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arxiv: 2507.19884 · v2 · submitted 2025-07-26 · 🧮 math.DS · math.OC

Structural Identifiability and Discrete Symmetries

Xabier Rey Barreiro , Nick Baberuxki , Meskerem Abebaw Mebratie , Alejandro F. Villaverde , Werner M. Seiler This is my paper

Pith reviewed 2026-05-19 03:28 UTC · model grok-4.3

classification 🧮 math.DS math.OC
keywords structural identifiabilitydiscrete symmetriescontrol systemsdynamical systemsobservabilityglobal identifiabilityparameter estimation
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0 comments X

The pith

Discrete symmetries in system equations cause parameters to be identifiable only locally, not globally.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that discrete symmetries, unlike continuous ones, produce parameters in control systems that admit local identification from data but multiple distinct global values. This matters because such non-uniqueness can yield equivalent model fits with divergent predictions in fields like biology and engineering. The authors introduce a methodology that works directly from the system equations to detect these parameters and characterize the discrete symmetries responsible. The approach is shown to apply to both identifiability and observability questions through four concrete case studies.

Core claim

Discrete symmetries are the origin of parameters which are structurally locally identifiable, but not globally. We exploit this fact to present a methodology for structural identifiability analysis that detects such parameters and characterizes the symmetries in which they are involved. We demonstrate the use of our methodology by applying it to four case studies.

What carries the argument

A symmetry-based methodology that detects discrete symmetries from the system equations alone and uses them to identify parameters that are locally but not globally structurally identifiable.

If this is right

  • Parameters involved in discrete symmetries will be structurally locally identifiable but not globally identifiable.
  • The methodology will characterize the precise discrete symmetries linked to each such parameter.
  • The same symmetry analysis extends to structural observability questions in control systems.
  • The approach requires no external data beyond the model equations themselves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Experiment design could deliberately break discrete symmetries to convert local identifiability into global identifiability.
  • Symmetry detection may offer a route to reduce model complexity by collapsing equivalent parameter sets.
  • The framework could be tested on larger networks or stochastic variants to check scalability.

Load-bearing premise

Discrete symmetries can be reliably identified and exploited for identifiability analysis using only the system equations without requiring additional external data or assumptions about their specific form.

What would settle it

A control system in which a parameter is locally but not globally identifiable yet possesses no discrete symmetries would disprove the claimed origin of the identifiability gap.

read the original abstract

We discuss the use of symmetries for analysing the structural identifiability and observability of control systems. Special emphasis is put on the role of discrete symmetries, in contrast to the more commonly studied continuous or Lie symmetries. We argue that discrete symmetries are the origin of parameters which are structurally locally identifiable, but not globally. We exploit this fact to present a methodology for structural identifiability analysis that detects such parameters and characterizes the symmetries in which they are involved. We demonstrate the use of our methodology by applying it to four case studies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a methodology for structural identifiability analysis of control systems that exploits discrete symmetries. The central claim is that discrete symmetries lead to parameters that are structurally locally identifiable but not globally identifiable. The authors develop a method to detect such parameters and characterize the involved symmetries using only the system equations, and demonstrate it on four case studies.

Significance. If the proposed methodology can autonomously identify discrete symmetries from the system equations and correctly characterize the associated identifiability issues, this would represent a useful advance in the field of structural identifiability for dynamical systems. It shifts focus from continuous Lie symmetries to discrete ones, which are relevant for many practical systems involving sign changes or permutations. The case studies provide initial evidence of applicability, though broader validation would be beneficial.

major comments (2)
  1. [§3] §3 (Methodology): The procedure for detecting discrete symmetries is not shown to generate candidate transformations (e.g., sign flips or permutations) systematically from the equations alone. If it instead requires the user to supply or suspect a list of candidates, the claim that the method 'detects such parameters and characterizes the symmetries' using only the system equations is undermined.
  2. [Case studies] Case studies (all four): Each example uses low-order discrete symmetries that are apparent once the equations are written. This does not yet demonstrate that the method discovers non-obvious symmetries in higher-dimensional or less transparent systems, which is load-bearing for the generality of the central claim.
minor comments (2)
  1. [Abstract] Abstract: The statement that the methodology works 'using only the system equations' could be qualified by noting the class of discrete symmetries considered (finite groups of sign changes, permutations, etc.).
  2. [Notation] Notation section: Define the action of a discrete symmetry on the parameter vector more explicitly, perhaps with a small example table, to aid readers bridging identifiability and symmetry literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their constructive feedback on our manuscript. We have carefully considered the major comments and provide point-by-point responses below. Where appropriate, we have revised the manuscript to address the concerns raised.

read point-by-point responses

  1. Referee: [§3] §3 (Methodology): The procedure for detecting discrete symmetries is not shown to generate candidate transformations (e.g., sign flips or permutations) systematically from the equations alone. If it instead requires the user to supply or suspect a list of candidates, the claim that the method 'detects such parameters and characterizes the symmetries' using only the system equations is undermined.

    Authors: We thank the referee for highlighting this important point regarding the systematic nature of our methodology. In the manuscript, Section 3 describes how discrete symmetries are detected by solving the invariance conditions derived directly from the system equations. Specifically, we set up the equations that a transformation must satisfy to leave the dynamics invariant and solve for the transformation parameters. This algebraic procedure generates the possible discrete symmetries from the equations without requiring the user to pre-specify candidates beyond considering the possible types of discrete transformations relevant to the system (such as sign changes for parameters that can be positive or negative). To make this clearer, we will revise Section 3 to include a step-by-step algorithmic outline of the detection procedure. revision: yes

  2. Referee: [Case studies] Case studies (all four): Each example uses low-order discrete symmetries that are apparent once the equations are written. This does not yet demonstrate that the method discovers non-obvious symmetries in higher-dimensional or less transparent systems, which is load-bearing for the generality of the central claim.

    Authors: The case studies were selected to provide clear and accessible illustrations of how discrete symmetries lead to parameters that are locally but not globally identifiable. While the symmetries in these examples are indeed relatively straightforward, they serve to validate the methodology on concrete systems where the identifiability issues can be verified independently. The methodology itself is general and does not depend on the symmetries being obvious; it applies the same algebraic procedure regardless of system dimension. We agree that additional examples in higher dimensions would further support the generality, and we will include a brief discussion of potential extensions to more complex systems in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation builds on standard symmetry-identifiability link without reduction to inputs

full rationale

The paper argues that discrete symmetries produce locally but not globally identifiable parameters and then presents a methodology to detect both using the system equations. No quoted step shows a prediction or result that equals its own fitted input or prior self-citation by construction. The four case studies apply the procedure to concrete systems, but the central logic remains independent of the target outputs and does not rely on enumerating pre-supplied symmetries as a hidden assumption. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted. The approach presumably rests on standard background results from algebraic geometry or differential algebra for identifiability and from group theory for symmetries.

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Reference graph

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