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Observable functions of rational ODE models and how to find them

Alexander Demin, Gleb Pogudin, Christopher Rackauckas

arXiv:2609.06134v3math.DScs.MScs.SCcs.SYeess.SY

TL;DR

When states and parameters are not individually observable, input-output data may still determine useful combinations, but finding a concise generating set for that information is a computational challenge. The paper develops an algorithm for rational ODE models using coefficient extraction from Lie derivatives and identifiable parameter combinations to reduce derivative orders. Applied to models from several domains, it produces efficient, interpretable generators.

  • Problem

    Even when individual states or parameters are not observable, useful observable combinations may remain, motivating a concise generating set for the observation field.

    Source: §1 · p. 1

  • Method

    The algorithm extracts observable functions from Lie-derivative coefficients and reduces derivative orders by starting from identifiable parameter combinations.

    Source: §1 · p. 2

  • Results

    The implemented algorithm is reported to be efficient and to produce concise, model-specific interpretable generators on literature models from epidemiology, chemistry, and cancer modeling.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Observation-field generators can include practically relevant quantities and guide system identification, selection of complementary data streams, and model reparametrization.

    Source: §1 · p. 1

    Smaller Lie-derivative orders may suffice than the stated upper bound, while the parametric-profile condition constrains that gap.

    Source: §4.2 · p. 9

Abstract

from arXiv · show

Consider a parametric ODE control model. A function of the states and parameters is called observable if its value can in principle be reconstructed from input-output data. The observable functions form a field, called the observation field, represented naturally by a set of generators. Even when the model is not fully observable, this field captures the information still accessible from input-output data. We present an algorithm for computing a concise generating set for the observation field of a model with rational dynamics. The algorithm relies on two new results: one allows observable functions to be extracted from the coefficients of repeated Lie derivatives of the outputs, while the other reduces the required orders of differentiation by exploiting identifiable parameter combinations. We implement the resulting algorithm in StructuralIdentifiability$.$jl (https://github.com/SciML/StructuralIdentifiability.jl). For computational efficiency, we employ recent techniques for differential elimination and rational function field simplification. Using models from epidemiology, chemical kinetics, and cancer modeling, we show that the algorithm produces generators with domain-specific interpretations that can inform model analysis and development.

1 Introduction

The paper develops a practical algorithm for generating the observation field of rational ODE models, addressing both non-affine input dependence and the cost of high-order Lie derivatives. Its generators can capture useful state-parameter combinations even when individual states or parameters are not observable.

1 Introduction

  • Observable functions can remain informative when individual states or parameters cannot be recovered, and their generators can support model analysis and development.The paper cites the basic reproduction number and binding density as examples, and notes uses in system identification, data-stream selection, and reparametrization.Source: §1 · p. 1
  • The paper presents an implemented algorithm for computing observation-field generators for rational dynamics with arbitrary rational dependence on inputs.It is applied to literature models in epidemiology, chemistry, and cancer modeling, producing efficient, concise generators with model-specific interpretations.Source: §1 · p. 2
  • The first technical contribution extracts observable functions from coefficients of Lie-derivative expressions even when inputs enter non-affinely.If an observable function is differential-rational in the inputs, its numerator and denominator coefficients are observable; this extends the input-affine result.Source: §1 · p. 2
  • The second contribution reduces the required Lie-derivative orders by completing an initial set of known observable functions with lower-order derivatives.The proposed starting set uses identifiable parameter-only functions obtained from input-output equations [35, 37].Source: §1 · p. 2
  • The implementation combines efficient input-output equation computation with field-generator simplification, which helps make the output concise and interpretable.The paper also situates its algorithm against a constructive input-affine procedure not known to have been implemented and local-observability methods without a complete algorithm for generating sets.Source: §1 · p. 2

2 Preliminaries and problem statement

The preliminaries formalize observability for rational ODE models and define the observation field as the state-and-parameter information recoverable from input-output data. The problem is to find generators for that field, which can be more concise and useful than other generating sets.

2 Preliminaries and problem statement

  • The model has states, time-independent parameters, inputs, and outputs, with rational state dynamics and output functions.The paper uses observability for states and parameters, including combinations of both.Source: §2 · p. 2
  • Observability asks whether a state can be uniquely recovered from known inputs and observed outputs under ideal conditions, while identifiability is the analogous property for parameters.The paper uses observability broadly for functions involving states and parameters.Source: §2 · p. 2
  • The paper's algebraic definition of observability is stated to be mathematically equivalent to the standard analytic definition.The formal definition uses polynomials in output and input derivatives to express reconstruction.Source: §2 · pp. 2–3
  • Observable functions form a field, and restricting it to rational functions of states and parameters defines the observation field.All observable functions involving inputs can be expressed using the observation field and input derivatives.Source: §2 · p. 3
  • A model can reveal parameter combinations without revealing an individual state, as illustrated by an output that determines μ1 and μ2x but not x.The example's observation field is generated by μ1 and μ2x, reflecting the output-preserving symmetry in the state and parameter.Source: §2 · p. 3
  • The formal problem is to find generators of the observation field for a given ODE model.Different generating sets can vary substantially in size and degree; the paper illustrates a four-generator, degree-two set versus a seven-generator set of degrees up to eleven.Source: §2 · p. 4

3 Informal description of solution

The informal method extracts input-free observable functions from output Lie derivatives, including for rational input dependence. It reduces differentiation work by starting from identifiable parameter combinations and simplifies the resulting generators.

3 Informal description of solution

  • All observable functions can be generated arithmetically from the output, its repeated Lie derivatives, and input derivatives, and finitely many Lie derivatives suffice.The general bound is order |x| + |µ|; the example's refined bound is order four rather than six.Source: §3 · p. 4
  • For rational input dependence, every numerator and denominator coefficient of an observable rational function in the input and its derivatives is itself observable.This extends coefficient extraction beyond the input-affine case, where the earlier result does not apply.Source: §3 · pp. 4–5
  • The Lie derivative LΣ(g) = A B + u shows that A and B are observable functions.Source: §3 · p. 5
  • Identifiable parameter-only observables can seed the process, allowing fewer Lie derivatives; the running example needs derivatives only through order three.The method alternates Lie differentiation with extracting observable coefficients.Source: §3 · p. 5
  • With a sufficiently exciting input, linear independence of y, y2, y′, and uy′ lets the input-output equation's coefficients be uniquely determined and treated as observable.Those coefficients can then be complemented using the output and its first Lie derivative.Source: §3 · p. 5
  • Generator simplification removes redundancies, while applying it to the large higher-order expressions can be much harder and sometimes out of reach.In the example, one generator can be expressed using two others.Source: §3 · p. 5

4 Formal description of solution

This section characterizes observable functions using coefficients of output Lie derivatives and conditions that bound the derivative orders required. It then describes how observable parameter functions can reduce those orders and be combined with Lie-derivative coefficients to generate and simplify the observation field.

4.1 Splitting with respect to inputs

  • If an observable function is written as a coprime numerator and denominator in input derivatives, with one coefficient normalized to one, all their coefficients are observable.This is the input-splitting result of Theorem 4.1.Source: §4.1 · pp. 5–6
  • In the example, a coefficient equal to one in the second output Lie derivative lets Theorem 4.1 establish that its other input-derivative coefficients are observable.The passage lists the resulting observable coefficients as 2µ1µ4(µ1x + µ2)3, µ4(µ1x + µ2)2, µ2^3, and 2µ3.Source: §4.1 · p. 6
  • The field-theoretic reformulation identifies observability with membership in the field generated by output derivatives and inputs.The proof of Lemma 4.4 shows that an identifiable function belongs to C(Y(∞), U(∞)); conversely, membership there implies identifiability.Source: §4.1 · p. 7

4.2 Generating all observable functions

  • Theorem 4.12 characterizes observable functions using repeated output Lie derivatives when each output satisfies an equation over a Lie-derivative-closed observation subfield with specified orders.For output i, the equation has order h_i + 1 in y_i and at most h_j in each other output y_j.Source: §4.2 · p. 8
  • The parametric profile depends on the ground field: for x′ = µ1x and y = x, it is 0 over C(µ1) but 1 over C.Over C(µ1), y′ − µ1y = 0; over C, the passage gives yy′′ − (y′)^2 = 0 as the smallest-order equation.Source: §4.2 · p. 9
  • Theorem 4.12 gives an upper bound of h + 1 on required Lie-derivative orders; when h is a parametric profile, any qualifying bound h* satisfies h ≤ h* ≤ h + 1.Some components of h* may be below h + 1 when h is not assumed to be a parametric profile.Source: §4.2 · p. 9

4.3 Where identifiability and observability meet

  • Starting from observable parameter functions c1, . . . , cp, the observation field is generated by them and coefficients of output Lie derivatives through the order set by the parametric profile.The corollary contrasts this with using Lie-derivative coefficients alone, which may require higher orders.Source: §4.3 · p. 10
  • For the DC motor, identifiable transfer-function coefficients c1, c2, c3 and coefficients from ω and its first two Lie derivatives generate the observation field, versus derivatives up to order 8 in the straightforward approach.The profile is bounded by 1 because the input-output variables satisfy a second-order differential equation over E = C(c1, c2, c3).Source: §4.3 · p. 10

4.4 Main algorithm

  • Algorithm 1 obtains parameter functions from a parametric profile and normalized input-output equations, checks their observability, computes Lie-derivative coefficients, and simplifies the generated field.If the parameter functions are not observable, it converts parameters into constant state variables and applies the algorithm to the modified model.Source: §4.4 · p. 11
  • The algorithm may alternate coefficient extraction with further Lie differentiation and stop when higher derivatives are algebraic over the field generated so far.This can require orders below h + 1; algebraicity can be tested using [14, Theorem 2.2].Source: §4.4 · p. 11
  • A state or parameter is observable exactly when it appears among the simplified generators returned by Algorithm 1.This follows from the simplification procedure’s property that a field containing a variable yields a simplified generator set containing that variable.Source: §4.4 · p. 11
  • Algorithm 1 is correct whether the parameter functions are observable or parameters are converted into state variables; both routes yield generators of the original observation field.In the modified model, the observation fields of the original and parameter-free models coincide.Source: §4.4 · p. 11
  • For the Lotka–Volterra variant, the input-output equation yields identifiable coefficients c1, . . . , c7, and the observation field is generated by those coefficients together with q1, . . . , q7.The example then applies algebraic simplification to obtain the algorithm’s output generators.Source: §4.4 · pp. 11–12

5 Implementation

This section evaluates the Julia implementation by comparing generator complexity and computation speed across algorithm variants. It reports that identifiable-function initialization and final simplification reduce complexity, while Step 1 typically speeds computation.

5 Implementation

  • The implementation is Monte Carlo probabilistic, but users can specify any positive upper bound on the probability of an incorrect result.Computations used StructuralIdentifiability.jl version 0.5.21.Source: §5 · p. 12
  • Table 1 compares highest generator degrees and term counts across prolongations-only, Step 1 plus prolongations without simplification, and the full algorithm.The prolongations-only variant uses Lie derivatives and Theorem 4.1; the full algorithm includes final simplification.Source: §5 · p. 12, Table 1 · p. 12
  • Identifiable-function initialization reduces Lie-derivative orders and expression complexity, while final simplification reduces complexity further and dramatically.The resulting generators are also often compact and admit domain-specific interpretations.Source: §5 · p. 12
  • Step 1 typically speeds computation and can turn intractable problems into tractable ones.Table 2 reports the speedup achieved by Step 1.Source: §5 · p. 12, Table 2 · p. 13

6 Examples

Across epidemiological, chemical-kinetics, and cancer examples, the algorithm returns observation-field generators with domain-specific interpretations, including combinations that remain observable when individual states or parameters are not.

Waterborne-disease model

  • For the waterborne-disease model, individual states and parameters are not observable, but the generators include force of infection, a transmission-population combination, and an identifiable effective reproduction number.The effective reproduction number has a product decomposition, and the generators can help determine complementary data sufficient to restore identifiability.Source: §6 · p. 13

Cancer modeling

  • In the cancer model, changing the second output from Q(t) to v(t) changes the observation-field generators, and both cases have observable functions that are products of integer powers of parameters and states.For outputs P(t) and Q(t), the generators include σx(t) and δσ; with v(t) as the second output, the listed remaining states and parameters include γ1, γ2, ε, d, and μm.Source: §6 · p. 14

Controlling enzyme kinetics

  • For competing substrates, generators include sums and products of substrate states and parameters; substrate-exchange symmetry explains these forms, while S1(t)S2(t) and V1S1(t) + V2S2(t) are also generated.The weighted sum uses the limiting rates, alongside the observed sum S1(t) + S2(t).Source: §6 · p. 14

Disease with relapse after remission

  • The relapse model’s generators include infectious individuals and their rate of change, while αS(t) and (α − σ)Q(t) represent tradeoffs involving population, latent period, and remission-stage counts.The rate of change of infectious individuals is interpreted as “new daily cases.”Source: §6 · p. 14

EAIHRD epidemiology model

  • With cumulative mortality observed, generators include mortality-scaled hospitalized and infected counts, force of infection, and parameter combinations agreeing with parameters introduced by hand in earlier models.The expressions dH(t) and dhI(t) scale hospitalized and infected individuals by mortality and hospitalization rates, respectively.Source: §6 · p. 15
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