  A variety of control tasks such as inverse kinematics (IK), trajectory
optimization (TO), and model predictive control (MPC) are commonly formulated
as energy minimization problems. Numerical solutions to such problems are
well-established. However, these are often too slow to be used directly in
real-time applications. The alternative is to learn solution manifolds for
control problems in an offline stage. Although this distillation process can be
trivially formulated as a behavioral cloning (BC) problem in an imitation
learning setting, our experiments highlight a number of significant
shortcomings arising due to incompatible local minima, interpolation artifacts,
and insufficient coverage of the state space. In this paper, we propose an
alternative to BC that is efficient and numerically robust. We formulate the
learning of solution manifolds as a minimization of the energy terms of a
control objective integrated over the space of problems of interest. We
minimize this energy integral with a novel method that combines Monte
Carlo-inspired adaptive sampling strategies with the derivatives used to solve
individual instances of the control task. We evaluate the performance of our
formulation on a series of robotic control problems of increasing complexity,
and we highlight its benefits through comparisons against traditional methods
such as behavioral cloning and Dataset aggregation (Dagger).
