[论文] Neural Harmonic Measure Operator

论文概要

研究领域: ML 作者: Jinjin He, Sinan Wang, Yuchen Sun, Bo Zhu 发布时间: 2026-09-28 arXiv: 2609.35752

中文摘要

我们引入神经调和测度算子（NHMO），一种用于可变形状域上椭圆 PDE 问题的神经求解器。域的调和测度是边界概率分布，对任何边界数据进行积分后返回 Dirichlet Laplace 方程的解。它仅依赖于几何，而不依赖于边界数据。NHMO 将该测度的密度参数化为基于 Transformer 的边界核，通过 Walk-on-Spheres 退出样本进行监督，因此一个训练好的核可以在不同边界值下处理同一形状而无需重新训练。我们通过经典分解将其扩展到 Poisson 方程，辅助网络分摊源项引起的修正，避免了直接评估时破坏计算的奇异体积求积。推理时，新的边界值和新的源项都通过对拟合核和升力的重新积分产生 PDE 解，无需重新训练。NHMO 在 MCB-B 3D 可变形状 Poisson 基准的全部五个类别上改进了四个先前基线，并在受控 2D 测试台上与主流神经算子基线具有竞争力。

原文摘要

We introduce Neural Harmonic Measure Operator (NHMO), a neural solver for elliptic PDE problems on variable-shape domains. The harmonic measure of a domain is the boundary probability distribution that, integrated against any boundary data, returns the Dirichlet Laplace solution. It depends only on the geometry, not on the boundary data. NHMO parameterizes the density of this measure as a transformer-based boundary kernel supervised by Walk-on-Spheres exit samples, so one trained kernel handles different boundary values on a shape with no retraining. We extend it to Poisson via a classical decomposition, with an auxiliary network amortizing the source-induced correction and avoiding the singular volume quadrature that breaks direct evaluation. At inference, new boundary values and new sour...

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