Neural Algorithmic Solvers for High-Dimensional Linear Systems
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| Formato: | Recurso digital |
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2025
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| _version_ | 1866901789275586560 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | High-dimensional linear systems are ubiquitous in scientific computing, engineering, and data science, arising from discretization of partial differential equations, statistical models, and large-scale optimization problems. Solving these systems efficiently and accurately presents significant computational challenges due due to the curse of dimensionality, memory constraints, and the inherent complexity of large-scale matrix operations. Traditional direct methods often become intractable, while iterative solvers, though more memory-efficient, can suffer from slow convergence or sensitivity to matrix conditioning. This paper explores the emerging paradigm of neural algorithmic solvers as a novel approach to tackle high-dimensional linear systems. We investigate how deep learning architectures, particularly those inspired by recurrent processes and graph neural networks, can learn to approximate or accelerate the solution process. The core idea is to leverage the approximation capabilities of neural networks to emulate, improve, or discover iterative solution algorithms that are robust, efficient, and scalable to high dimensions. We present a conceptual framework for designing such solvers, discussing potential architectures, training methodologies, and evaluation metrics. The envisioned benefits include reduced computational complexity, enhanced adaptability to various matrix properties, and the potential for superior generalization compared to traditional methods. Furthermore, we delve into the challenges associated with interpretability, theoretical guarantees, and the generation of appropriate training data for these data-driven solvers. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17687854 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Neural Algorithmic Solvers for High-Dimensional Linear Systems SÉRGIO DE ANDRADE, PAULO High-dimensional linear systems are ubiquitous in scientific computing, engineering, and data science, arising from discretization of partial differential equations, statistical models, and large-scale optimization problems. Solving these systems efficiently and accurately presents significant computational challenges due due to the curse of dimensionality, memory constraints, and the inherent complexity of large-scale matrix operations. Traditional direct methods often become intractable, while iterative solvers, though more memory-efficient, can suffer from slow convergence or sensitivity to matrix conditioning. This paper explores the emerging paradigm of neural algorithmic solvers as a novel approach to tackle high-dimensional linear systems. We investigate how deep learning architectures, particularly those inspired by recurrent processes and graph neural networks, can learn to approximate or accelerate the solution process. The core idea is to leverage the approximation capabilities of neural networks to emulate, improve, or discover iterative solution algorithms that are robust, efficient, and scalable to high dimensions. We present a conceptual framework for designing such solvers, discussing potential architectures, training methodologies, and evaluation metrics. The envisioned benefits include reduced computational complexity, enhanced adaptability to various matrix properties, and the potential for superior generalization compared to traditional methods. Furthermore, we delve into the challenges associated with interpretability, theoretical guarantees, and the generation of appropriate training data for these data-driven solvers. |
| title | Neural Algorithmic Solvers for High-Dimensional Linear Systems |
| url | https://doi.org/10.5281/zenodo.17687854 |