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Finite Expression Approximation of High-Dimensional PDEs Without the Curse of Dimensionality — arXiv:2609.32229

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.32229
License (as archived): CC0-1.0
Archived: 2026-10-07T23:25:07.138Z
SHA-256 of the archived text: 272c47edc4d231250a39f4e327614f69fed209adb919f686e00ef669b40716fc
Stable URL of this page: https://metraipolis.com/documento/dc_bb555180bbc3

How to cite

Finite Expression Approximation of High-Dimensional PDEs Without the Curse of Dimensionality — arXiv:2609.32229 — arXiv (en), https://arxiv.org/abs/2609.32229 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bb555180bbc3 (Metraipolis archive dc_bb555180bbc3, sha256 272c47edc4d231250a39f4e327614f69fed209adb919f686e00ef669b40716fc)

BibTeX

@misc{dcbb555180bbc3,
  title = {Finite Expression Approximation of High-Dimensional PDEs Without the Curse of Dimensionality — arXiv:2609.32229},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.32229},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bb555180bbc3; sha256 272c47edc4d231250a39f4e327614f69fed209adb919f686e00ef669b40716fc},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.32229. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We establish that finite expressions form a symbolic representation class capable of overcoming the curse of dimensionality for several classes of high-dimensional partial differential equations. For semilinear heat equations, we show how finite expression approximations of the terminal condition and nonlinearity can be propagated through the multilevel Picard framework to produce randomized pointwise approximations with prescribed root-mean-square accuracy. For semilinear Kolmogorov equations with Laplacian diffusion and zero drift, diagonal Black--Scholes equations, and the Laplace Dirichlet problem on a half-space, we construct deterministic finite expression approximations with arbitrarily small spatial $L^p$ error. Under suitable growth and regularity assumptions, the evaluation cost of the resulting approximants is bounded polynomially in the dimension and the reciprocal accuracy. A key ingredient in the nonlinear setting is the construction of finite expressions that approximate the PDE solution while preserving the growth and Lipschitz structures required by the stochastic solution theory. More broadly, our results show that dimension-robust approximation of high-dimensional PDEs is not restricted to conventional neural-network architectures: structured symbolic expressions generated from a fixed dictionary can achieve comparable polynomial-complexity guarantees. This provides a rigorous foundation for finite expression methods as a representation paradigm for high-dimensional scientific computing.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-29 in math.NA).
Categorie arXiv: Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2609.32229 — 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 2026-10-07T23:04:09.445Z dalla pagina https://arxiv.org/catchup/math.NA/2026-09-29?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).