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Observability of complex systems via conserved quantities

Bhargav Karamched, Jack Schmidt, David Murrugarra

TL;DR

This work addresses the problem of inferring the full state of nonlinear dynamical systems from partial measurements by linking observability to conserved quantities. It introduces a theorem showing that if conserved quantities involve a sufficient subset of variables and satisfy invertibility conditions, one can reduce the system via a differential embedding to a form that is observable even when the original outputs are not. The authors apply the framework to a constant-population SIR model and to Michaelis–Menten kinetics, demonstrating that conserved quantities can expand the set of practically measurable outputs that render the system observable, sometimes allowing measurements that are easier to obtain in practice. The results provide a principled way to choose observables in biological and physical systems by leveraging inherent invariants, with clear guidance on when conserved quantities aid observability and when they do not.

Abstract

Many systems in biology, physics, and engineering are modeled by nonlinear dynamical systems where the states are usually unknown and only a subset of the state variables can be physically measured. Can we understand the full system from what we measure? In the mathematics literature, this question is framed as the observability problem. It has to do with recovering information about the state variables from the observed states (the measurements). In this paper, we relate the observability problem to another structural feature of many models relevant in the physical and biological sciences: the conserved quantity. For models based on systems of differential equations, conserved quantities offer desirable properties such as dimension reduction which simplifies model analysis. Here, we use differential embeddings to show that conserved quantities involving a set of special variables provide more flexibility in what can be measured to address the observability problem for systems of interest in biology. Specifically, we provide conditions under which a collection of conserved quantities make the system observable. We apply our methods to provide alternate measurable variables in models where conserved quantities have been used for model analysis historically in biological contexts.

Observability of complex systems via conserved quantities

TL;DR

This work addresses the problem of inferring the full state of nonlinear dynamical systems from partial measurements by linking observability to conserved quantities. It introduces a theorem showing that if conserved quantities involve a sufficient subset of variables and satisfy invertibility conditions, one can reduce the system via a differential embedding to a form that is observable even when the original outputs are not. The authors apply the framework to a constant-population SIR model and to Michaelis–Menten kinetics, demonstrating that conserved quantities can expand the set of practically measurable outputs that render the system observable, sometimes allowing measurements that are easier to obtain in practice. The results provide a principled way to choose observables in biological and physical systems by leveraging inherent invariants, with clear guidance on when conserved quantities aid observability and when they do not.

Abstract

Many systems in biology, physics, and engineering are modeled by nonlinear dynamical systems where the states are usually unknown and only a subset of the state variables can be physically measured. Can we understand the full system from what we measure? In the mathematics literature, this question is framed as the observability problem. It has to do with recovering information about the state variables from the observed states (the measurements). In this paper, we relate the observability problem to another structural feature of many models relevant in the physical and biological sciences: the conserved quantity. For models based on systems of differential equations, conserved quantities offer desirable properties such as dimension reduction which simplifies model analysis. Here, we use differential embeddings to show that conserved quantities involving a set of special variables provide more flexibility in what can be measured to address the observability problem for systems of interest in biology. Specifically, we provide conditions under which a collection of conserved quantities make the system observable. We apply our methods to provide alternate measurable variables in models where conserved quantities have been used for model analysis historically in biological contexts.
Paper Structure (21 sections, 1 theorem, 34 equations, 3 figures)

This paper contains 21 sections, 1 theorem, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Result
  3. Applications
  4. Constant Population SIR Model
  5. Relating to the Graphical Approach
  6. Michaelis-Menten Kinetics
  7. Enzyme Conservation
  8. Substrate Conservation
  9. Enzyme and Substrate Conservation
  10. Conclusions
  11. Acknowledgments
  12. A very small example
  13. Explicit
  14. Algebraic
  15. Graphical
  16. ...and 6 more sections

Key Result

Theorem 4

Let be an observable system, where $g:\mathbb{R}^m\to\mathbb{R}^m$. If $G:\mathbb{R}^{n}\to\mathbb{R}^\ell$ is a collection of conserved quantities involving sufficient nodes $\textbf{s}$ and other variables $\textbf{r}$ where $\frac{\partial G}{\partial\textbf{s}}(\textbf{r},\textbf{s})$ is invertible is observable.

Figures (3)

  • Figure 1: Schematic of the components and result of our work. (A) A conserved quantity projects dynamics of a dynamical system to a lower-dimensional submanifold. (B) A differential embedding is a transformation of the phase space of the original system. (C) By projecting the differential embedding onto the submanifold given by the conserved quantity, scalar observables that are not predicted to render the full system observable do make the system observable.
  • Figure 2: Reaction graphs corresponding to the SIR system. (A) The original model. (B) The transformed model from invoking the transformation $I = N - S - R$.
  • Figure 3: Reaction graphs corresponding to the Michaelis-Menten system. (A) The full model. (B) The model with enzyme conservation imposed. (C) The model with enzyme conservation and substrate conservation imposed. (D) The model with substrate conservation imposed.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Example 3
  • Theorem 4
  • proof
  • proof