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Tensor-Train Compressed Separable PINNs: A Curvature-Aware Optimization Framework for Parametric PDEs in High Dimensions — arXiv:2609.36165

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.36165
License (as archived): CC0-1.0
Archived: 2026-10-07T17:32:30.797Z
SHA-256 of the archived text: 44763c44529831ceda615c6a31a43e55b6291886abb695770b1707cf45f79981
Stable URL of this page: https://metraipolis.com/documento/dc_d350d84fa7fa

How to cite

Tensor-Train Compressed Separable PINNs: A Curvature-Aware Optimization Framework for Parametric PDEs in High Dimensions — arXiv:2609.36165 — arXiv (en), https://arxiv.org/abs/2609.36165 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_d350d84fa7fa (Metraipolis archive dc_d350d84fa7fa, sha256 44763c44529831ceda615c6a31a43e55b6291886abb695770b1707cf45f79981)

BibTeX

@misc{dcd350d84fa7fa,
  title = {Tensor-Train Compressed Separable PINNs: A Curvature-Aware Optimization Framework for Parametric PDEs in High Dimensions — arXiv:2609.36165},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.36165},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_d350d84fa7fa; sha256 44763c44529831ceda615c6a31a43e55b6291886abb695770b1707cf45f79981},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.36165. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In this work, we develop a second-order optimization framework for physics-informed neural networks (PINNs) applied to high-dimensional parametric partial differential equations (PDEs). The framework is built on the Gauss--Newton pullback metric, which provides an operator-informed notion of curvature in parameter space and connects the method to the broader family of natural gradient schemes. We show that, for coordinate-separable neural architectures and linear differential operators (or linearized operators in the nonlinear case) admitting a finite separable representation, the residual Jacobian inherits a structured separable factorization. This yields an exact compressed formulation of the Gauss--Newton step in a reduced space, without assembling the full residual Jacobian on the exponentially large tensor-product collocation grid. The dimension of the reduced space (the effective compressed dimension) is determined by the local collocation grid sizes, the separable operator structure, and the contraction pattern of the architecture, thereby avoiding dependence on the full tensor-product grid size and replacing dense linear algebra in parameter space by a substantially smaller structured problem. Within our framework, we investigate canonical polyadic and tensor-train parametrizations and derive their full algebraic characterization relevant to the Gauss--Newton method, including the structure of the residual Jacobian, the resulting compressed system, and its effective compressed dimension. Numerical experiments on high-dimensional PDEs, including parametric problems, demonstrate the high efficiency of the proposed compressed Gauss--Newton method, which achieves substantially lower errors than tensor-compressed first-order baselines with orders of magnitude fewer iterations and only a fraction of the computing time.

Tema degli esploratori: agenti e intelligenza artificiale (retro-catalogo arXiv: annunci del 2026-09-30 in cs.LG).
Categorie arXiv: Optimization and Control (math.OC) ; Machine Learning (cs.LG); Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2609.36165 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T16:59:48.474Z dalla pagina https://arxiv.org/catchup/cs.LG/2026-09-30?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).