融合全纯神经网络求解 2D 与 3D Laplace 问题

Enriched Holomorphic Neural Networks for 2D and 3D Laplace Problems

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研究者将全纯神经网络与由局部几何和边界条件导出的奇异函数结合，构建出按构造即调和的表示，因此无需内部残差，网络仅凭边界数据拟合光滑余项、由富集项捕捉奇异行为。该方法支持奇异指数可训练，并扩展到三维，使用独立网络沿复零方向求值。在 CdZnTe 辐射探测器等多奇异位点问题上，该模型在所有测试问题中精度最高，二维和三维均比次优方法提升约 1.7-2.1 倍。

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Abstract:Boundary-value problems for the Laplace equation develop singular behaviour near corners, edges and junctions that smooth approximators resolve poorly. We combine holomorphic neural networks with singular functions derived from the local geometry and boundary conditions. The representation is harmonic by construction, so no interior residual is needed: the network fits the smooth remainder from boundary data alone, and the enrichment terms capture the singular behaviour. Singular exponents are trainable, so the construction also applies when the local expansion is not known in advance. We extend the approach to three dimensions using independent networks evaluated along complex null directions. The enriched model is the most accurate tested method on every problem. Its margin is several orders of magnitude where the enrichment captures the singularity exactly, and a factor of about 1.7-2.1 over the next-best method on a CdZnTe radiation detector with many singular sites, in both two and three dimensions.

Subjects:	Machine Learning (cs.LG)
Cite as:	arXiv:2610.12541 [cs.LG]
	(or arXiv:2610.12541v1 [cs.LG] for this version)
	https://doi.org/10.48550/arXiv.2610.12541 arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Alejandro Valverde Mahou [view email]
[v1] Thu, 8 Oct 2026 14:48:38 UTC (1,890 KB)

来源：arXiv：cs.LG · arxiv.org