Multitask multivariate regression?
You may want to model your problem using bayesian regression, which may allow you to introduce your prior knowledge in the form of priors (a priori distributions) of the model parameters. They would also allow you to model latent variables that govern the dynamics of the interactions (and impose priors on them as well).

The specific approach may be based on sampling (e.g. Markov Chain Monte Carlo) or optimization (e.g. Variational Bayes).

One of the most popular bayesian frameworks is Stan, which has bindings to R (rstan) and python (pystan). In R there are other alternatives such a BUGS and JAGS. In the python realm other options are PyMC (which is also pretty popular) or Edward.