Self-learning Emulators and Eigenvector Continuation
This work addresses the problem of computational expense in scientific calculations for researchers in fields like physics and engineering, offering a method that is incremental by building on existing emulator techniques.
The paper tackles the challenge of efficiently solving constraint equations by introducing a self-learning emulator that uses active learning and machine learning to estimate and correct emulator errors, demonstrating its application in solving transcendental equations, parameterized differential equations, and large Hamiltonian matrices with high accuracy and speed.
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.