There are many physical processes that have inherent discontinuities in their mathematical formulations. This paper is motivated by the specific case of collisions between two rigid or deformable bodies and the intrinsic nature of that discontinuity. The impulse response to a collision is discontinuous with the lack of any response when no collision occurs, which causes difficulties for numerical approaches that require differentiability which are typical in machine learning, inverse problems, and control. We theoretically and numerically demonstrate that the derivative of the collision time with respect to the parameters becomes infinite as one approaches the barrier separating colliding from not colliding, and use lifting to complexify the solution space so that solutions on the other side of the barrier are directly attainable as precise values. Subsequently, we mollify the barrier posed by the unbounded derivatives, so that one can tunnel back and forth in a smooth and reliable fashion facilitating the use of standard numerical approaches. Moreover, we illustrate that standard approaches fail in numerous ways mostly due to a lack of understanding of the mathematical nature of the problem (e.g. typical backpropagation utilizes many rules of differentiation, but ignores L'Hopital's rule).

翻译：许多物理过程在数学表述中固有地存在不连续性。本文的动机源于两个刚体或可变形体碰撞这一具体情形及其不连续性的内在本质。碰撞的脉冲响应具有不连续性——当无碰撞发生时响应完全缺失，这给机器学习、反问题与控制等需要可微性的数值方法带来了困难。我们从理论及数值上证明，当参数趋近于分离碰撞与非碰撞的势垒时，碰撞时间对参数的导数趋于无穷大，进而通过提升（lifting）将解空间复化，使得势垒另一侧的解可直接作为精确值获得。随后，我们对由无界导数构成的势垒进行光滑化处理（mollify），使得人们能够以平滑可靠的方式来回穿越势垒，从而便于使用标准数值方法。此外，我们指出标准方法在多个方面失败的根本原因在于对该问题数学本质的理解不足（例如，典型反向传播虽运用了众多微分法则，却忽略了洛必达法则）。