Self-learning Emulators and Eigenvector Continuation

TL;DR

This paper introduces a self-learning emulator framework that efficiently solves constraint equations by actively estimating and improving error accuracy, demonstrated through examples including eigenvector continuation.

Contribution

It presents a novel active learning protocol for emulators that enhances accuracy and efficiency in solving complex scientific equations.

Findings

Effective error estimation improves emulator accuracy over iterations

Demonstrated success with spline, reduced basis, and eigenvector continuation methods

Enables faster solutions to parameter-dependent scientific problems

Abstract

Emulators that can bypass computationally expensive scientific calculations with high accuracy and speed can enable new studies of fundamental science as well as more potential applications. In this work we discuss solving a system of constraint equations efficiently using a self-learning emulator. A self-learning emulator is an active learning protocol that can be used with any emulator that faithfully reproduces the exact solution at selected training points. The key ingredient is a fast estimate of the emulator error that becomes progressively more accurate as the emulator is improved, and the accuracy of the error estimate can be corrected using machine learning. We illustrate with three examples. The first uses cubic spline interpolation to find the solution of a transcendental equation with variable coefficients. The second example compares a spline emulator and a reduced basis method emulator to find solutions of a parameterized differential equation. The third example uses eigenvector continuation to find the eigenvectors and eigenvalues of a large Hamiltonian matrix that depends on several control parameters.