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Structural identifiability of linear-in-parameter parabolic PDEs through auxiliary elliptic operators

Yurij Salmaniw, Alexander P Browning

TL;DR

This work develops a framework for structural identifiability of fully observed parabolic PDEs that are linear in parameters by recasting identifiability as a question about the existence and uniqueness of solutions to an auxiliary elliptic problem. The key tool is the Fredholm alternative, which constrains identifiability through the kernel of the elliptic operator $L[A]$ under various boundary conditions, yielding unconditional identifiability for many linear homogeneous cases and precise conditions for non-identifiability tied to initial data in $\ ext{ker}(L[A])$. The analysis extends to nonlinear reaction-diffusion equations that are linear in parameters (notably the logistic case) and to systems, where identifiability can be unconditional or compromised by spectral properties of the auxiliary problem. Overall, the approach connects PDE identifiability to spectral theory, enabling rigorous conclusions about when parameters can be uniquely recovered and clarifying how boundary and initial conditions influence identifiability in spatially extended models.

Abstract

Parameter identifiability is often requisite to the effective application of mathematical models in the interpretation of biological data, however theory applicable to the study of partial differential equations remains limited. We present a new approach to structural identifiability analysis of fully observed parabolic equations that are linear in their parameters. Our approach frames identifiability as an existence and uniqueness problem in a closely related elliptic equation and draws, for homogeneous equations, on the well-known Fredholm alternative to establish unconditional identifiability, and cases where specific choices of initial and boundary conditions lead to non-identifiability. While in some sense pathological, we demonstrate that this loss of structural identifiability has ramifications for practical identifiability; important particularly for spatial problems, where the initial condition is often limited by experimental constraints. For cases with nonlinear reaction terms, uniqueness of solutions to the auxiliary elliptic equation corresponds to identifiability, often leading to unconditional global identifiability under mild assumptions. We present analysis for a suite of simple scalar models with various boundary conditions that include linear (exponential) and nonlinear (logistic) source terms, and a special case of a two-species cell motility model. We conclude by discussing how this new perspective enables well-developed analysis tools to advance the developing theory underlying structural identifiability of partial differential equations.

Structural identifiability of linear-in-parameter parabolic PDEs through auxiliary elliptic operators

TL;DR

This work develops a framework for structural identifiability of fully observed parabolic PDEs that are linear in parameters by recasting identifiability as a question about the existence and uniqueness of solutions to an auxiliary elliptic problem. The key tool is the Fredholm alternative, which constrains identifiability through the kernel of the elliptic operator under various boundary conditions, yielding unconditional identifiability for many linear homogeneous cases and precise conditions for non-identifiability tied to initial data in . The analysis extends to nonlinear reaction-diffusion equations that are linear in parameters (notably the logistic case) and to systems, where identifiability can be unconditional or compromised by spectral properties of the auxiliary problem. Overall, the approach connects PDE identifiability to spectral theory, enabling rigorous conclusions about when parameters can be uniquely recovered and clarifying how boundary and initial conditions influence identifiability in spatially extended models.

Abstract

Parameter identifiability is often requisite to the effective application of mathematical models in the interpretation of biological data, however theory applicable to the study of partial differential equations remains limited. We present a new approach to structural identifiability analysis of fully observed parabolic equations that are linear in their parameters. Our approach frames identifiability as an existence and uniqueness problem in a closely related elliptic equation and draws, for homogeneous equations, on the well-known Fredholm alternative to establish unconditional identifiability, and cases where specific choices of initial and boundary conditions lead to non-identifiability. While in some sense pathological, we demonstrate that this loss of structural identifiability has ramifications for practical identifiability; important particularly for spatial problems, where the initial condition is often limited by experimental constraints. For cases with nonlinear reaction terms, uniqueness of solutions to the auxiliary elliptic equation corresponds to identifiability, often leading to unconditional global identifiability under mild assumptions. We present analysis for a suite of simple scalar models with various boundary conditions that include linear (exponential) and nonlinear (logistic) source terms, and a special case of a two-species cell motility model. We conclude by discussing how this new perspective enables well-developed analysis tools to advance the developing theory underlying structural identifiability of partial differential equations.

Paper Structure

This paper contains 18 sections, 6 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Identifiability of fully observed systems of linear ODEs
  3. Identifiability of linear parabolic PDE models
  4. The general framework
  5. Linear, homogeneous parabolic equations
  6. The Dirichlet boundary operator
  7. Case I: No drift.
  8. Case II: Constant drift.
  9. The Neumann, Robin & periodic boundary conditions
  10. The Neumann condition.
  11. The Robin condition.
  12. The periodic condition.
  13. Nonlinear reaction-diffusion equations and systems
  14. Nonlinear reaction-diffusion equations
  15. Systems of reaction-diffusion equations
  16. ...and 3 more sections

Key Result

Proposition 2.1

Fix $M_1 \in \mathbb{R}^{n \times n}$ and suppose $M \in \mathbb{R}^{n \times n}$ satisfies the following: Then, for any initial condition $X_0 \in \ker (M)$, $X(t)$ simultaneously solves eq:linear_homogeneous_ODE_proto with parameter point $M_1$ and where $M_2 := M_1 + M$. In particular, the matrix $M_2$ encodes possible parameter point combinations that are indistinguishable from the point $M_

Figures (4)

  • Figure 1: Structural non-identifiability in a (a) ordinary- and (b) partial differential equation. Shown are indistinguishable solutions from two distinct parameter sets. In (a) we show the first component of $X$ subject to $\dot X = M_1 X$ (red solid) and $\dot X = M_2 X$ (black dashed) where $M_1 = (1001)$, $M_2 = (2-110)$, and $X(0) = (1,1)^\intercal$. In (b), we show the solution a linear reaction-diffusion equation (given by \ref{['eq:linear_rd_sol']}) at $t \in \{0,0.4,0.8,1.2,1.6,2\}$ for $(c_1,d_1) = (1,0.05)$ (blue solid) and $(c_1,d_1) \approx (2,0.15)$ (black dashed). Full details are given in Sections \ref{['secODEs']} and \ref{['dirichlet']} for the ODE and PDE models, respectively.
  • Figure 2: A flow chart diagram outlining Steps 1-3 described in Section \ref{['sec:general_framework']}.
  • Figure 3: A depiction of the set $\mathcal{A}$ for the Dirichlet problem without drift (a) and with drift (b).
  • Figure 4: Practical identifiability of parameters in the linear reaction-diffusion equation subject to Dirichlet boundary conditions. (a--c) Initial conditions used to produce synthetic data. Initial conditions are based on a Gaussian centred at $x = 1/2$ with standard deviations $\omega = 0.1,0.2,0.3$, translated and normalized such that $u_0(1/2) = 1$ and that the boundary conditions are satisfied (black solid). To simplify the analysis, the initial conditions used are constructed from a truncated eigenfunction expansion of each Gaussian (red dashed). The leading order term is shown in blue. (d--f) Bivariate profile log-likelihoods for each initial condition, normalized to a maximum of zero. A 95% confidence region is calculated based on a normalized log-likelihood threshold of approximately 2.997, arising from the likelihood-ratio test. We also show the true value (orange diamond) and the theoretically indistinguishable set for an initial condition exactly matching the dominant eigenfunction (black dashed).

Theorems & Definitions (22)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Example 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • ...and 12 more