We present an algorithm for learning operators between Banach spaces, based on the use of Leray-Schauder mappings to learn a finite-dimensional approximation of compact subspaces. We show that the resulting method is a universal approximator of (possibly nonlinear) operators. We demonstrate the efficiency of the approach on two benchmark datasets showing it achieves results comparable to state of the art models.

We obtain a new universal approximation theorem for continuous operators on arbitrary Banach spaces using the Leray-Schauder mapping. Moreover, we introduce and study a method for operator learning in Banach spaces $L^p$ of functions with multiple variables, based on orthogonal projections on polynomial bases. We derive a universal approximation result for operators where we learn a linear projection and a finite dimensional mapping under some additional assumptions. For the case of $p=2$, we give some sufficient conditions for the approximation results to hold. This article serves as the theoretical framework for a deep learning methodology whose implementation will be provided in subsequent work.

Neural integral equations are deep learning models based on the theory of integral equations, where the model consists of an integral operator and the corresponding equation (of the second kind) which is learned through an optimization procedure. This approach allows to leverage the nonlocal properties of integral operators in machine learning, but it is computationally expensive. In this article, we introduce a framework for neural integral equations based on spectral methods that allows us to learn an operator in the spectral domain, resulting in a cheaper computational cost, as well as in high interpolation accuracy. We study the properties of our methods and show various theoretical guarantees regarding the approximation capabilities of the model, and convergence to solutions of the numerical methods. We provide numerical experiments to demonstrate the practical effectiveness of the resulting model.

Battery performance datasets are typically non-normal and multicollinear. Extrapolating such datasets for model predictions needs attention to such characteristics. This study explores the impact of data normality in building machine learning models. In this work, tree-based regression models and multiple linear regressions models are each built from a highly skewed non-normal dataset with multicollinearity and compared. Several techniques are necessary, such as data transformation, to achieve a good multiple linear regression model with this dataset; the most useful techniques are discussed. With these techniques, the best multiple linear regression model achieved an R^2 = 81.23% and exhibited no multicollinearity effect for the dataset used in this study. Tree-based models perform better on this dataset, as they are non-parametric, capable of handling complex relationships among variables and not affected by multicollinearity. We show that bagging, in the use of Random Forests, reduces overfitting. Our best tree-based model achieved accuracy of R^2 = 97.73%. This study explains why tree-based regressions promise as a machine learning model for non-normally distributed, multicollinear data.

Americans prefer a space program that focuses on potential asteroid impacts, scientific research and using robots to explore the cosmos over sending humans back to the moon or on to Mars, a poll shows. The poll by The Associated Press and the NORC Center for Public Affairs Research, released Thursday, one month before the 50th anniversary of the Apollo 11 moon landing, lists asteroid and comet monitoring as the No. 1 desired objective for the United States space program. About two-thirds of Americans call that very or extremely important, and about a combined 9 in 10 say it's at least moderately important. The poll comes as the White House pushes to get astronauts back on the moon, but only about a quarter of Americans said moon or Mars exploration by astronauts should be among the space program's highest priorities. About another third called each of those moderately important.

CAPE CANAVERAL, FLORIDA - Americans prefer a space program that focuses on potential asteroid impacts, scientific research and using robots to explore the cosmos over sending humans back to the moon or on to Mars, a poll shows. The poll by The Associated Press-NORC Center for Public Affairs Research, released Thursday, one month before the 50th anniversary of the Apollo 11 moon landing, lists asteroid and comet monitoring as the No. 1 desired objective for the U.S. space program. About two-thirds of Americans call that very or extremely important, and about a combined 9 in 10 say it's at least moderately important. The poll comes as the White House pushes to get astronauts back on the moon, but only about a quarter of Americans said moon or Mars exploration by astronauts should be among the space program's highest priorities. About another third called each of those moderately important.

CAPE CANAVERAL, Fla. -- Americans prefer a space program that focuses on potential asteroid impacts, scientific research and using robots to explore the cosmos over sending humans back to the moon or on to Mars, a poll shows. The poll by The Associated Press and the NORC Center for Public Affairs Research, released Thursday, one month before the 50th anniversary of the Apollo 11 moon landing, lists asteroid and comet monitoring as the No. 1 desired objective for the U.S. space program. About two-thirds of Americans call that very or extremely important, and about a combined 9 in 10 call it at least moderately important. The poll comes as the White House pushes to get astronauts back on the moon, but only about a quarter of Americans said moon or Mars exploration by astronauts should be among the space program's highest priorities. About another third called each of those moderately important.