learning preconditioners for the conjugate gradient method [2.899792823251184]
  We develop a data-driven approach to accelerate the generation of effective preconditioners. We replace the typically hand-engineered preconditioners by the output of graph neural networks. Our method generates an incomplete factorization of the matrix and is, therefore, referred to as neural incomplete factorization (NeuralIF)
  arXiv  Detail & Related papers  (2023-05-25T11:45:46Z)
• A Unified Algebraic Perspective on Lipschitz Neural Networks [88.14073994459586]
  This paper introduces a novel perspective unifying various types of 1-Lipschitz neural networks. We show that many existing techniques can be derived and generalized via finding analytical solutions of a common semidefinite programming (SDP) condition. Our approach, called SDP-based Lipschitz Layers (SLL), allows us to design non-trivial yet efficient generalization of convex potential layers.
  arXiv  Detail & Related papers  (2023-03-06T14:31:09Z)
• Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs [0.8594140167290099]
  We perform the convergence analysis of unsupervised Legendre--Galerkin neural networks (ULGNet) ULGNet is a deep-learning-based numerical method for solving partial differential equations (PDEs)
  arXiv  Detail & Related papers  (2022-11-16T13:31:03Z)
• Learning via nonlinear conjugate gradients and depth-varying neural ODEs [5.565364597145568]
  The inverse problem of supervised reconstruction of depth-variable parameters in a neural ordinary differential equation (NODE) is considered. The proposed parameter reconstruction is done for a general first order differential equation by minimizing a cost functional. The sensitivity problem can estimate changes in the network output under perturbation of the trained parameters.
  arXiv  Detail & Related papers  (2022-02-11T17:00:48Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Semi-Implicit Neural Solver for Time-dependent Partial Differential Equations [4.246966726709308]
  We propose a neural solver to learn an optimal iterative scheme in a data-driven fashion for any class of PDEs. We provide theoretical guarantees for the correctness and convergence of neural solvers analogous to conventional iterative solvers.
  arXiv  Detail & Related papers  (2021-09-03T12:03:10Z)
• Fourier Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  We formulate a new neural operator by parameterizing the integral kernel directly in Fourier space. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. It is up to three orders of magnitude faster compared to traditional PDE solvers.
  arXiv  Detail & Related papers  (2020-10-18T00:34:21Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.