Solution of Linear Systems in High Dimensions with Application to Chemical Reaction Networks

For biochemical reaction networks containing species with small copy numbers, it is necessary to model the correlated fluctuations of the copy numbers of different species to obtain an accurate picture of the dynamics. This makes it tempting to model systems by time-evolving the joint probability distribution over all possible microstates. While the curse of dimensionality would appear to preclude such a strategy, we illustrate how it is possible to use the tensor train (TT) format to controllably approximate that exponentially large distribution. Moreover, we show how by combining the Doi-Peliti formalism, the Crank-Nicholson method, and the tensor train (TT) and quantized tensor train (QTT) formats, it is possible to obtain a linear system that can be solved to obtain a single tensor network representation for the joint distribution at multiple times in a chosen interval. We argue that this leads to a logarithmic scaling of computational resources with the time evolved, an exponential improvement over time stepping schemes using uniform time steps, which scale linearly with time. We demonstrate our method by application to a genetic toggle switch (GTS) gene regulatory network.