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Algebraic identifiability of partial differential equation models

Helen Byrne, Heather Harrington, Alexey Ovchinnikov, Gleb Pogudin, Hamid Rahkooy, Pedro Soto

TL;DR

This work develops a differential‑algebraic framework for the global identifiability of polynomial PDEs by constructing input–output (IO) equations and analyzing their coefficients. It proves that, for generic PDE solutions, identifiable parameter functions are generated by IO‑equation coefficients, and that a nonsingular IO‑Wronskian suffices to identify those coefficients, enabling two Maple‑implemented algorithms for IO‑identifiability and strong identifiability. The authors extend the ODE theory to PDEs by handling initial/boundary conditions via differential ideals and provide practical demonstrations on parabolic, elliptic, and hyperbolic PDEs—ranging from Fisher’s equation to cancer‑invasion models—often with symbolic results supported by numeric verification when necessary. The results offer a systematic, algebraic route to determine when PDE parameters can be uniquely recovered from observed inputs and outputs, with broad implications for spatio‑temporal modeling in applied sciences.

Abstract

Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.

Algebraic identifiability of partial differential equation models

TL;DR

This work develops a differential‑algebraic framework for the global identifiability of polynomial PDEs by constructing input–output (IO) equations and analyzing their coefficients. It proves that, for generic PDE solutions, identifiable parameter functions are generated by IO‑equation coefficients, and that a nonsingular IO‑Wronskian suffices to identify those coefficients, enabling two Maple‑implemented algorithms for IO‑identifiability and strong identifiability. The authors extend the ODE theory to PDEs by handling initial/boundary conditions via differential ideals and provide practical demonstrations on parabolic, elliptic, and hyperbolic PDEs—ranging from Fisher’s equation to cancer‑invasion models—often with symbolic results supported by numeric verification when necessary. The results offer a systematic, algebraic route to determine when PDE parameters can be uniquely recovered from observed inputs and outputs, with broad implications for spatio‑temporal modeling in applied sciences.

Abstract

Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.
Paper Structure (19 sections, 7 theorems, 72 equations, 2 algorithms)

This paper contains 19 sections, 7 theorems, 72 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Differential equations to differential algebra
  3. Results
  4. Identifiability for generic solutions
  5. Identifiability for solutions with specified initial and boundary conditions
  6. Algorithms
  7. Examples of PDE identifiability
  8. PDEs arising in mathematical biology
  9. Case 1. $\partial_tc(x,t)=0$
  10. Case 2. $\partial_xc(x,t)=0$
  11. Case 1. $\partial_xn(x,t)=0$.
  12. Case 2. $\partial_tn(x,t)=0$.
  13. Case 1. $\partial_x u(x,t)=0$
  14. Case 2. $\partial_t u(x, t) = \partial_t v(x, t)=0$
  15. Case 2.1. $-2a+b=-a+b$
  16. ...and 4 more sections

Key Result

Theorem 1

(cf. Ovchinnikov-Identifiability2023) For a model $\Sigma$ of the form eq:sigma_def, the identifiable functions in $\mathbb{C} (\mathbf{k})$ form a subfield, and this subfield is generated by the coefficients of any set of IO-equations of the model.

Theorems & Definitions (33)

  • Definition 1: Ring of differential polynomials
  • Definition 2: Strong identifiability
  • Remark 1
  • Definition 3: Differential ideals
  • Definition 4: Generic solution
  • Definition 5: Identifiability
  • Definition 6: Differential rankings and characteristic sets
  • Definition 7: IO-equations
  • Theorem 1
  • Remark 2
  • ...and 23 more