Model Reduction for Complex Systems with Exponential Nonlinearity via Geometric Singular Perturbation Theory

Important aspects of nonlinear complex systems like the Earth's climate, gene regulatory networks or the global telecommunication network can be modelled mathematically by systems of ordinary differential equations.

However, the dynamics of such systems can rarely be understood by direct numerical simulation due to significant problems including high dimensionality, strong nonlinearity and processes occurring over a wide range of timescales.

This leads to the necessity of model reduction, i.e. the identification of reliable methods for discarding unnecessary details to obtain a simpler 'reduced system', whilst preserving the salient dynamical features of the original system.

Existing approaches to model reduction rely on a combination of methods which (i) exploit multi-scale structure in order to decompose the problem into lower dimensional subsystems, and (ii) reduce nonlinearity by replacing highly nonlinear terms by piecewise-smooth (PWS) approximations.

Despite substantial progress in particular applications, a sound mathematical basis is often lacking and existing methodologies sometimes lead to qualitatively different results.This project addresses a number of limitations to existing model reduction techniques, by developing a mathematical formalism which we call ""GSPTexp"".

We begin by building upon recent developments in Geometric Singular Perturbation Theory, with the aim to formalise recently identified connections between PWS systems and multi-scale systems with exponential nonlinearities.

Via multiple novel adaptations of a geometric method known as blow-up, the GSPTexp formalism will be developed and used as a mathematically sound method for model reduction of (i) classical model problems from combustion and (ii) gene regulatory networks.

Finally, in an ambitious interdisciplinary collaboration with systems biologists, we aim to couple GSPTexp with emerging techniques based on tropical geometry to study and simplify (iii) biochemical networks.