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arxiv: 2606.30289 · v1 · pith:RO6BYLYQnew · submitted 2026-06-29 · 🧮 math.ST · math.DS· stat.TH

Structural functional identifiability and model discovery in differential equation models

Torkel E Loman , Alexander P Browning , Ruth E Baker This is my paper

Pith reviewed 2026-06-30 03:43 UTC · model grok-4.3

classification 🧮 math.ST math.DSstat.TH
keywords functional identifiabilitystructural identifiabilitydifferential equationsdifferential algebramodel discoveryinverse problemsmachine learning
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The pith

Functional identifiability generalizes parameter identifiability so that unknown functions in differential equation models can be checked for unique recovery using differential algebra.

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

The paper extends the classical notion of structural parameter identifiability to functional identifiability, which asks whether unknown functions or constitutive relationships in a differential equation model can be uniquely recovered from ideal observations. It shows that differential algebra methods already used for parameters apply to this functional setting and identifies broad model classes where unique recovery is impossible. The work reveals phenomena that appear only when moving from parameters to functions and have no counterpart in the older theory. It then characterizes functional identifiability for several standard classes of models. The results supply a theoretical basis for inverse problems that use machine learning to discover unknown system components.

Core claim

We generalise the classical notion of structural parameter identifiability to functional identifiability. Functional identifiability can be assessed for differential equation models using differential algebra-based techniques which are well-established as a means of assessing structural identifiability for ordinary differential equation-based models. Our framework reveals new phenomena that arise in the transition from parametric to functional inference and have no analogue in the classical setting. We characterise functional identifiability in several common model classes.

What carries the argument

Differential algebra techniques extended from parameter identifiability to determine whether unknown functional components in differential equation models admit unique recovery from observations.

If this is right

  • Unique functional recovery is impossible in broad classes of models.
  • New phenomena without analogue in the parameter setting appear during the shift to functional inference.
  • Functional identifiability can be characterized explicitly in several common model classes.
  • Machine learning representations of unknown system components now have a theoretical foundation for assessing whether they can be recovered uniquely.

Where Pith is reading between the lines

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

  • The same differential algebra approach could be used to test whether neural-network or other flexible representations of unknown terms remain identifiable after training.
  • Experiment design or choice of observation locations could be guided by functional identifiability results to guarantee unique recovery is possible in principle.
  • The framework might be adapted to check identifiability for constitutive relations in partial differential equation models or stochastic differential equations.

Load-bearing premise

Differential algebra techniques developed for parameter identifiability in ordinary differential equation models extend directly to functional identifiability without new foundational results or insurmountable obstacles.

What would settle it

A concrete differential equation model with an unknown function for which the differential algebra procedure predicts functional identifiability, yet two distinct functions generate identical observations under noise-free conditions.

Figures

Figures reproduced from arXiv: 2606.30289 by Alexander P Browning, Ruth E Baker, Torkel E Loman.

Figure 1
Figure 1. Figure 1: Typical function learned workflow. Example workflow where a functional form is learnt using a universal differential equation combined with data. (A) A differential equation model is formulated, including unknown parameters (here c and d) and functions (here f). (B) The unknown function is replaced with a universal function approximator (here, a neural network) which represents an arbitrary function throug… view at source ↗
Figure 2
Figure 2. Figure 2: A simple self-activation loop model is inherently structurally non-identifiable. (A) A self-activation loop where a single component, X, activates its own production according to an unknown function f(X) and decays at a linear (unknown) rate d. According to Proposition 3.1, this model is structurally non-identifiable. The reason is that, given any base pair (f1(X), d1), then any alternative pair d2 ∈ R, f2… view at source ↗
read the original abstract

Differential equation models are widely used to describe, interpret, and predict dynamical phenomena across science and engineering. In practice, however, the governing dynamics are rarely fully known and must be inferred from observational data. Traditionally, inverse problems in differential equation modelling have focused on estimating unknown parameter values. In this setting, structural identifiability determines whether parameter values can, in principle, be uniquely recovered from ideal observations and is, therefore, a prerequisite for meaningful inference. More recently, the integration of machine learning with mechanistic modelling has enabled the discovery of unknown equations, functions, and constitutive relationships, substantially expanding the space of admissible models. This raises a fundamental question: under what conditions can unknown functional components be uniquely recovered from data? In this paper, we generalise the classical notion of structural parameter identifiability to functional identifiability. We first identify broad classes of models for which unique functional recovery is impossible. We then show how functional identifiability can be assessed for differential equation models using differential algebra-based techniques which are well-established as a means of assessing structural identifiability for ordinary differential equation-based models. Our framework reveals new phenomena that arise in the transition from parametric to functional inference and have no analogue in the classical setting. Finally, we characterise functional identifiability in several common model classes. Taken together, our results demonstrate that functional identifiability provides a theoretical foundation for modern inverse problems in differential equation modelling, particularly those that use machine learning representations of unknown system components.

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

1 major / 2 minor

Summary. This paper generalizes the classical notion of structural parameter identifiability to functional identifiability for differential equation models. It identifies classes of models where unique recovery of unknown functions is impossible, demonstrates the use of differential algebra techniques for assessing functional identifiability, highlights new phenomena arising in the functional setting, and characterizes identifiability for several common model classes. The work aims to provide a theoretical foundation for inverse problems in DE modeling that involve machine learning for unknown components.

Significance. If the technical extension holds, the results would offer a valuable theoretical framework for ensuring unique recovery in functional model discovery, which is increasingly relevant with the integration of machine learning in mechanistic modeling. The identification of impossible cases and the discussion of new phenomena without parametric analogues represent potential strengths. Credit is due for attempting to bridge classical differential algebra methods with modern functional inference problems.

major comments (1)
  1. [§3 (framework for assessment)] §3 (framework for assessment): The assertion that differential algebra techniques extend directly to functional identifiability lacks an explicit re-derivation of the existence of a finite characteristic set or the rank conditions on the differential ideal when the ring is enlarged to include function-valued indeterminates rather than constant parameters. The classical pipeline assumes finite transcendence degree over the base field; the functional case requires a redefined notion of generic input/initial condition and a proof that the ideal membership problem remains decidable in this setting. This is load-bearing for the central claim that the same techniques suffice.
minor comments (2)
  1. [Abstract] The abstract refers to 'broad classes of models' where recovery is impossible; a concrete example or forward reference to the relevant theorem would improve clarity.
  2. [Introduction] Notation distinguishing functional unknowns from parameters could be introduced earlier to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: The assertion that differential algebra techniques extend directly to functional identifiability lacks an explicit re-derivation of the existence of a finite characteristic set or the rank conditions on the differential ideal when the ring is enlarged to include function-valued indeterminates rather than constant parameters. The classical pipeline assumes finite transcendence degree over the base field; the functional case requires a redefined notion of generic input/initial condition and a proof that the ideal membership problem remains decidable in this setting. This is load-bearing for the central claim that the same techniques suffice.

    Authors: We agree that the manuscript would be strengthened by an explicit re-derivation of the relevant differential-algebraic results in the functional setting. While the framework in §3 builds directly on the standard theory by adjoining function-valued indeterminates to the differential ring and applying the same elimination procedures, we acknowledge that the existence of a finite characteristic set, the appropriate rank conditions, and decidability of ideal membership require careful justification when the transcendence degree is no longer finite in the classical sense. In the revised version we will add a dedicated subsection (or appendix) that (i) redefines generic inputs and initial conditions for function-valued unknowns, (ii) establishes the existence of a finite characteristic set under the paper’s standing assumptions on the differential ideal, and (iii) confirms that the ideal-membership problem remains decidable. This addition will make the central claim self-contained without altering the overall results. revision: yes

Circularity Check

0 steps flagged

No circularity: extension of external differential algebra techniques to functional identifiability

full rationale

The paper's central claim is an extension of classical differential algebra methods (characteristic sets, Rosenfeld-Groebner bases) already established for parametric structural identifiability in ODEs. The abstract and description explicitly frame the functional case as an application of these pre-existing techniques rather than a self-derived result. No equations or steps reduce the claimed assessment procedure to a fitted parameter, a self-definition, or a load-bearing self-citation chain; the algebraic pipeline is treated as an independent external tool whose applicability is asserted but not re-derived from the paper's own inputs. This satisfies the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; the central claim rests on the unexamined extension of differential algebra methods.

axioms (1)
  • domain assumption Differential algebra techniques established for parameter identifiability extend to functional components without additional foundational obstacles
    The abstract states that these techniques can be used to assess functional identifiability.

pith-pipeline@v0.9.1-grok · 5802 in / 1175 out tokens · 46828 ms · 2026-06-30T03:43:04.358864+00:00 · methodology

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    Case 2:fis affine.Writef i(X2) =a iX2 +b i

    Thus dis identifiable, but onlyf ′ (notfitself) is identifiable. Case 2:fis affine.Writef i(X2) =a iX2 +b i. Substituting into Equations (49)–(50) and equating coefficients gives (1−d 1)a1 = (1−d 2)a2, (1−d 1)b1 = (1−d 2)b2, d1 +a 1 =d 2 +a 2. The first and third relations show thata+dand (1−d)aare globally identifiable. Substituting a= (a+d)−dinto (1−d)a...