  This paper addresses the problem of modeling and estimating dynamic
multi-valued mappings. While most mathematical models provide a unique solution
for a given input, real-world applications often lack deterministic solutions.
In such scenarios, estimating dynamic multi-valued mappings is necessary to
suggest different reasonable solutions for each input. This paper introduces a
deep neural network framework incorporating a generative network and a
classification component. The objective is to model the dynamic multi-valued
mapping between the input and output by providing a reliable uncertainty
measurement. Generating multiple solutions for a given input involves utilizing
a discrete codebook comprising finite variables. These variables are fed into a
generative network along with the input, producing various output
possibilities. The discreteness of the codebook enables efficient estimation of
the output's conditional probability distribution for any given input using a
classifier. By jointly optimizing the discrete codebook and its uncertainty
estimation during training using a specially designed loss function, a highly
accurate approximation is achieved. The effectiveness of our proposed framework
is demonstrated through its application to various imaging problems, using both
synthetic and real imaging data. Experimental results show that our framework
accurately estimates the dynamic multi-valued mapping with uncertainty
estimation.
