GSPTexp · Model Reduction for Complex Systems with Exponential Nonlinearity via Geometric Singular Perturbation Theory
Horizon Europe — Marie Skłodowska-Curie Actions
- Duration
- 2024-04-02 → 2026-04-01
- EU contribution
- €199,441
- Participants
- 3
- Scheme
- HORIZON-TMA-MSCA-PF-EF
Lines connect the coordinator with its partners.
Results in brief
Model Reduction for Complex Systems with Exponential Nonlinearity via Geometric Singular Perturbation Theory
Important aspects of the dynamics of complex systems ranging from the Earth’s climate to the cell cycle can be mathematically modeled using ordinary differential equations. However, direct analysis or numerical simulation is rarely possible due to the fact that these models are typically highly nonlinear, high dimensional and multi-scale. This leads to the problem of model reduction, in which the modeler is faced with the task of deciding which and how much information should be discarded in order to obtain a reduced model which is more amenable to analysis, while preserving the salient features of the original system. Unfortunately, model reductions are rarely justified in mathematical terms, which is problematic because they are often used in order to make predictions about complex dynamical systems of social and economic significance. The aim of this project was to develop systematic mathematical theory for model reduction, and to apply it to important problems in combustion theory, gene regulatory dynamics and selected biological and biochemical networks which are, from a mathematical point of view, highly non-trivial because of the presence of 'severe' exponential nonlinearities. The primary innovation was to combine different approaches to model reductions in a single mathematical framework using adaptations of an established analytical tool known to experts as the "geometric blow-up method". By developing a sound mathematical theory for model reduction, we hoped to identify conditions for the validity or invalidity of commonly used model reductions in applications that are well-known within the modeling community, thereby raising awareness and - hopefully - preventing incorrect predictions about important complex dynamical systems in the future which result from unjustified reliance of the 'wrong' reduced model.
Data: CORDIS, © European Union
Project objective
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.""
Original text from CORDIS.
Participants
Links
- View on CORDIS
- DOI: 10.3030/101103827
- https://ec.europa.eu/research/participants/documents/downloadPublic?documentIds=080166e5139202c1&appId=PPGMS
- https://ec.europa.eu/research/participants/documents/downloadPublic?documentIds=080166e5182f04a9&appId=PPGMS
Data: CORDIS, © European Union