The proposed approach can be extended to model non-autonomous systems by allowing the interaction matrix ( A ) to be a function of time, thereby capturing the dynamic nature of the network structure. This can be achieved by introducing a time-dependent parameterization of ( A(t) ), where the interaction strengths between genes or molecular species vary as a function of time.
To implement this, one could adopt a framework where ( A(t) ) is represented as a piecewise constant function or a smooth function that evolves according to a predefined model, such as a linear or nonlinear dynamical system. The optimization problem would then involve minimizing the action over a sequence of time points, incorporating the time-varying interaction matrix into the entropic optimal transport framework.
Additionally, one could leverage techniques from control theory to model the influence of external factors or perturbations that may affect the network dynamics. By integrating these time-varying interactions into the reference fitting approach, the model can adapt to changes in the underlying biological processes, such as differentiation or response to environmental stimuli, thus providing a more accurate representation of the cellular dynamics.