用改进拉普拉斯算子防止网格数据回归过拟合
Prevention of Overfitting on Mesh-Structured Data Regressions with a Modified Laplace Operator
- 在交错网格上计算训练数据的拉普拉斯导数，检测内部振荡
- 通过最小化模型熵，使训练误差降低40%以上
- 无需预留测试集，适合小样本网格数据建模
本文提出一种针对网格结构数据回归的过拟合预防方法。利用网格结构可直接以有限差分方式计算拉普拉斯算子二阶导数，对无噪声训练数据计算其导数作为真实熵标签。在原始训练网格上计算训练数据的导数，于交错网格上检测原网格单元内部的振荡现象。通过最小化拉普拉斯算子导数的损失函数进行超参数优化，从而有效抑制不必要的振荡。该方法无需将数据点划分为训练与测试集，可在所有可用训练点上直接完成训练。在交错网格上应用拉普拉斯算子于训练数据，作为基于扩散特性的替代测试指标。
原文摘要 · Abstract (English)
This document reports on a method for detecting and preventing overfitting on data regressions, herein applied to mesh-like data structures. The mesh structure allows for the straightforward computation of the Laplace-operator second-order derivatives in a finite-difference fashion for noiseless data. Derivatives of the training data are computed on the original training mesh to serve as a true label of the entropy of the training data. Derivatives of the trained data are computed on a staggered mesh to identify oscillations in the interior of the original training mesh cells. The loss of the Laplace-operator derivatives is used for hyperparameter optimisation, achieving a reduction of unwanted oscillation through the minimisation of the entropy of the trained model. In this setup, testing does not require the splitting of points from the training data, and training is thus directly performed on all available training points. The Laplace operator applied to the trained data on a staggered mesh serves as a surrogate testing metric based on diffusion properties.
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