发表机构
NED University of Engineering and Technology（NED工程技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探究了物理信息神经网络中程函正则化的二维设计原理在三维水平集平流中的可迁移性，通过多种子实验验证了权重选择规律，并与WENO求解器对比了性能。
AI 中文摘要
应用于界面平流水平集形式的物理信息神经网络（PINNs），通常会在残差损失和初始条件损失中加入程函正则化项，以惩罚梯度范数∇φ与1的偏差。此前一项二维研究发现，该权重是主导超参数，且在刚体流动与变形流动之间，其最优值会发生四个数量级的偏移，但未明确这些原理是否可迁移至三维，以及单种子结果是否能在运行间变异性中保持。我们通过在四个三维基准案例（平移球、旋转球、开槽球、反向涡）中重复权重选择来回答这两个问题，在预注册的选择规则下，以完整训练预算对六个权重进行三次种子的扫描。排序可迁移：所选权重跟踪精确解与符号距离属性的偏离程度，从精确成立的10⁻¹跨越至界面拉伸时的10⁻⁵，共四个数量级。数值仅在基准案例间可迁移：四个案例中有两个保持不变，两个发生变化，因此必须验证可继承性。多种子协议显示，在小权重下，种子间标准差等于误差本身，而正则化项将其降低一个数量级以上，同时提升了可复现性与准确性。我们在相同网格和误差度量下，与五阶WENO求解器进行基准对比：经典格式在所有四个问题上更准确，在光滑刚体平流中高出两个数量级，差距随几何难度缩小，体积守恒中的差距小于场范数。最后，我们证明相对L₂误差无法验证薄特征的保持情况，并提出了可验证此情况的特征受限度量。
英文摘要
Adaptive loss-balancing schemes for physics-informed neural networks rest on a premise that every residual should be driven to zero. For level-set advection with an eikonal regulariser that premise fails: the eikonal term penalises deviation of $\lVert\nablaϕ\rVert$ from unity, a property transport preserves only under rigid motion; where the exact solution departs from a signed-distance function the eikonal residual of the correct answer is nonzero, and driving it to zero moves the network away from that answer. We show that standard gradient-norm balancing fails in exactly this way, its weight remaining near its initial value throughout training on benchmarks where the property is violated, and we introduce SDF-Aware Weighting (SAW), which combines a residual-quantile gate with a gradient-norm ratio so that points exhibiting legitimate departure are excluded before the surviving term is scaled. Across four three-dimensional benchmarks SAW selects an eikonal weight within an order of magnitude of the value located by an eighteen-run manual sweep, spanning four decades from $10^{-1}$ to $10^{-5}$ with a single fixed configuration. On the slotted sphere, where the initial field is non-differentiable at reentrant edges, SAW attains a lower error than any weight in that sweep. Two smooth rigid benchmarks serve as controls: SAW is worse there, as expected when its premise does not hold. An ablation with the gate disabled shows the slot is nearly entirely filled while the relative $L_2$ error reads $1.06\%$, indistinguishable from a field that never represented the slot. We give a feature-restricted measure that separates the two cases.
Comments37 pages, 15 figures, 7 tables, 1 algorithm. Submitted to Machine Learning: Science and Technology; under peer review
