Neural networks can approximate data because of owning many compact non-linear layers. In high-dimensional space, due to the curse of dimensionality, data distribution becomes sparse, causing that it is difficulty to provide sufficient information. Hence, the task becomes even harder if neural networks approximate data in high-dimensional space. To address this issue, according to the Lipschitz condition, the two deviations, i.e., the deviation of the neural networks trained using high-dimensional functions, and the deviation of high-dimensional functions approximation data, are derived. This purpose of doing this is to improve the ability of approximation high-dimensional space using neural networks. Experimental results show that the neural networks trained using high-dimensional functions outperforms that of using data in the capability of approximation data in high-dimensional space. We find that the neural networks trained using high-dimensional functions more suitable for high-dimensional space than that of using data, so that there is no need to retain sufficient data for neural networks training. Our findings suggests that in high-dimensional space, by tuning hidden layers of neural networks, this is hard to have substantial positive effects on improving precision of approximation data.
A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks
