NHMO：一个神经网络求解器搞定变形状域椭圆偏微分方程
Neural Harmonic Measure Operator
arXiv 上这篇论文挺有意思：把调和测度做成可学习的边界核，训练一次，换边界条件换源项都不用重训，直接重新积分就出新解。做科学计算或 PDE 求解的可以看看。
论文提出 Neural Harmonic Measure Operator（NHMO），把域的调和测度密度参数化为 transformer 边界核，用 Walk-on-Spheres 出口样本监督训练。由于调和测度只依赖几何形状而不依赖边界数据，一个训练好的核无需重训就能处理同一形状上不同的边界值。Poisson 情形通过经典分解扩展，辅助网络摊销源项修正，避免直接求体积积分的奇异性问题。推理时新的边界值和新的源项都只需对拟合核重新积分即可得到解，无需重训。NHMO 在 MCB-B 3D 变形状 Poisson 基准的全部五个类别上超过四个先前基线，并在受控 2D 测试平台上与主流 neural-operator 基线相当。
Neural Harmonic Measure Operator
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.