Copulas with Regression Copulas are joint distribution of uniform marginal distributions. Traditionally I have seen examples of fitting a Copula to the data and then simulating from the data. 
I haven't seen much on Copula based regression models. Assuming you fit a copula to m variables and response, 


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*How can I construct the conditional density of this fitted copula given all the m variables. Is it possible to differentiate an arbitrary copula wrt to the m variables

*If I can construct a conditional density by transforming from copula basis to the original basis, can I numerically integrate it to get a conditional expectation of the response given the dependent variables


Are there any examples of fitting a non-linear regression model using copulas
 A: In my opinion the two methods (copula, regression) answer quite different questions. The copula approach is much more general than regression and one of the reasons why you have not seen regression models based on copulas, might be that using copulas is much harder than using regression. Two observations why this is so:


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*For a copula fit you need to know or estimate the joint distribution of all variables involved. You do not need this for regression.

*If you are only interested in the response, regression gives you the answer more or less directly. But from the joint distribution you need to manufacture the conditional expectation of the response with additional effort.


This extra effort for estimating the joint distribution and only then finding the expected response would need to be justified by the specific problem you are interested in. Two justifications I can think of are: You are actually interested in the joint distribution (that is what you called "traditionally") or you know that your model does not allow for the standard assumptions of regression (additive independent errors, say).
On your questions 1. and 2.: Sure you can do this in theory (if the copula is differentiable and has a density). If you know the joint distribution, you can calculate all marginals and conditional expectations. The problems start when you want to estimate this from data. Unless your problem prescribes a specific, nice parametric copula, you might need special samples or lots of them to do this. 
A: I have recently  devised a curve fitting method for the relationship between two random variables based on copulas:
Regression by Integration demonstrated on Ångström-Prescott-type relations,
Renewable Energy,
Volume 127,
2018,
Pages 713-723,
ISSN 0960-1481,
https://doi.org/10.1016/j.renene.2018.05.004.
(http://www.sciencedirect.com/science/article/pii/S0960148118305238)
Abstract: We present a novel approach for the determination of the relationship between two random variables, which we call Regression by Integration. The resulting curve is a least absolute error estimate. Compared to other regression methods, it has the advantage that, instead of a sample of simultaneously taken pairs of the two random variables, only a separate sample of each of the random variables is required. We demonstrate the practicability of the method on Ångström-Prescott-type relations and compare the results with those obtained by least square error fits. We present supporting theoretical background information based on copulas. We show that Regression by Integration leads to the strict interdependence of the two random variables; Spearman's rho is equal to one.
Keywords: Ångström-Prescott relation; Copula; Curve fits; Regression by Integration; Random variable
