Neural Network Approximation of PDEs - Efficiency, Reliability and Quantifiable Accuracy

Machine learning and neural networks (NN) are impacting everyday our lives in a way that few, if any, other mathematical tool has ever done before. Machine learning is also attracting fevered interest in the scientific computing community where neural network techniques have been applied to all manner of applications in scientific computing including the numerical solution of differential equations (PDEs). However, the bar is set much higher when it comes to applying neural networks to scientific applications, where there is an expectation that numerical methods are capable of producing high accuracy approximations, supported by a solid theoretical foundation, give results that make sense, are reproducible, intelligible to the analyst, and (ideally) come with reliable numerical bounds on the accuracy. Numerical simulation is routinely used in science and engineering applications to form the basis for a critical decision for which certain minimal levels of confidence in the numerical results are required. Machine learning and neural network techniques will not be accepted until and unless methods are developed that address these minimal requirements. Current machine learning and neural network methods for scientific applications in general, and numerical solution of PDEs in particular, are not at this level. The project includes graduate student training on the fundamental research questions relating to the project and implementation of numerical algorithms that verify the theoretical analysis. This project builds on foundations in machine learning and neural network approximation with the overall objective of developing effective computational tools supported by rigorous theory that provide the analyst with the confidence to apply neural network based techniques to scientific applications. A key feature of the current project is the importance attached to developing rigorous theoretical foundations for the approach including a priori results on the convergence and accuracy of the methods. In addition, the proposal aims to provide quantitative estimates for the accuracy and fidelity of the resulting approximations through the provision of computable a posteriori estimators for the error. The techniques developed in this project will be applicable to a broad class of PDEs arising across a wide range of disciplines. The tools that will be developed will find immediate application in these areas. An important feature of the project is the development of effective training algorithms for neural networks that will have impact beyond the numerical approximation of PDEs that are the focus of the project. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.