提出SDF感知加权,解决3D水平集物理神经网络中梯度惩罚失效问题。
SDF-Aware Weighting: Adaptive Eikonal Regularisation for Three-Dimensional Level-Set Physics-Informed Neural Networks

- 引入残差分位数门与梯度比结合的自适应加权机制
- 在4个三维基准上仅用固定配置就逼近人工调参最优权重
- 对非光滑几何结构误差更低,适合复杂边界场景
物理信息神经网络中的自适应损失平衡依赖于所有残差应趋近零的假设。但在含eikonal正则项的水平集平流中,这一假设不成立:eikonal项惩罚∇ϕ的范数偏离1,而该性质仅在刚性运动下保持;当精确解偏离符号距离函数时,其eikonal残差非零,强制归零反而使网络远离真解。我们发现标准梯度范数平衡在此情形下失败,权重始终维持初始值。为此提出SDF-Aware Weighting(SAW),通过残差分位数门剔除合法偏离点后,再基于梯度范数比进行缩放。在四个三维基准上,SAW仅用单组固定配置即可获得接近十八次人工调参的结果,权重范围覆盖10⁻¹至10⁻⁵四数量级。在槽球测试中,初始场在回形边处不可微,SAW误差低于所有人工调参权重。两个光滑刚性基准作为对照,SAW表现更差,符合预期。消融实验关闭门控后,槽几乎完全填充,相对L₂误差达1.06%,与从未表示槽的场无异。我们还提供一种特征受限度量以区分这两种情况。
原文摘要 · Abstract (English)
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.
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