几何分析中的PINNs用户指南：来自渐近普拉托问题的经验

中文摘要 AI 辅助

本会议论文阐述了arXiv:2605.26234v2的研究成果，这是与Marco Usula合作的工作，我们引入了基于物理信息神经网络（PINNs）的机器学习框架，旨在构造双曲空间中在无穷远处渐近于指定纽结的近极小圆盘。我们使用该方法为Joel Fine的一个猜想提供了数值证据，该猜想将H⁴中的极小曲面与HOMFLY多项式的系数联系起来。本文是上述论文的方法学配套内容，基于2026年“DANGER：数据、数字与几何”研讨会的报告撰写。我们不详细回顾上述预印本中已充分呈现的结果，而是讨论根据我们的经验决定该方法是否有效的框架的两个方面：第一，必须将问题的几何结构编码到模型的架构中，使得对于每个可学习参数值，边界条件和无穷远处的渐近性都能严格满足，从而得到单分量损失函数；第二，必须精心设计PDE残差的计算，以确保能在合理时间内完成完整训练。关于后一点，我们描述了两种在原论文中未详细说明的实现技术：用二阶喷流的前向传播替换嵌套的反向模式自动微分，以及一次性编译残差的计算图而非在每个优化步骤中重建它。在相同硬件上，这两项改动共同将训练步骤的成本降低了约40至50倍。我们希望这些方法学讨论对希望在自身问题中部署PINNs的微分几何和几何分析领域研究者有所助益。

英文摘要

This proceedings contribution elaborates on the findings of arXiv:2605.26234v2: a joint work with Marco Usula, where we introduced a machine learning framework based on physics-informed neural networks (PINNs), aimed at constructing near-minimal discs in hyperbolic space asymptotic to a prescribed knot at infinity. We used this method to provide numerical evidence for a conjecture of Joel Fine relating minimal surfaces in $H^{4}$ to the coefficients of the HOMFLY polynomial. This is a methodological companion to that paper, based on a presentation given at the 2026 edition of the workshop "DANGER: Data, Numbers, and Geometry". Rather than reviewing the results, which are presented extensively in the preprint above, we discuss the two aspects of the framework which, in our experience, determined whether the method worked at all. First, the geometry of the problem must be encoded in the architecture of the model, so that the boundary condition and asymptotics at infinity hold exactly for every value of the learnable parameters - leaving us with a single-component loss function; second, the evaluation of the PDE residual must be engineered with care to ensure that complete trainings can be performed in a reasonable time. On the latter point, we describe two implementation techniques which are not spelled out in detail in the original paper: replacing nested reverse-mode automatic differentiation with the forward propagation of second-order jets, and compiling the computational graph of the residual once instead of rebuilding it at every optimisation step. Together, on identical hardware, these two changes reduce the cost of a training step by a factor of roughly forty to fifty. We hope these methodological discussions can be useful for researchers in differential geometry and geometric analysis who wish to deploy PINNs on problems of their own.

发表机构

• University of Bonn（波恩大学）
• Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）

机构由 AI 辅助整理，请以论文原文为准。

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10 pages; 1 figure