  This paper investigates the learning, or system identification, of a class of
piecewise-affine dynamical systems known as linear complementarity systems
(LCSs). We propose a violation-based loss which enables efficient learning of
the LCS parameterization, without prior knowledge of the hybrid mode
boundaries, using gradient-based methods. The proposed violation-based loss
incorporates both dynamics prediction loss and a novel complementarity -
violation loss. We show several properties attained by this loss formulation,
including its differentiability, the efficient computation of first- and
second-order derivatives, and its relationship to the traditional prediction
loss, which strictly enforces complementarity. We apply this violation-based
loss formulation to learn LCSs with tens of thousands of (potentially stiff)
hybrid modes. The results demonstrate a state-of-the-art ability to identify
piecewise-affine dynamics, outperforming methods which must differentiate
through non-smooth linear complementarity problems.
