Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact hierarchical reductions of dynamical models via linear transformations

Alexander Demin, Elizaveta Demitraki, Gleb Pogudin

TL;DR

The paper addresses exact linear reductions of ODEs by computing a longest chain of lumpings that refine each other, enabling flexible retention of dynamical detail. It builds a novel algorithm that reduces the problem to finding common invariant subspaces of a Jacobian algebra $\mathcal{A} = \langle I_n, J_1, \ldots, J_N \rangle$, combining sparsity-aware algebra basis generation with finite-dimensional algebra theory (radical, semisimple, center, centralizer). The authors implement the method in Julia (ExactODEReduction.jl) and demonstrate its effectiveness on BioModels benchmarks and two life-science case studies, achieving long chains (up to 23) and interpretable macro-variables. This work enables exact, mechanistic reductions suitable for formal analysis and preprocessing for other modeling techniques, with future directions including approximate lumpings and extensions to rational right-hand sides.

Abstract

Dynamical models described by ordinary differential equations (ODEs) are a fundamental tool in the sciences and engineering. Exact reduction aims at producing a lower-dimensional model in which each macro-variable can be directly related to the original variables, and it is thus a natural step towards the model's formal analysis and mechanistic understanding. We present an algorithm which, given a polynomial ODE model, computes a longest possible chain of exact linear reductions of the model such that each reduction refines the previous one, thus giving a user control of the level of detail preserved by the reduction. This significantly generalizes over the existing approaches which compute only the reduction of the lowest dimension subject to an approach-specific constraint. The algorithm reduces finding exact linear reductions to a question about representations of finite-dimensional algebras. We provide an implementation of the algorithm, demonstrate its performance on a set of benchmarks, and illustrate the applicability via case studies. Our implementation is freely available at https://github.com/x3042/ExactODEReduction.jl

Exact hierarchical reductions of dynamical models via linear transformations

TL;DR

The paper addresses exact linear reductions of ODEs by computing a longest chain of lumpings that refine each other, enabling flexible retention of dynamical detail. It builds a novel algorithm that reduces the problem to finding common invariant subspaces of a Jacobian algebra , combining sparsity-aware algebra basis generation with finite-dimensional algebra theory (radical, semisimple, center, centralizer). The authors implement the method in Julia (ExactODEReduction.jl) and demonstrate its effectiveness on BioModels benchmarks and two life-science case studies, achieving long chains (up to 23) and interpretable macro-variables. This work enables exact, mechanistic reductions suitable for formal analysis and preprocessing for other modeling techniques, with future directions including approximate lumpings and extensions to rational right-hand sides.

Abstract

Dynamical models described by ordinary differential equations (ODEs) are a fundamental tool in the sciences and engineering. Exact reduction aims at producing a lower-dimensional model in which each macro-variable can be directly related to the original variables, and it is thus a natural step towards the model's formal analysis and mechanistic understanding. We present an algorithm which, given a polynomial ODE model, computes a longest possible chain of exact linear reductions of the model such that each reduction refines the previous one, thus giving a user control of the level of detail preserved by the reduction. This significantly generalizes over the existing approaches which compute only the reduction of the lowest dimension subject to an approach-specific constraint. The algorithm reduces finding exact linear reductions to a question about representations of finite-dimensional algebras. We provide an implementation of the algorithm, demonstrate its performance on a set of benchmarks, and illustrate the applicability via case studies. Our implementation is freely available at https://github.com/x3042/ExactODEReduction.jl
Paper Structure (14 sections, 10 theorems, 30 equations, 3 figures, 1 table, 3 algorithms)

This paper contains 14 sections, 10 theorems, 30 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Algorithm
  4. Reduction to the search for common invariant subspaces
  5. Generating the algebra
  6. Search for invariant subspaces: algebraic preliminaries
  7. Search for invariant subspaces: how to find one
  8. Search for invariant subspaces: how to find a chain
  9. Putting everything together
  10. Implementation and performance
  11. Case studies
  12. Inactivation of factor Va
  13. Model of cell death
  14. Conclusions

Key Result

Lemma 1

Using the notation above, the linear transformation $\mathbf{y} = \mathbf{x}L$, where $L \in \mathbb{C}^{n \times m}$, is a lumping of $\mathbf{x}' = \mathbf{f}(\mathbf{x})$ if and only if the column space of $L$ is invariant with respect to $J_1, \ldots, J_N$.

Figures (3)

  • Figure 1: Maximal chain of lumpings for \ref{['eq:ex_ode']} and the corresponding reductions
  • Figure 2: Numerical simulation for the model from FactorV and its reduction using the initial conditions and parameter values from FactorV
  • Figure 3: The relevant chemical species and dependencies between them

Theorems & Definitions (28)

  • definition 1: Lumping
  • definition 2: Chain of lumpings
  • remark 1: Connection to symmetries of a rule-based representation
  • remark 2: Connection to quiver-equivariant ODEs
  • Lemma 1
  • proof
  • remark 3
  • corollary 1
  • definition 3: Matrix algebra
  • corollary 2
  • ...and 18 more