  The saddle point (SP) calculation is a grand challenge for computationally
intensive energy function in computational chemistry area, where the saddle
point may represent the transition state (TS). The traditional methods need to
evaluate the gradients of the energy function at a very large number of
locations. To reduce the number of expensive computations of the true
gradients, we propose an active learning framework consisting of a statistical
surrogate model, Gaussian process regression (GPR) for the energy function, and
a single-walker dynamics method, gentle accent dynamics (GAD), for the
saddle-type transition states. SP is detected by the GAD applied to the GPR
surrogate for the gradient vector and the Hessian matrix. Our key ingredient
for efficiency improvements is an active learning method which sequentially
designs the most informative locations and takes evaluations of the original
model at these locations to train GPR. We formulate this active learning task
as the optimal experimental design problem and propose a very efficient
sample-based sub-optimal criterion to construct the optimal locations. We show
that the new method significantly decreases the required number of energy or
force evaluations of the original model.
