Do we really need complex models?

Biological systems are often analyzed using mathematical models. These models describe the interactions between different biological components. There does not exist a single generic model that can include all the components in a biological due to their highly complex nature. Also, due to incomplete understanding and limited measurements, the number of candidate models and equations can be almost as large as the number of modelers out there. The resulting models then also greatly vary in terms of their complexity, as more complex models can capture key processes and feedbacks. Mathematically, these biological systems are often modeled using ordinary differential equations (ODEs). For example, consider marine ecosystems. In broad terms, they can be seen as the flow of food energy from nutrients, to phytoplanktons, to zooplanktons, to fishes, and finally recycling back to nutrients. Here is a simple 3 component, nutrients (N), phytoplankton (P), and zooplankton (Z) model (Newberger et. al., 2003),

In ODE based models, such as the one above, it is implicitly assumed that information is passed from one variable to the other instantaneously. In reality, however, there is often a time-delay because changes in populations or reactions have non-negligible time-scales. For example, it takes time for the dead phytoplankton and zooplankton to convert to nutrients. Such time-scales can be introduced in ODE models by modeling intermediate biological states, thus leading to complex models. Continuing with the previous example, detritus would be such an intermediate state. Unfortunately, increasing the number of components however can come at a great computational cost, can increase the overall uncertainty, and can lead to loss of accuracy or stability. However, based on the above arguments, one could hope that,

by explicitly introducing delays, the time response of lower-complexity models could be made comparable to that of high-complexity models.

Thus, capturing the effects of unmodeled intermediate components, without actually simulating them (Glass et. al., 2020).

Recently, I have worked on developing a delay differential equations (DDEs) based deep learning framework, which could be used to learn such augmentations for low complexity models that exhibit explicit delays. We call our framework, neural closure models. DDE based models are not only used in biology, but also in fields such as pharmacokinetic-pharmacodynamics, chemistry, economics, transportation, control theory, climate dynamics, etc. And, our framework enables one to learn the underlying DDE based model from the available data. We have proposed novel architectures to handle both, discrete and distributed time-delays, thus putting no constraint on the functional form of the DDE models which could be learned using our framework.

For more information and applications, please read our arXiv preprint: Gupta, A. and Lermusiaux, P.F., 2020. Neural Closure Models for Dynamical Systems. arXiv preprint arXiv:2012.13869.