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There are many physical processes that have inherent discontinuities in theirmathematical formulations. This paper is motivated by the specific case ofcollisions between two rigid or deformable bodies and the intrinsic nature ofthat discontinuity. The impulse response to a collision is discontinuous withthe lack of any response when no collision occurs, which causes difficultiesfor numerical approaches that require differentiability which are typical inmachine learning, inverse problems, and control. We theoretically andnumerically demonstrate that the derivative of the collision time with respectto the parameters becomes infinite as one approaches the barrier separatingcolliding from not colliding, and use lifting to complexify the solution spaceso that solutions on the other side of the barrier are directly attainable asprecise values. Subsequently, we mollify the barrier posed by the unboundedderivatives, so that one can tunnel back and forth in a smooth and reliablefashion facilitating the use of standard numerical approaches. Moreover, weillustrate that standard approaches fail in numerous ways mostly due to a lackof understanding of the mathematical nature of the problem (e.g. typicalbackpropagation utilizes many rules of differentiation, but ignores L'Hopital'srule).

Addressing Discontinuous Root-Finding for Subsequent Differentiability in Machine Learning, Inverse Problems, and Control