论文标题
级别方法中三维均值曲率计算的机器学习算法
Machine learning algorithms for three-dimensional mean-curvature computation in the level-set method
论文作者
论文摘要
我们为级别集方法提出了一个数据驱动的均值曲线求解器。这项工作是我们在[DOI：10.1007：10.1007/s10915-022-01952-2] [1]和[doi：10.1016/j.jcp.2022.111291] [doi：10.1016/j.jcp.2016/j.jcp.2022.111291] [doi：10.1007/s10915-022-01952-2]中的$ \ mathbb {r}^3 $的自然扩展。但是，与[1,2]相比，它构建了依赖分辨率的神经网络词典，在这里，我们在$ \ mathbb {r}^3 $中开发了两对模型，无论网格大小如何。我们的前馈网络摄入的水平集，梯度和曲率数据转换为固定接口节点的数值均值曲率近似值。为了降低问题的复杂性，我们使用高斯曲率对模板进行了分类，并将模型分别适合于非堆肥和鞍模式。非插曲模板更容易处理，因为它们表现出以单调性和对称性为特征的曲率误差分布。尽管后者只允许我们在平均曲面频谱的一半上进行训练，但前者帮助我们融合了数据驱动的，并且基线估计在平坦的区域附近无缝地融合。另一方面，马鞍模式误差结构不太清楚。因此，我们没有利用超出已知信息的潜在信息。在这方面，我们不仅在球形和正弦和双曲线抛物面斑块上训练了我们的模型。我们构建他们的数据集的方法是系统的，但是随机收集样品，同时确保均衡。我们还采用了标准化和降低维度的降低和综合正则化，以最大程度地减少异常值。此外，我们利用曲率旋转/反射不变性在推理时提高精度。几项实验证实，与现代粒子界面的重建和隔离不足区域周围的水平集合相比，我们提出的系统可以产生更准确的均值映射估计。
We propose a data-driven mean-curvature solver for the level-set method. This work is the natural extension to $\mathbb{R}^3$ of our two-dimensional strategy in [DOI: 10.1007/s10915-022-01952-2][1] and the hybrid inference system of [DOI: 10.1016/j.jcp.2022.111291][2]. However, in contrast to [1,2], which built resolution-dependent neural-network dictionaries, here we develop a pair of models in $\mathbb{R}^3$, regardless of the mesh size. Our feedforward networks ingest transformed level-set, gradient, and curvature data to fix numerical mean-curvature approximations selectively for interface nodes. To reduce the problem's complexity, we have used the Gaussian curvature to classify stencils and fit our models separately to non-saddle and saddle patterns. Non-saddle stencils are easier to handle because they exhibit a curvature error distribution characterized by monotonicity and symmetry. While the latter has allowed us to train only on half the mean-curvature spectrum, the former has helped us blend the data-driven and the baseline estimations seamlessly near flat regions. On the other hand, the saddle-pattern error structure is less clear; thus, we have exploited no latent information beyond what is known. In this regard, we have trained our models on not only spherical but also sinusoidal and hyperbolic paraboloidal patches. Our approach to building their data sets is systematic but gleans samples randomly while ensuring well-balancedness. We have also resorted to standardization and dimensionality reduction and integrated regularization to minimize outliers. In addition, we leverage curvature rotation/reflection invariance to improve precision at inference time. Several experiments confirm that our proposed system can yield more accurate mean-curvature estimations than modern particle-based interface reconstruction and level-set schemes around under-resolved regions.
