  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.
