神经调和测度算子
Neural Harmonic Measure Operator
- Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出神经调和测度算子NHMO，以Transformer核参数化调和测度，通过Walk-on-Spheres训练，实现变形状域椭圆PDE的免重训求解，并在基准上超越多个基线。
AI中文摘要：
我们提出了神经调和测度算子（NHMO），一种用于变形状域上椭圆偏微分方程问题的神经求解器。域的调和测度是边界概率分布，将其与任意边界数据积分，即可返回Dirichlet Laplace解。它仅依赖于几何形状，而与边界数据无关。NHMO将该测度的密度参数化为基于Transformer的边界核，并通过Walk-on-Spheres退出样本进行监督训练，因此一个训练好的核即可处理同一形状上的不同边界值，无需重新训练。我们通过经典分解将其扩展到Poisson方程，并利用辅助网络摊销源项引起的修正，避免了破坏直接评估的奇异体积求积。在推理时，新的边界值和新的源项均可通过对拟合核和提升函数重新积分得到PDE解，无需重新训练。在MCB-B三维变形状Poisson基准上，NHMO在所有五个类别中均优于四个先前基线，并在受控的二维测试平台上与主要神经算子基线竞争。
英文摘要：
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 sources both yield PDE solutions by re-integration against the fitted kernel and lift, with no retraining. NHMO improves over four prior baselines on the MCB-B 3D variable-shape Poisson benchmark across all five categories, and is competitive with major neural-operator baselines on a controlled 2D testbed.
