  The Copula is widely used to describe the relationship between the marginal
distribution and joint distribution of random variables. The estimation of
high-dimensional Copula is difficult, and most existing solutions rely either
on simplified assumptions or on complicating recursive decompositions.
Therefore, people still hope to obtain a generic Copula estimation method with
both universality and simplicity. To reach this goal, a novel neural
network-based method (named Neural Copula) is proposed in this paper. In this
method, a hierarchical unsupervised neural network is constructed to estimate
the marginal distribution function and the Copula function by solving
differential equations. In the training program, various constraints are
imposed on both the neural network and its derivatives. The Copula estimated by
the proposed method is smooth and has an analytic expression. The effectiveness
of the proposed method is evaluated on both real-world datasets and complex
numerical simulations. Experimental results show that Neural Copula's fitting
quality for complex distributions is much better than classical methods. The
relevant code for the experiments is available on GitHub. (We encourage the
reader to run the program for a better understanding of the proposed method).
