无数据物理信息神经算子用于水平集界面平流
A Data-Free Physics-Informed Neural Operator for Level-Set Interface Advection
浏览论文内容
中文总结 AI 辅助
提出一种无数据物理信息神经算子，仅用输运残差和几何约束训练，将初始界面映射到完整时空轨迹，在约束有效时比监督基线更优，且混合分支用更少参考解超越监督分支。
中文摘要 AI 辅助
针对界面问题的算子是在它们旨在替代的求解器产生的参考解上训练的。本工作开发了一种无数据物理信息神经算子，用于水平集界面平流，其中界面是方程中的未知量，算子将初始界面映射到指定流场下的完整时空轨迹。训练仅使用输运残差和几何约束；在目标函数中任何阶段都不引入参考解。时空傅里叶骨干网络一次性输出整个轨迹，初始条件通过构造而非惩罚施加，这消除了在没有解数据时锚定项与残差之间可能出现的竞争。在相同架构、族、预算和测试集下训练了监督和混合算子，并作为基线报告，以量化拒绝标签的代价。在反转单涡上，无数据算子在100个留出初始界面上达到1.614±0.067%的相对L2误差，而监督基线为0.369±0.035%，相差4.4倍；在刚体旋转上，相应数值为3.804±1.075%和2.576±0.159%，相差1.5倍。两个基准对两种方法的排名不同，差异归因于eikonal约束：精确解在旋转下违反|grad phi|=1的区域占0.3%，在涡流下占86.9%。在约束有效的情况下，物理训练的算子尽管场误差较大，但其封闭区域守恒比监督基线好2.7倍，而使用八个参考解的混合分支优于使用十六个参考解的监督分支。
英文摘要
Operators for interfacial problems are trained on reference solutions produced by the solver they are intended to replace. This work develops a data-free physics-informed neural operator for level-set interface advection, in which the interface is the equation's unknown and the operator maps an initial interface to the full spatiotemporal trajectory under a prescribed flow. Training uses only the transport residual and a geometric constraint; no reference solution enters the objective at any point. A spacetime Fourier backbone emits the entire trajectory in one pass, and the initial condition is imposed by construction rather than by penalty, which removes the competition between the anchoring term and the residual that otherwise arises when no solution data are available. Supervised and hybrid operators are trained under an identical architecture, family, budget and test set, and are reported throughout as baselines that quantify what refusing labels costs. On a reversed single vortex the data-free operator reaches 1.614 +/- 0.067% relative L2 error on 100 held-out initial interfaces against 0.369 +/- 0.035% for the supervised baseline, a factor of 4.4; on solid-body rotation the corresponding figures are 3.804 +/- 1.075% and 2.576 +/- 0.159%, a factor of 1.5. The two benchmarks rank the arms differently, and the difference is attributable to the eikonal constraint: the exact solution violates |grad phi| = 1 over 0.3% of the domain under rotation and 86.9% under the vortex. Where the constraint is valid the physics-trained operator conserves enclosed area 2.7 times better than the supervised baseline despite a larger field error, and a hybrid arm using eight reference solutions outperforms a supervised arm using sixteen.
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。
