论文标题
通过神经网络估算样本值的未知多元函数的雅各布矩阵
Estimating the Jacobian matrix of an unknown multivariate function from sample values by means of a neural network
论文作者
论文摘要
我们描述，实施和测试一种训练神经网络的新方法，以估算未知多元功能$ f $的雅各布矩阵$ j $。培训集由有限的多对$（x，f（x））构建，并且不包含有关$ j $的明确信息。反向传播的损耗函数基于线性近似值和示例数据中最近的邻居搜索。我们正式建立了误差规范统一规范的上限，在操作员规范中，算法估计的雅各布矩阵与算法所提供的实际雅各布矩阵，在自然假设上，对训练集的自然假设以及训练期间的神经网络的丧失。 多元功能的Jacobian矩阵包含有关该功能的大量信息，并且在科学和工程中具有许多应用。此处给出的方法代表了从神经网络的函数函数的近似值转变为提供有关该功能的一些结构信息的近似值。
We describe, implement and test a novel method for training neural networks to estimate the Jacobian matrix $J$ of an unknown multivariate function $F$. The training set is constructed from finitely many pairs $(x,F(x))$ and it contains no explicit information about $J$. The loss function for backpropagation is based on linear approximations and on a nearest neighbor search in the sample data. We formally establish an upper bound on the uniform norm of the error, in operator norm, between the estimated Jacobian matrix provided by the algorithm and the actual Jacobian matrix, under natural assumptions on the function, on the training set and on the loss of the neural network during training. The Jacobian matrix of a multivariate function contains a wealth of information about the function and it has numerous applications in science and engineering. The method given here represents a step in moving from black-box approximations of functions by neural networks to approximations that provide some structural information about the function in question.
