There has been a great interest in the scientific community in using machine learning to build emulators that can accurately predict scientific processes using only a fraction of the time needed for direct calculations. The computational advantage of emulators allows us to study processes that are beyond what is possible with direct calculations. Eigenvector continuation is one such emulation technique that was introduced recently. It is a variational method that finds the extremal eigenvalues and eigenvectors of a Hamiltonian matrix with one or more control parameters. The computational advantage comes from projecting the Hamiltonian onto a much smaller subspace of basis vectors corresponding to eigenvectors at some chosen training values of the control parameters. The method has proven to be very efficient and accurate for interpolating and extrapolating eigenvectors. In this work, we present a study on the error convergence properties of eigenvector continuation. With the insights we gain from learning the convergence properties, we then propose a self-learning algorithm to efficiently select training eigenvectors for eigenvector continuation. Self-learning is an active-learning process that relies on a fast estimate of the emulator error and a greedy local optimization algorithm that becomes more accurate as the emulator approximation improves. We show that self-learning emulators are highly efficient algorithms that offer both high speed and high accuracy, and it can be applied to any emulator that emulates the solution to a system of constraint equations, such as solutions of algebraic or transcendental equations, linear and nonlinear differential equations, and linear and nonlinear eigenvalue problems.
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