Non-injective functions are not invertible. However, non-injective functions can be restricted to sub-domains on which they are locally injective and surjective and thus invertible if the dimensionality between input and output spaces are the same. Further, even if the dimensionalities do not match it is often possible to choose a preferred solution from many possible solutions. Twin neural network regression is naturally capable of incorporating these properties to invert non-injective functions. Twin neural network regression is trained to predict adjustments to well known input variables $\mathbf{x}^{\text{anchor}}$ to obtain an estimate for an unknown $\mathbf{x}^{\text{new}}$ under a change of the target variable from $\mathbf{y}^{\text{anchor}}$ to $\mathbf{y}^{\text{new}}$. In combination with k-nearest neighbor search, I propose a deterministic framework that finds input parameters to a given target variable of non-injective functions. The method is demonstrated by inverting non-injective functions describing toy problems and robot arm control that are a) defined by data or b) known as mathematical formula.

翻译：非单射函数不可逆。然而，当输入与输出空间维度相同时，可将非单射函数限制在局部单射且满射的子域上，从而获得可逆性。此外，即使维度不匹配，通常也能从众多可能解中选取一个优选解。孪生神经网络回归天然具备整合这些特性以实现非单射函数求逆的能力。该方法通过训练来预测已知输入变量$\mathbf{x}^{\text{anchor}}$的调整量，从而在目标变量从$\mathbf{y}^{\text{anchor}}$变为$\mathbf{y}^{\text{new}}$时，获得未知变量$\mathbf{x}^{\text{new}}$的估计值。结合k近邻搜索，本文提出一种确定性框架，可为非单射函数的给定目标变量寻找输入参数。该方法通过求逆两类非单射函数得到验证：a) 数据定义的玩具问题；b) 数学公式描述的机器人手臂控制问题。