Quantification of Errors Generated by Uncertain Data in a Linear Boundary Value Problem Using Neural Networks

SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 4, Page 1258-1277, December 2023.
Abstract. Quantifying errors caused by indeterminacy in data is currently computationally expensive even in relatively simple PDE problems. Efficient methods could prove very useful in, for example, scientific experiments done with simulations. In this paper, we create and test neural networks which quantify uncertainty errors in the case of a linear one-dimensional boundary value problem. Training and testing data is generated numerically. We created three training datasets and three testing datasets and trained four neural networks with differing architectures. The performance of the neural networks is compared to known analytical bounds of errors caused by uncertain data. We find that the trained neural networks accurately approximate the exact error quantity in almost all cases and the neural network outputs are always between the analytical upper and lower bounds. The results of this paper show that after a suitable dataset is used for training even a relatively compact neural network can successfully predict quantitative effects generated by uncertain data. If these methods can be extended to more difficult PDE problems they could potentially have a multitude of real-world applications.