Neural Preconditioning via Krylov Subspace Geometry
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908458778886144 |
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| author | Dimola, Nunzio Coclite, Alessandro Zunino, Paolo |
| author_facet | Dimola, Nunzio Coclite, Alessandro Zunino, Paolo |
| contents | We propose a geometry-aware strategy for training neural preconditioners tailored to parametrized linear systems arising from the discretization of mixed-dimensional partial differential equations (PDEs). These systems are typically ill-conditioned because of the presence of embedded lower-dimensional structures and are solved using Krylov subspace methods. Our approach yields an approximation of the inverse operator employing a learning algorithm consisting of a two-stage training framework: an initial static pre-training phase, based on residual minimization, followed by a dynamic fine-tuning phase that incorporates solver convergence dynamics into training via a novel loss functional. This dynamic loss is defined by the principal angles between the residuals and the Krylov subspaces. It is evaluated using a differentiable implementation of the Flexible GMRES algorithm, which enables backpropagation through both the Arnoldi process and Givens rotations. The resulting neural preconditioner is explicitly optimized to improve early-stage convergence and reduce iteration counts in a family of 3D-1D mixed-dimensional problems with geometric variability of the 1D domain. Numerical experiments show that our solver-aligned approach significantly improves convergence rate, robustness, and generalization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Neural Preconditioning via Krylov Subspace Geometry Dimola, Nunzio Coclite, Alessandro Zunino, Paolo Numerical Analysis 65F08, 65F10, 68T07, 65Y20 We propose a geometry-aware strategy for training neural preconditioners tailored to parametrized linear systems arising from the discretization of mixed-dimensional partial differential equations (PDEs). These systems are typically ill-conditioned because of the presence of embedded lower-dimensional structures and are solved using Krylov subspace methods. Our approach yields an approximation of the inverse operator employing a learning algorithm consisting of a two-stage training framework: an initial static pre-training phase, based on residual minimization, followed by a dynamic fine-tuning phase that incorporates solver convergence dynamics into training via a novel loss functional. This dynamic loss is defined by the principal angles between the residuals and the Krylov subspaces. It is evaluated using a differentiable implementation of the Flexible GMRES algorithm, which enables backpropagation through both the Arnoldi process and Givens rotations. The resulting neural preconditioner is explicitly optimized to improve early-stage convergence and reduce iteration counts in a family of 3D-1D mixed-dimensional problems with geometric variability of the 1D domain. Numerical experiments show that our solver-aligned approach significantly improves convergence rate, robustness, and generalization. |
| title | Neural Preconditioning via Krylov Subspace Geometry |
| topic | Numerical Analysis 65F08, 65F10, 68T07, 65Y20 |
| url | https://arxiv.org/abs/2507.15452 |