Function Extrapolation with Neural Networks and Its Application for Manifolds

TL;DR

This paper proposes a neural network-based method for function extrapolation on manifolds, providing error bounds, a difficulty measure, and improved robustness, with demonstrated effectiveness through numerical experiments.

Contribution

It introduces a novel neural network approach for function extrapolation on manifolds, including error bounds and a condition number to quantify problem difficulty.

Findings

Effective in various extrapolation scenarios

Provides error bounds and a condition number

Outperforms standard methods in tests

Abstract

This paper addresses the problem of accurately estimating a function on one domain when only its discrete samples are available on another domain. To answer this challenge, we utilize a neural network, which we train to incorporate prior knowledge of the function. In addition, by carefully analyzing the problem, we obtain a bound on the error over the extrapolation domain and define a condition number for this problem that quantifies the level of difficulty of the setup. Compared to other machine learning methods that provide time series prediction, such as transformers, our approach is suitable for setups where the interpolation and extrapolation regions are general subdomains and, in particular, manifolds. In addition, our construction leads to an improved loss function that helps us boost the accuracy and robustness of our neural network. We conduct comprehensive numerical tests and comparisons of our extrapolation versus standard methods. The results illustrate the effectiveness of our approach in various scenarios.