# Algebraic identifiability of partial differential equation models

Helen M. Byrne  Mathematical Institute, Radcliffe Observatory Quarter,
Woodstock Road, Oxford OX2 6GG, UK and
Ludwig Institute for Cancer Research, Oxford, Nuffield Department of Clinical
Medicine, Old Road Campus Research Building, off Roosevelt Drive, Headington,
Oxford, OX3 7DQ, UK<EMAIL_ADDRESS> ,  Heather A. Harrington
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford
OX2 6GG, UK and
Faculty of Mathematics, Technische Universitat Dresden, 01062 Dresden, Germany
and
Centre for Systems Biology Dresden (CSBD), Pfotenhauerstrasse 108, 01307
Dresden, Germany and
Max Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG),
Pfotenhauerstrasse 108, 01307 Dresden, Germany<EMAIL_ADDRESS> ,
Alexey Ovchinnikov  Department of Mathematics, CUNY Queens College and Ph.D.
Programs in Mathematics and Computer Science, CUNY Graduate Center, New York,
USA<EMAIL_ADDRESS> ,  Gleb Pogudin  LIX, CNRS, École
Polytechnique, Institute Polytechnique de Paris, 1 rue Honoré d’Estienne
d’Orves, 91120, Palaiseau, France<EMAIL_ADDRESS> ,  Hamid
Rahkooy  Mathematical Institute, Radcliffe Observatory Quarter, Woodstock
Road, Oxford OX2 6GG, UK<EMAIL_ADDRESS> and  Pedro Soto  Wellcome
Centre for Human Genetics, Roosevelt Drive, Oxford OX3 7BN UK
<EMAIL_ADDRESS>
###### 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.

Byrne and Harrington are members of the Centre for Topological Data Analysis,
which is funded by the EPSRC grant ‘New Approaches to Data Science:
Application Driven Topological Data Analysis’ EP/R018472/1. This work was also
partially supported by the NSF grants CCF-2212460, DMS-1760448, DMS-1853650,
and DMS-1853482 and PSC-CUNY grant #65605-00 53, a Royal Society University
Research Fellowship, Renewal and Enhancement grant. For the purpose of Open
Access, the authors have applied a CC BY public copyright licence to any
Author Accepted Manuscript (AAM) version arising from this submission.

Harrington is supported by a Royal Society University Research Fellowship, URF
Renewal, URF Enhancement grant and EPSRC EP/R018472/1.

The authors are grateful for the stimulating research environment at MPI-MiS
Leipzig. The work started at the MPI-MiS Workshop on Differential Algebra.

##  1\. Introduction

Differential equations form the bedrock of numerous scientific and engineering
models, particularly in the realms of biological and chemical interactions.
Indeed, systems of ordinary and partial differential equations are integral to
our understanding of such scientific phenomena. These models invariably
incorporate time-varying dependent variables, input functions, and output
functions, along with parameter vectors. Without loss of generality,
parameters are positive scalars, which are independent of time, and whose
values are often unknown. Parameter estimation or inference is determining the
unknown parameter values from observations. Identifiability is a necessary
condition for well-posed parameter estimation and inference.

There are different notions of parameter identifiability for differential
equation models [13]. A parameter is called _globally structurally
identifiable_ if its value can be uniquely determined from the input and
output functions. If a parameter takes finitely many values, it is called
locally identifiable. A parameter that is neither globally, nor locally
identifiable, is called unidentifiable. In this paper, we only consider global
identifiability and, for brevity, use the notation identifiable for global
identifiability. Given a model with observables, the _parameter
identifiability problem_ is determining whether the parameters are
identifiable.

For ordinary differential equation (ODE) models, the parameter identifiability
problem has been extensively studied. Various approaches have emerged in areas
ranging from control theory and dynamical systems (e.g., Taylor series
approximations) to computational algebraic geometry and differential algebra.
Starting from Ritt [26], several theoretical results and algorithms have been
developed within the last decades. The Rosenfeld-Gröbner algorithm and Gröbner
bases are the two crucial concepts that underpin existing identifiability
algorithms. Based on these, several software have been designed to test
identifiability, e.g., DAISY [3], SIAN [12], COMBOS [17], Structural
Identifiability Toolbox [15], and StructuralIdentifiability.jl [9]. An
increase in the availability of spatio-temporal data enables the investigation
of parameter values in PDE models.

A key aim of this work is to adapt and extend the successful algebraic
approaches for studying ODE identifiability to spatial systems. PDEs are
inherently more complex than ODEs. There are additional challenges associated
with the boundary conditions. With the exception of the work by [28, 25, 8,
7], there is no systematic study of PDE identifiability, to the best of our
knowledge. Authors in [28] investigate the identifiability and estimation of
parameters of a chikungunya epidemic transmission model. Structural
identifiability of age-structured PDE models using a differential algebra
framework has been studied In [25]. Structural and practical identifiability
of PDE models of fluorescence recovery after photobleaching has been studied
in [8]. In [7], the authors present a differential algebra approach to
structural identifiability analysis on partially observed linear reaction-
advection-diffusion PDE models.

We extend the identifiability problem to spatio-temporal models

(1)
$\Sigma=\begin{cases}\partial_{t}\mathbf{v}=\mathbf{f}\left(\mathbf{k},\mathbf{w},\mathbf{v},\partial_{x}\mathbf{v},\ldots,\partial_{x}^{h}\mathbf{v}\right)\\\
\mathbf{y}=\mathbf{g}\left(\mathbf{k},\mathbf{w},\mathbf{v},\partial_{x}\mathbf{v},\ldots,\partial_{x}^{h}\mathbf{v}\right),\end{cases}$

which have broad applications in applied mathematics. Here $\mathbf{w}(x,t)$,
$\mathbf{v}(x,t)$, $\mathbf{y}(x,t)$, and $\mathbf{k}$ are vectors of inputs,
state variables, outputs, and constant parameters, respectively and
$\mathbf{f}$ and $\mathbf{g}$ are vectors of rational functions. We present an
algebraic approach for the PDE identifiability problem, focusing on models
arising in applied mathematics. Our main results state that for a given model
of the form System (1), one can construct certain differential polynomials
equations called Input Output (IO) equations such that the identifiability of
the parameters can be obtained from the identifiability of the coefficients of
the IO-equations (Theorem 1), and that the coefficients of the IO-equations
are identifiable if their Wronskian is nonsingular (Proposition 1). Based on
our results, we present two algorithms (for IO-identifiability and strong
identifiability), with implementation in Maple for our illustrating examples,
that provide a sufficient condition for solving the PDE identifiability
problem. The main steps of our algorithms are the following:

  * •

calculating input-output (IO) equations of the PDE model (Step 1 in Algorithms
1 and 2),

  * •

checking if the parameters can be uniquely determined from the coefficients of
the IO-equations (Step 3 in Algorithm 1, Steps 2 & 3 in Algorithm 2), and

  * •

verifying that the coefficients of the IO-equations are themselves
identifiable (Step 3 in Algorithm 1, Step 4 in Algorithm 2).

In the ODE case, there is a key subtlety: even for generic initial conditions,
the coefficients of the IO-equations are not always identifiable, see e.g.
[13, Example 2.14]. Surprisingly, this subtlety does not occur in PDEs with
generic initial/boundary conditions, as we prove in Theorem 1 and exploit in
Algorithm 1.

Initial and boundary conditions in practical examples are not necessarily
generic (cf. the ODE case [27, 11]), which adds an additional layer of
difficulty. To account for these conditions, the last step of our method finds
potential linear dependencies between the monomials present in the input-
output equations corresponding to the PDE models. The linear dependencies are
then tested by computing the Wronskian of the monomials and using differential
algebra tools to determine if the Wronskian is non-singular. Using the
Rosenfeld-Gröbner algorithm [5], we compute the normal form of the determinant
of the Wronskian with respect to the differential ideal of the model. We then
use the initial and boundary conditions of the model in order to refute the
vanishing of the determinant (see Proposition 1).

We demonstrate these results in Algorithm 2, for testing the identifiability
of standard models arising in applied mathematics. We consider different types
of PDEs (parabolic, elliptic and hyperbolic), with a particular focus on
parabolic PDEs that arise in mathematical biology. We show that a scalar
reaction-diffusion equation, Fisher’s equation, the coupled reaction-diffusion
equations system, and a reaction-diffusion system [21, 10] are all
identifiable. We also demonstrate the wider applicability of this framework on
Laplace’s equation (elliptic) and the wave equation (hyperbolic). In Example
4, we show that following our symbolic-computation based algorithm directly
could be too demanding on the computational resources. We demonstrate how
numeric computation with random values of parameters gives evidence for
identifiability at a generic point in reasonable computing time.

The organization of the paper is as follows. Section 2 outlines the
preliminaries on differential algebra, detailing precise definitions and
required results. Section 3 presents our results, offering the theoretical
foundation for our PDE identifiability procedure using the Wronskian and
generalizing the literature on ODE identifiability. Section 4 provides our two
identifiability algorithms. In Section 5, we showcase Algorithm 2 on the above
suite of PDE models arising in mathematical biology.

##  2\. Differential equations to differential algebra

We start the preliminaries by recalling differential polynomials, which
provide a general framework for polynomial PDEs.

######  Definition 1 (Ring of differential polynomials).

  1.   2. (1)

A differential ring $(R,\Delta)$ is a commutative ring with a set
$\Delta=\\{\partial_{1},\ldots,\partial_{m}\\}$ of pairwise-commuting
derivations $\partial_{i}:R\to R$, that is, maps such that, for all $a,b\in
R$, $\partial_{i}(a+b)=\partial_{i}(a)+\partial_{i}(b)$ and
$\partial_{i}(ab)=\partial_{i}(a)b+a\partial_{i}(b)$.

  3. (2)

A differential ring that is a field is called a differential field.

  4. (3)

For a differential field $K$, the ring of differential polynomials in the
variables $x_{1},\ldots,x_{n}$ over a differential field $K$ is the polynomial
ring in infinitely many variables

$K[\partial_{1}^{n_{1}}\cdots\partial_{m}^{n_{m}}x_{j}\mid n_{i}\geqslant
0,\,1\leqslant j\leqslant n]$

with the derivations extended from $K$ by

$\partial_{i}\left(\partial_{1}^{n_{1}}\cdots\partial_{i}^{n_{i}}\cdots\partial_{m}^{n_{m}}(x_{j})\right):=\partial_{1}^{n_{1}}\cdots\partial_{i}^{n_{i}+1}\cdots\partial_{m}^{n_{m}}(x_{j}).$

This differential ring is denoted by $K\\{x_{1},\ldots,x_{n}\\}$.

######  Definition 2 (Strong identifiability).

Let $\mathbb{K}$ be one of the fields $\mathbb{R}$ and $\mathbb{C}$. We fix a
domain $\mathcal{D}$ in $\mathbb{K}^{m}$ on which a PDE system will be
defined. We will consider a PDE system in $n$ variables
$\mathbf{v}=(v_{1},\ldots,v_{n})$ and, for each $1\leqslant i\leqslant n$, we
fix a class $\mathcal{C}_{i}$ of function in $\mathcal{D}$ where the solutions
for $v_{i}$ will be sought. The requirements on functions from
$\mathcal{C}_{i}$ may involve, for example, regularity conditions (e.g., twice
or infinitely differentiable) or boundary conditions on $\partial\mathcal{D}$.

Now we consider a system of PDEs in $\mathcal{D}$ of the form

(2) $\mathbf{F}\left(\mathbf{k},\mathbf{v}\right)=0,$

where $\mathbf{F}=(F_{1},\ldots,F_{s})$ and
$F_{1},\ldots,F_{s}\in\mathbb{K}(\mathbf{k})\\{\mathbf{v}\\}$ are differential
polynomials in $\mathbf{v}$ with coefficients that are rational functions in
the scalar parameters $\mathbf{k}=(k_{1},\ldots,k_{\ell})$. Let us fix a
domain $\Omega\subset\mathbb{K}^{\ell}$, which will play a role of domain for
the parameters (in our examples, if $\Omega$ is not specified, we will assume
that $\Omega=\mathbb{K}^{\ell}$). Furthermore, we fix outputs
$\mathbf{y}=(y_{1},\ldots,y_{r})$ defined by given formulas

$y_{i}=\frac{G_{i}(\mathbf{k},\mathbf{v})}{Q(\mathbf{k},\mathbf{v})},\quad
1\leqslant i\leqslant r,$

where $G_{1},\ldots,G_{r},H\in\mathbb{K}(\mathbf{k})\\{\mathbf{v}\\}$. We will
say that a rational function $h(\mathbf{k})\in\mathbb{K}(\mathbf{k})$ is
_identifiable_ if, for all $\mathbf{k}_{1},\mathbf{k}_{2}\in\Omega$ and all
$\mathbf{v}_{1},\mathbf{v}_{2}\in\mathcal{C}_{1}\times\ldots\times\mathcal{C}_{n}$
such that
$\mathbf{F}(\mathbf{k}_{1},\mathbf{v}_{1})=\mathbf{F}(\mathbf{k}_{2},\mathbf{v}_{2})=0$,
we have

$\mathbf{y}(\mathbf{k}_{1},\mathbf{v}_{1})=\mathbf{y}(\mathbf{k}_{2},\mathbf{v}_{2})\implies
h(\mathbf{k}_{1})=h(\mathbf{k}_{2}).$

######  Remark 1.

While the definition above is very general and stated in natural analytic
terms, some of its properties make it challenging to use it in practice:
First, since it allows for arbitrary function classes and arbitrary polynomial
PDEs, a complete constructive approach to verifying this property seems to be
out of reach at the moment. Nevertheless, we will show that strong
identifiability can be established in a variety of practical cases by a
uniform approach (see Section 5).

Second, a system that is “almost always” identifiable can become
nonidentifiable according to this definition because of some special
degenerate cases. For example, a parameter $k$ in the ODE model
$x^{\prime}(t)=kx(t)$ with output $y(t)=x(t)$ will be considered
nonidentifiable because the zero solution $x(t)=0$ does not distinguish
between different parameter values. However, as long as $x(t)$ is nonzero, the
value of $k$ is uniquely determined. The standard mitigation of this issue in
the ODE case is to restrict the discussion to generic solutions. The problem
is that because of significantly more involved conditions on the existence and
uniqueness of solutions of PDEs, the topology (and thus the notion of
genericity) of the solution space of a PDE system may be much more involved
even if the function classes $\mathcal{C}_{1},\ldots,\mathcal{C}_{n}$ are
infinitely differentiable functions subject to some boundary conditions.

Therefore, to obtain more refined results in a more restricted context, we
will give another definition, which is a direct analogue of the notion of
algebraic identifiability in the ODE case [2, Section 2.2]. Note that, in the
ODE case, this definition is equivalent to the analytic one [13, Proposition
3.4], and we expect a similar equivalence result to hold in the PDE case
(perhaps in the class of power series or analytic solutions).

We will recall some relevant notions from differential algebra.

######  Definition 3 (Differential ideals).

An ideal $I$ of a differential ring $(R,\Delta)$ is called a differential
ideal if, for all $a\in I$ and $\partial\in\Delta$, $\partial(a)\in I$. For
$F\subset R$, the smallest differential ideal containing the set $F$ is
denoted by $[F]$.

For an ideal $I$ and set $S$ in a ring $R$, we denote $S^{\infty}$ to be the
multiplicatively closed subset of $R$ generated by $S$ and

$I\colon S^{\infty}=\\{r\in R\mid\exists a\in S^{\infty}\,\colon ar\in I\\}.$

The set $I:S^{\infty}$ is also an ideal in $R$. If $S=\\{a\\}$ for some $a\in
R$, we also denote $I:S^{\infty}$ by $I:a^{\infty}$.

As mentioned in the introduction, we will mostly focus on evolutionary PDEs.
More precisely, we will have one distinguished derivation $\partial_{t}$ with
respect to the time and one or several spatial derivations
$\partial_{1},\ldots,\partial_{m}$, and we will consider systems of the form

(3)
$\Sigma=\begin{cases}\partial_{t}\mathbf{v}=\frac{\mathbf{f}(\mathbf{k},\mathbf{v},\mathbf{w})}{Q(\mathbf{k},\mathbf{v},\mathbf{w})}\\\
\mathbf{y}=\frac{\mathbf{g}(\mathbf{k},\mathbf{v},\mathbf{w})}{Q(\mathbf{k},\mathbf{v},\mathbf{w})},\end{cases}$

where $\mathbf{v}$ and $\mathbf{w}$ are the state and input variables of the
model, $\mathbf{y}$ are outputs, $\mathbf{k}$ are scalar parameters, and
$\mathbf{f},\mathbf{g},Q$ are differential polynomials in
$\mathbf{v},\mathbf{w}$ with coefficients in $\mathbb{C}(\mathbf{k})$ of order
zero with respect to $\partial_{t}$. Note that we have focused on complex
numbers to unlock powerful algebraic tools we will use, and identifiability
over complex numbers implies identifiability over reals. Now we will give a
formal definition of what generic solution of (3) is and then give our second
definition of identifiability.

######  Definition 4 (Generic solution).

Given $\Sigma$ as in (3), we define the differential ideal of $\Sigma$ as

$I_{\Sigma}=[Q\partial_{t}\mathbf{v}-\mathbf{f},Q\mathbf{y}-\mathbf{g}]:Q^{\infty}\subset\mathbb{C}(\mathbf{k})\\{\mathbf{v},\mathbf{y},\mathbf{w}\\}.$

In the same way as [13, Lemma 3.1 and 3.2], one can show that $I_{\Sigma}$ is
a prime ideal and

(4)
$I_{\Sigma}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{w}\\}\left[\partial_{1}^{i_{1}}\ldots\partial_{m}^{i_{m}}\mathbf{v},\
i_{1},\ldots,i_{m}\geqslant 0\right]=\\{0\\}.$

Since $I_{\Sigma}$ is prime, we can consider the field of fractions of
$R/I_{\Sigma}$, which we denote by $\mathcal{F}$. We will denote the images of
$\mathbf{v},\mathbf{y},\mathbf{w}$ in $\mathcal{F}$ by
$\hat{\mathbf{v}},\hat{\mathbf{y}},\hat{\mathbf{w}}$, respectively. We will
call them the generic solutions of (3).

######  Definition 5 (Identifiability).

In the notation of the previous definition, we say that a parameter
$k_{i}\in\mathbf{k}$ (or, more generally, a rational function of parameters)
is identifiable if

$k_{i}\in\mathbb{C}\langle\hat{\mathbf{y}},\hat{\mathbf{w}}\rangle,$

where $\mathbb{C}\langle\hat{\mathbf{y}},\hat{\mathbf{w}}\rangle$ is the
smallest field extension of $\mathbb{C}$ containing
$\hat{\mathbf{y}},\hat{\mathbf{w}}$ and their derivatives.

##  3\. Results

###  3.1. Identifiability for generic solutions

We will prove that, unlike the ODE case, all of the identifiable functions of
parameters can be read off from particular relations between the input and
output variables called input-output equations. Below we recall some necessary
notions and constructions from constructive differential algebra.

######  Definition 6 (Differential rankings and characteristic sets).

  1.   2. (1)

A differential ranking on $K\\{x_{1},\ldots,x_{n}\\}$ is a total order $>$ on

$X:=\\{\partial_{1}^{n_{1}}\cdots\partial_{n}^{n_{m}}(x_{j})\mid
n_{i}\geqslant 0,\,1\leqslant j\leqslant n\\}$

satisfying:

     * •

for all $x\in X$ and $\partial\in\Delta$, $\partial(x)>x$ and

     * •

for all $x,y\in X$ and $\partial\in\Delta$, if $x>y$, then
$\partial(x)>\partial(y)$.

It can be shown that a differential ranking on $K\\{x_{1},\ldots,x_{n}\\}$ is
always well-ordered.

  3. (2)

For $f\in K\\{x_{1},\ldots,x_{n}\\}\backslash K$ and differential ranking $>$,

     * •

$\operatorname{\text{lead}}(f)$ is the element of $X$ appearing in $f$ that is
maximal with respect to $>$.

     * •

The _leading coefficient_ of $f$ considered as a polynomial in
$\operatorname{\text{lead}}(f)$ is denoted by $\operatorname{in}(f)$ and
called the _initial_ of $f$.

     * •

The _separant_ of $f$ is $\frac{\partial
f}{\partial\operatorname{\text{lead}}(f)}$, the partial derivative of $f$ with
respect to $\operatorname{\text{lead}}(f)$.

     * •

The _rank_ of $f$ is
$\rank(f)=\operatorname{\text{lead}}(f)^{\deg_{\operatorname{\text{lead}}(f)}f}$.

     * •

For $S\subset K\\{x_{1},\ldots,x_{n}\\}\backslash K$, the set of _initials_
and _separants_ of $S$ is denoted by $H_{S}$.

     * •

For $g\in K\\{x_{1},\ldots,x_{n}\\}\backslash K$, say that $f<g$ if
$\operatorname{\text{lead}}(f)<\operatorname{\text{lead}}(g)$ or
$\operatorname{\text{lead}}(f)=\operatorname{\text{lead}}(g)$ and
$\deg_{\operatorname{\text{lead}}(f)}f<\deg_{\operatorname{\text{lead}}(g)}g$.

  4. (3)

For $f,g\in K\\{x_{1},\ldots,x_{n}\\}\backslash K$, $f$ is said to be reduced
w.r.t. $g$ if no proper derivative of $\operatorname{\text{lead}}(g)$ appears
in $f$ and
$\deg_{\operatorname{\text{lead}}(g)}f<\deg_{\operatorname{\text{lead}}(g)}g$.

  5. (4)

A subset $\mathcal{A}\subset K\\{x_{1},\ldots,x_{n}\\}\backslash K$ is called
autoreduced if, for all $p\in\mathcal{A}$, $p$ is reduced w.r.t. every element
of $\mathcal{A}\setminus\\{p\\}$. One can show that every autoreduced set is
always finite.

  6. (5)

Let $\mathcal{A}=\\{A_{1},\ldots,A_{r}\\}$ and
$\mathcal{B}=\\{B_{1},\ldots,B_{s}\\}$ be autoreduced sets such that
$A_{1}<\ldots<A_{r}$ and $B_{1}<\ldots<B_{s}$. We say that
$\mathcal{A}<\mathcal{B}$ if

     * •

$r>s$ and $\rank(A_{i})=\rank(B_{i})$, $1\leqslant i\leqslant s$, or

     * •

there exists $q$ such that $\rank(A_{q})<\rank(B_{q})$ and, for all $i$,
$1\leqslant i<q$, $\rank(A_{i})=\rank(B_{i})$.

  7. (6)

An autoreduced subset of the smallest rank of a differential ideal $I\subset
K\\{x_{1},\ldots,x_{n}\\}$ is called a characteristic set of $I$. One can show
that every non-zero differential ideal in $K\\{x_{1},\ldots,x_{n}\\}$ has a
characteristic set. Note that a characteristic set does not necessarily
generate the ideal.

  8. (7)

For elements $r_{1},\ldots,r_{n}$ is a differential ring $(R,\Delta)$, the
Wronskian matrix $\operatorname{Wr}(r_{1},\ldots,r_{n})$ with respect to a
given $\partial\in\Delta$ is the matrix

$\begin{pmatrix}r_{1}&\ldots&r_{n}\\\ \partial r_{1}&\ldots&\partial r_{n}\\\
\vdots&\ddots&\vdots\\\
\partial^{n-1}r_{1}&\ldots&\partial^{n-1}r_{n}\end{pmatrix}.$

######  Definition 7 (IO-equations).

Given a differential ranking on the differential variables $\mathbf{y}$ and
$\mathbf{w}$, the _IO-equations_ are defined as the monic characteristic
presentation of the prime differential ideal
$I_{\Sigma}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$ with respect
to this ranking (see [23, Definition 6 and Section 5.2] for more details). For
each differential ranking, such a monic characteristic presentation is unique
[4, Theorem 3].

Now we are ready to state the main result of this section, namely, that all
the identifiable functions can be read off the coefficients of the IO-
equations. Interestingly, the corresponding statement is not true for ODEs
[23, Example 2].

######  Theorem 1.

_(cf.[23, Theorems 1 & 2])_ For a model $\Sigma$ of the form (3), 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.

######  Remark 2.

Note that using Theorem 1, one obtains identifiability of the coefficients $C$
of the IO-equations $F_{j}(\mathbf{w},\mathbf{y})$, $1\leqslant j\leqslant s$,
considered as differential polynomials in $\mathbf{w},\mathbf{y}$. Elements of
$C$ are in fact the generators of the field of identifiable functions. For
each parameter $k_{i}$ in $\mathbf{k}$, one can check its identifiability by
verifying whether $k_{i}\in\mathbb{Q}(C)$. This can be reduced to the ideal
membership problem as described in [20, Section 1.3] (for a randomized
version, see [9, Theorem 3.3]). The Maple code [1] has been implemented in
[24] for testing this ideal membership.

Before proving Theorem 1, we will proceed with some preparation. Consider a
PDE model $\Sigma$ as in (1), its prime differential ideal $I_{\Sigma}$, and
the corresponding field of fractions $\mathcal{F}$ (see Definition 4). Since
$I_{\Sigma}$ is stable under $\partial_{t}$ and $\partial_{x}$, the
derivations $\partial_{t}$ and $\partial_{x}$ can be transferred to
$\mathcal{F}$ in a natural way.

######  Definition 8.

We will call an element $c\in\mathcal{F}$ a _constant_ if
$\partial_{t}c=\partial_{x}c=0$.

The key difference with the ODE case will be the following lemma.

######  Lemma 1.

If $c\in\mathcal{F}$ and $\partial_{x}(c)=0$, then
$c\in\mathbb{C}(\mathbf{k})$. In particular, the set of constants of
$\mathcal{F}$ is $\mathbb{C}(\mathbf{k})$.

###### Proof.

Let $c=\frac{p}{q}\in\mathcal{F}$ be such $\partial_{x}(c)=0$, $p,q\in
R=\mathbb{C}(\mathbf{k})\\{\mathbf{v},\mathbf{y},\mathbf{w}\\}$ are coprime
and at least one of them does not belong to $\mathbb{C}(\mathbf{k})$. There
exists a positive integer $N$ such that, using equations (1) and their
derivatives, we can replace $Q^{N}p$ and $Q^{N}q$ by elements of
$\mathbb{C}(\mathbf{k})\\{\mathbf{w}\\}[\partial_{x}^{i}\mathbf{v},\
i\geqslant 0]$ equivalent to $Q^{N}p$ and $Q^{N}q$ modulo $I_{\Sigma}$ and,
thus, yielding the same element of $\mathcal{F}$. By (4), $\mathcal{F}$ has a
subfield isomorphic to the field of rational functions

$\mathcal{F}_{0}:=\mathbb{C}(\mathbf{k})\langle\mathbf{w}\rangle(\partial_{x}^{i}\mathbf{v},\
i\geqslant 0),$

and $\frac{p}{q}$ belongs to this subfield. With respect to $\partial_{x}$,
the field $\mathcal{F}_{0}$ is isomorphic to the field of differential
rational functions over $\mathbf{k}$ in infinitely many variables
$\mathbf{v},\mathbf{w},\partial_{t}\mathbf{w},\partial_{t}^{2}\mathbf{w},\ldots$.
Since the constants of a field of differential rational functions are exactly
the constants of the ground field, we deduce that
$\frac{p}{q}\in\mathbb{C}(\mathbf{k})$. ∎

######  Lemma 2.

Let $\Delta$ be a finite set of derivations, $L\subset K$ differential fields,
and $X$ a finite set of variables. Let $P$ be a prime non-zero differential
ideal of $K\\{X\\}$ such that the ideal generated by $P$ in
$\overline{K}\\{X\\}$ is prime, where $\overline{K}$ is the algebraic closure
of $K$. If $\mathcal{C}$ is a monic characteristic presentation of $P$, then
the field of definition of $P$ over $L$ is the field extension of $L$
generated by the coefficients of $\mathcal{C}$.

###### Proof.

Mutatis mutandis proof of [23, Proposition 2]. ∎

######  Corollary 1 (cf. [23, Corollary 1]).

If $\mathcal{C}$ is a monic characteristic presentation of the prime
differential ideal
$J:=I_{\Sigma}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$, then
field of definition of $J$ over $\mathbb{C}$ is generated over $\mathbb{C}$ by
the coefficients of $\mathcal{C}$.

###### Proof.

This follows from Lemma 2 because the differential ideal generated by $J$ in
$\overline{\mathbb{C}(\mathbf{k})}\\{\mathbf{y},\mathbf{w}\\}$ is prime. ∎

######  Lemma 3 (cf. [23, Lemma 1]).

Consider a polynomial $P\in
I_{\Sigma}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$ with at least
one of the coefficients being one. If there is no element in
$I_{\Sigma}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$ whose
support is a subset of the support of $P$, then all coefficients of $P$ are
identifiable.

###### Proof.

We write $P=m_{0}+\sum_{i=1}^{\ell}c_{i}m_{i}$, where $m_{0},\ldots,m_{\ell}$
are monomials in

$\left\\{\partial_{t}^{i}\partial_{x}^{j}\mathbf{y},\partial_{t}^{i}\partial_{x}^{j}\mathbf{w}\mid
i,j\geqslant 0\right\\}$

and $c_{1},\ldots,c_{\ell}\in\mathbb{C}(\mathbf{k})$. Let
$(\hat{\mathbf{v}},\hat{\mathbf{y}},\hat{\mathbf{w}})$ be the generic solution
of $I_{\Sigma}$. We consider the Wronskian

$W(\hat{\mathbf{y}},\hat{\mathbf{w}}):=\operatorname{Wr}(m_{1}(\hat{\mathbf{y}},\hat{\mathbf{w}}),\ldots,m_{\ell}(\hat{\mathbf{y}},\hat{\mathbf{w}}))$

with respect to $\partial_{x}$. If this Wronskian was singular, then, by [16,
Theorem 3.7], there would exist $a_{1},\ldots,a_{\ell}\in F$ constant with
respect to $\partial_{x}$ so that

$a_{1}m_{1}(\hat{\mathbf{y}},\hat{\mathbf{w}})+\ldots+a_{\ell}m_{\ell}(\hat{\mathbf{y}},\hat{\mathbf{w}})=0.$

By Lemma 1, $a_{1},\ldots,a_{\ell}\in\mathbb{C}(\mathbf{k})$. This yields a
nonzero element

$a_{1}m_{1}+\ldots+a_{\ell}m_{\ell}\in I_{\Sigma}$

with the support being a subset of the support of $P$. Thus, $W$ is
nonsingular. By taking the derivatives of
$P(\hat{\mathbf{y}},\hat{\mathbf{w}})=0$ of orders from $0$ to $\ell-1$ with
respect to $\partial_{x}$, we obtain:

$\left(\begin{array}[]{c}m_{0}(\hat{\mathbf{y}},\hat{\mathbf{w}})\\\
\partial_{x}m_{0}(\hat{\mathbf{y}},\hat{\mathbf{w}})\\\ \vdots\\\
\partial_{x}^{\ell-1}m_{0}(\hat{\mathbf{y}},\hat{\mathbf{w}})\end{array}\right)=W(\hat{\mathbf{y}},\hat{\mathbf{w}})\left(\begin{array}[]{c}c_{1}\\\
c_{2}\\\ \vdots\\\ c_{\ell}\end{array}\right).$

Viewing this as a nonsingular linear system in $c_{1},\ldots,c_{\ell}$ over
$\mathbb{C}\langle\hat{\mathbf{y}},\hat{\mathbf{w}}\rangle$ and applying
Cramer’s rule, we deduce that each of $c_{1},\ldots,c_{\ell}$ belongs to
$\mathbb{C}\langle\hat{\mathbf{y}},\hat{\mathbf{w}}\rangle$. ∎

######  Corollary 2.

If system (1) is of the form

$\Sigma=\begin{cases}\partial_{t}\mathbf{v}=\mathbf{f}\left(\mathbf{k},\mathbf{w},\mathbf{v},\partial_{x}\mathbf{v},\ldots,\partial_{x}^{h}\mathbf{v}\right)\\\
\mathbf{y}=\mathbf{v},\end{cases}$

that is, all states are observable, then all coefficients of $\mathbf{f}$ as
polynomials in
$\mathbf{w},\mathbf{v},\partial_{x}\mathbf{v},\ldots,\partial_{x}^{h}\mathbf{v}$
are identifiable.

###### Proof.

It is sufficient to prove that polynomials $\partial_{t}\mathbf{v}-\mathbf{f}$
satisfy the condition of Lemma 3. Assume that one of them, say
$\partial_{t}v_{1}-f_{1}$ does not. Therefore, there is a polynomial $g\in
I_{\Sigma}$ with the support being a proper subset of the support of
$\partial_{t}v_{1}-f_{1}$. By cancelling $\partial_{t}v_{1}$, we obtain a
$\mathbb{C}(\mathbf{k})$-linear combination $\tilde{g}$ of
$\partial_{t}v_{1}-f_{1}$ and $g$ with the support being a subset of the
support of $f_{1}$. Then $\tilde{g}$ is a nonzero element of
$I\cap\mathbb{C}(\mathbf{k})\\{\mathbf{w}\\}[\partial_{x}^{i}\mathbf{v},\
i\geqslant 0]$, contradicting (4). ∎

We are now ready to prove our main result, Theorem 1.

###### Proof of Theorem 1.

Let

$S=\\{c\in\mathbb{C}(\mathbf{k})\mid c\text{ is
identifiable}\\}=\mathbb{C}(\mathbf{k})\cap\mathbb{C}\langle\hat{\mathbf{y}},\hat{\mathbf{w}}\rangle,$

and so $S$ is a subfield in $\mathbb{C}(\mathbf{k})$. Let $\mathcal{C}$ be a
set of IO-equations of $\Sigma$ and $C$ the set of coefficients of
$\mathcal{C}$. We will prove that $S=\mathbb{C}(C)$.

We first show that $S\subset\mathbb{C}(C)$. For this, let
$c\in\mathbb{C}(\mathbf{k})$ be identifiable. We will show how the proof of
[23, Theorem 1] extends to PDEs to prove that $c\in\mathbb{C}(C)$. By
Corollary 1, the field of definition of
$I_{\Sigma}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$ is equal to
$\mathbb{C}(C)$. In particular,
$I_{\Sigma}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$ is generated
over $\mathbb{C}(\mathbf{k})$ by
$I_{\Sigma}\cap\mathbb{C}(C)\\{\mathbf{y},\mathbf{w}\\}$.

Since $c\in\mathbb{C}\langle\hat{\mathbf{y}},\hat{\mathbf{w}}\rangle$, there
exist $g\in\mathbb{C}\\{\mathbf{y},\mathbf{w}\\}\setminus I_{\Sigma}$ and
$h\in\mathbb{C}\\{\mathbf{y},\mathbf{w}\\}$ such that $gc+h\in I_{\Sigma}$.
Therefore, there exist
$m_{1},\ldots,m_{r}\in\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$ and
$p_{1},\ldots,p_{r}\in I_{\Sigma}\cap\mathbb{C}(C)\\{\mathbf{y},\mathbf{w}\\}$
such that

$gc+h=m_{1}p_{1}+\ldots+m_{r}p_{r}.$

Now assume that $c\not\in\mathbb{C}(C)$. By [19, Theorem 9.29, p. 117], there
exists an automorphism $\sigma$ on $\overline{\mathbb{C}(\mathbf{k})}$ that
fixes $\mathbb{C}(C)$ pointwise, but does not fix $c$, i.e., $\sigma(c)\neq
c$. Extend $\sigma$ to
$\overline{\mathbb{C}(\mathbf{k})}\\{\mathbf{v},\mathbf{y},\mathbf{w}\\}$ by
letting $\sigma$ fix $\mathbf{v},\mathbf{y}$, and $\mathbf{w}$. We have in
$\overline{\mathbb{C}(\mathbf{k})}\\{\mathbf{v},\mathbf{y},\mathbf{w}\\}$ that

(5) $\displaystyle\left(gc+h\right)-\sigma(gc+h)$
$\displaystyle=\left(m_{1}p_{1}+\ldots+m_{r}p_{r}\right)-\sigma(m_{1}p_{1}+\ldots+m_{r}p_{r})$
$\displaystyle g(c-\sigma(c))$
$\displaystyle=(m_{1}-\sigma(m_{1}))p_{1}+\ldots+(m_{r}-\sigma(m_{r}))p_{r}.$

Let $\overline{I_{\Sigma}}$ denote the differential ideal generated by
$I_{\Sigma}$ in
$\overline{\mathbb{C}(\mathbf{k})}\\{\mathbf{v},\mathbf{y},\mathbf{w}\\}$.
Since $\overline{I_{\Sigma}}$ is a prime differential ideal and the right-hand
side of (5) belongs to $\overline{I_{\Sigma}}$, it follows that either
$g\in\overline{I_{\Sigma}}$ or $c-\sigma(c)\in\overline{I_{\Sigma}}$. As
$\sigma(c)\neq c$, we have that $c-\sigma(c)$ is a non-zero element of
$\overline{\mathbb{C}(\mathbf{k})}$. Since $\overline{I_{\Sigma}}$ is a proper
ideal, therefore $c-\sigma(c)\not\in\overline{I_{\Sigma}}$. Therefore,
$g\in\overline{I_{\Sigma}}$. Hence,

$g\in\overline{I_{\Sigma}}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{v},\mathbf{y},\mathbf{w}\\}=I_{\Sigma},$

contradicting the assumption on $g\not\in I_{\Sigma}$.

For the proof of the converse, note that we have shown in Lemma 1 that the set
of constants of $\mathcal{F}$ coincides with $\mathbb{C}(\mathbf{k})$.
Therefore, the assumption of [23, Theorem 2] holds. So we follow the proof of
[23, Theorem 2] for PDEs.

Let $J:=I_{\Sigma}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$. Let
$\mathcal{B}$ be the set of differential monomials in $\mathbf{y}$ and
$\mathbf{w}$ indexed by $\mathbb{N}$ such that the indexing respects a
ranking. Then $\mathcal{B}$ is a basis for
$\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$ as a vector space over
$\mathbb{C}(\mathbf{k})$. Consider a basis $\mathcal{B}_{J}$ for $J$, as a
$\mathbb{C}(\mathbf{k})$-subspace of
$\mathbb{C}(\mathbf{k})\\{\mathbf{y},\mathbf{w}\\}$. Each element of
$\mathcal{B}_{J}$ is a finite linear combination of the element of
$\mathcal{B}$. Autoreduce $\mathcal{B}_{J}$ as a set of vectors expressed in
the basis $\mathcal{B}$ and call it $\mathcal{B}_{J}^{\prime}$. Note that
$\mathcal{B}_{J}^{\prime}$ is still a basis of $J$ as a vector space. The
field of definition of $J$ over $\mathbb{C}$ is contained in the field
generated by the coefficient of $\mathcal{B}_{J}^{\prime}$ written as linear
combinations of $\mathcal{B}$. Therefore, it is sufficient to prove that these
coefficients are identifiable.

Let $P\in J$ be the differential polynomial corresponding to a member of
$\mathcal{B}_{J}^{\prime}$ of $J$. Let $q$ be a polynomial whose support is a
subset of $P$ and whose monomials are linearly dependent modulo $J$ over
$\mathbb{C}(\mathbf{k})$. The representation of $q$ in basis $\mathcal{B}$ can
be reduced to zero by $\mathcal{B}_{J}$, however, $q$ cannot be reduced to
zero by $P$. Also, $q$ reduced by $P$ cannot be further reduced by other
elements of $\mathcal{B}_{J}^{\prime}$, as $\mathcal{B}_{J}^{\prime}$ is a
reduced basis. This proves that there is no $q$ whose support is a subset of
the support of $P$ and whose support is linearly dependent. Therefore, by
Lemma 3, the coefficients of $P$ are identifiable. ∎

###  3.2. Identifiability for solutions with specified initial and boundary
conditions

In this section, we are in the setup of Definition 2. Consider a PDE system of
the form (2) and each $\mathcal{C}_{i}$ contained in the
$C^{\infty}$-functions on a domain $\mathcal{D}$. Fix a differential ranking
such that any derivative of $\mathbf{v}$ is greater than any derivative of
$\mathbf{y}$, and let

(6)
$I_{\Sigma}=[C_{1}]:H_{C_{1}}^{\infty}\cap\ldots\cap[C_{r}]:H_{C_{r}}^{\infty}$

be a decomposition computed by [4, Section 4] or [14, Algorithm 7.2] with
respect to the differential ranking. For each $i$, $1\leqslant i\leqslant r$,
let

$\widetilde{C_{i}}=C_{i}\cap\mathbb{C}(\mathbf{k})\\{\mathbf{y}\\}.$

If the system $\Sigma$ is as in (1), then, as we discussed, $I_{\Sigma}$ is a
prime differential ideal and, as a result, one can remove redundant components
in (6) and obtain $r=1$ and $\widetilde{C_{1}}\subset C_{1}\subset
I_{\Sigma}$, see [6, Theorem 3.2.1]. If $\Sigma$ is more general and
$I_{\Sigma}$ is not necessarily a prime differential ideal, consider the set
$\widetilde{C}=\prod_{i=1}^{r}\widetilde{C_{i}}\subset I_{\Sigma}$. By
dividing every element of $\widetilde{C}$ by an element of
$\mathbb{C}(\mathbf{k})$, for every $c\in\widetilde{C}$, we pick a
representation of the form

(7) $c=p_{0,c}+\sum_{j=1}^{q_{c}}a_{j,c}\cdot p_{j,c},$

where $p_{0},\ldots,p_{q_{c}}\in\mathbb{C}\\{\mathbf{y}\\}$ and
$a_{1}\ldots,a_{q_{c}}\in\mathbb{C}(\mathbf{k})$. Denote
$\operatorname{Wr}_{c}=\operatorname{Wr}(p_{1,c},\ldots,p_{q_{c},c})$.

######  Remark 3.

Such a grouping (7) is implemented in Maple in [1], the function
DecomposePolynomial.

######  Proposition 1.

Let $c\in\widetilde{C}$, defined above. If, for all
$\widehat{\mathbf{k}}\in\Omega$ and solutions
$(\widehat{\mathbf{v}},\widehat{\mathbf{y}})$ of (2) with the parameter values
$\widehat{\mathbf{k}}$, there exists a point $(t_{0},x_{0})\in\mathcal{D}$
such that the matrix
$\operatorname{Wr}_{c}\left(\widehat{\mathbf{y}}\right)(t_{0},x_{0})$ with
respect to $\partial_{x}$ or $\partial_{t}$ is invertible, then

  * •

all $a_{1,c},\ldots,a_{q_{c},c}$ are identifiable.

  * •

Moreover, let $k_{i_{1}},\ldots,k_{i_{c}}$ be the parameters that explicitly
appear in $a_{1,c},\ldots,a_{q_{c},c}$. If the “coefficient map”

$\varphi_{c}:(k_{i_{1}},\ldots,k_{i_{c}})\mapsto(a_{1,c},\ldots,a_{q_{c},c})$

is injective, then the parameters $k_{i_{1}},\ldots,k_{i_{c}}$ are
identifiable.

###### Proof.

Suppose that there exists $j$ such that $a_{j,c}$ is not identifiable
according to Definition 2. Let
$\widehat{\mathbf{k}}_{1},\widehat{\mathbf{k}}_{2}\in\Omega$ and solutions
$(\widehat{\mathbf{v}}_{1},\widehat{\mathbf{y}}_{1})$ and
$(\widehat{\mathbf{v}}_{2},\widehat{\mathbf{y}}_{2})$ of (2) be such that

(8) $\widehat{\mathbf{y}}_{1}=\widehat{\mathbf{y}}_{2}\quad\text{but}\quad
a_{j,c}(\widehat{\mathbf{k}}_{1})\neq a_{j,c}(\widehat{\mathbf{k}}_{2}).$

Consider the square system of linear equations

(9) $\begin{cases}\sum\limits_{j=1}^{q_{c}}a_{j,c}\cdot p_{j,c}=-p_{0,c}\\\
\sum\limits_{j=1}^{q_{c}}a_{j,c}\cdot\partial(p_{j,c})=-\partial(p_{0,c})\\\
\qquad\vdots\\\
\sum\limits_{j=1}^{q_{c}}a_{j,c}\cdot\partial^{q_{c}-1}(p_{j,c})=-\partial^{q_{c}-1}(p_{0,c})\end{cases}$

in the unknowns $a_{1,c},\ldots,a_{q_{c},c}$, whose matrix is
$\operatorname{Wr}_{c}(\mathbf{y})$. Assume that, for all
$\widetilde{\mathbf{k}}\in\Omega$ and solutions
$(\widetilde{\mathbf{v}},\widetilde{\mathbf{y}})$ of (2) with parameters
$\widetilde{\mathbf{k}}$, there exists $(t_{0},x_{0})\in\mathcal{D}$ such that
the matrix
$\operatorname{Wr}_{c}\left(\widetilde{\mathbf{y}}\right)(t_{0},x_{0})$ with
respect to $\partial_{x}$ ($\partial_{t}$ is considered similarly) is
invertible. Substituting $\widehat{\mathbf{y}}_{1}$ or
$\widehat{\mathbf{y}}_{2}$ and the corresponding point $(t_{0},x_{0})$ into
(9) and solving returns
$\left(a_{1,c}(\widehat{\mathbf{k}}_{1}),\ldots,a_{q_{c},c}(\widehat{\mathbf{k}}_{1})\right)$
or
$\left(a_{1,c}(\widehat{\mathbf{k}}_{2}),\ldots,a_{q_{c},c}(\widehat{\mathbf{k}}_{2})\right)$,
respectively. By the first part of (8), these tuples are equal, which
contradicts with the second part of (8).

For the proof of the second item, suppose that the coefficient map
$\varphi_{c}$ is injective. If $k_{i_{j}}$ is not identifiable, then there
exist $\widehat{\mathbf{k}}_{1},\widehat{\mathbf{k}}_{2}\in\Omega$ and
solutions $(\widehat{\mathbf{v}}_{1},\widehat{\mathbf{y}}_{1})$ and
$(\widehat{\mathbf{v}}_{2},\widehat{\mathbf{y}}_{2})$ of (2) with parameter
values $\widehat{\mathbf{k}}_{1}$ and $\widehat{\mathbf{k}}_{2}$ such that

$\widehat{\mathbf{y}}_{1}=\widehat{\mathbf{y}}_{2}\quad\text{but}\quad\operatorname{proj}_{k_{i_{j}}}(\widehat{\mathbf{k}}_{1})\neq\operatorname{proj}_{k_{i_{j}}}(\widehat{\mathbf{k}}_{2}).$

Hence, by the injectivity of $\varphi_{c}$, there exists $\ell$ such that
$a_{\ell,c}(\widehat{\mathbf{k}}_{1})\neq
a_{\ell,c}(\widehat{\mathbf{k}}_{2})$, which contradicts the identifiability
of $a_{\ell,c}$ established above. ∎

##  4\. Algorithms

The correctness of Algorithms 1 and 2, which we present in this section,
follows from Theorem 1 and Proposition 1, respectively.

Algorithm 1 Assessing identifiability in the sense of Defintion 5

Input:

A rational PDE system of the form

$\begin{cases}\partial_{t}{\mathbf{v}}=\mathbf{f}\left(\mathbf{k},\mathbf{w},\mathbf{v},\partial_{x}{\mathbf{v}},\ldots,\partial_{x}^{h}\mathbf{v}\right)\\\
\mathbf{y}=\mathbf{g}\left(\mathbf{k},\mathbf{w},\mathbf{v},\partial_{x}{\mathbf{v}},\ldots,\partial_{x}^{h}\mathbf{v}\right).\end{cases}$

Output:

A list of generators of the field of identifiable functions of the system and,
for each parameter, $k_{i}$ whether it is identifiable or not.

  1. (Step 1)

Using RosenfeldGroebner with an elimination ranking
$\mathbf{v}>\mathbf{y},\mathbf{w}$, eliminate the variables $\mathbf{v}$ and
obtain a set $\mathcal{S}$ of IO-equations of the input PDE system.

  2. (Step 2)

Let $\mathcal{C}\subset\mathbb{C}(\mathbf{k})$ be the set of the coefficients
of $\mathcal{S}$.

  3. (Step 3)

Return:

     * •

For the generators of the field of identifiable functions, the set
$\mathcal{S}$. It can be additionally simplified using [1] by applying
FieldToIdeal and then FilterGenerators (or [18, Section 5]);

     * •

For each $k_{i}\in\mathbf{k}$, the result of the test
$k_{i}\in\mathbb{C}(\mathbf{k})$ (using, e.g., [20, Section 1.3], see [1] for
implementation).

Algorithm 2 Approach to establishing strong identifiability (Defintion 2)

Input:

A system of rational PDEs of the form

$\mathbf{F}\left(\mathbf{k},\mathbf{v}\right)=0,\quad\mathbf{y}=\mathbf{G}(\mathbf{k},\mathbf{v}).$

together with some set of requirements $\mathcal{R}$ (regularity, boundary
conditions, etc.) on the states $\mathbf{v}$ and a domain $\Omega$ for the
parameter values.

Output:

For each parameter $k_{i}$, either returns that it is strongly identifiable,
or that the test was inconclusive.

  1. (Step 1)

Using RosenfeldGroebner with an elimination ranking $\mathbf{v}>\mathbf{y}$,
eliminate the variables $\mathbf{v}$, and obtain a set
$\mathcal{S}=\\{S_{1},\ldots,S_{\ell}\\}$ of IO-equations of the input PDE
system. We denote the set of their coefficients by $\mathcal{C}$.

  2. (Step 2)

For each $j$, $1\leqslant j\leqslant\ell$, write $S_{j}$ in the form

$F_{j}=p_{0,j}+\sum_{i=1}^{L_{j}}a_{i,j}\cdot p_{i,j},$

with $a_{i,j}\in\mathbb{C}(\mathbf{k})$ and
$p_{i,j}\in\mathbb{C}\\{\mathbf{y}\\}$ (see Remark 3). Compute the Wronskians
$\operatorname{Wr}_{j,x}$ and $\operatorname{Wr}_{j,t}$ of
$p_{1,j},\ldots,p_{L_{j},j}$ with respect to $\partial_{x}$ and
$\partial_{t}$, respectively.

  3. (Step 3)

For each $j$, $1\leqslant j\leqslant\ell$, compute the normal forms $N_{j,x}$
and $N_{j,t}$ of $\operatorname{Wr}_{j,x}$ and $\operatorname{Wr}_{j,t}$,
respectively, w.r.t. the input PDE system, and obtain a sufficient condition
$\mathcal{I}=\bigwedge_{j}(N_{j,x}\neq 0\lor N_{j,t}\neq 0)$ for the
identifiability of $\mathcal{C}$.

  4. (Step 4)

Check if $\mathcal{I}$ holds under the requirements $\mathcal{R}$ and for the
parameters in $\Omega$111Since there are almost no assumptions on the
requirements, this step is non-algorithmic. We show, how to do this in
practice in the next section..

  5. (Step 5)

If $\mathcal{I}$ does not hold, return “inconclusive” for every parameter.

  6. (Step 6)

If $\mathcal{I}$ does hold, return “identifiable” for every parameter $k_{i}$
such that $k_{i}\in\mathbb{C}(\mathbf{k})$ (can be checked with, e.g., [20,
Section 1.3], see [1] for implementation), and “inconclusive” for the rest of
the parameters.

##  5\. Examples of PDE identifiability

In this section, we consider several models arising in mathematical biology
(Section 5.1), as well as other natural phenomena (Section 5.2). We use
Proposition 1 and Algorithm 2, to test the identifiability of the parameters
in these models.

###  5.1. PDEs arising in mathematical biology

We study four well-known, and increasingly complex, PDE systems in
mathematical biology [21]. All of these examples use parabolic PDEs.

######  Example 1 (Scalar Reaction–Diffusion Equation).

We start by considering the following reaction–diffusion equation in which $c$
represents the concentration of a diffusible nutrient, such as oxygen or
glucose, which is consumed at a rate which is an increasing, saturating
function of its concentration [21]:

(10) $\partial_{t}c(x,t)=d\partial_{x}^{2}c(x,t)+\frac{\lambda
c(x,t)}{c_{0}+c(x,t)},$

where the set of parameters is $\\{d,c_{0},\lambda\\}$, and the boundary
conditions are given by:

(11) $\displaystyle\partial_{x}c(x,t)(0,t)=$ $\displaystyle 0$ (12)
$\displaystyle c(R,t)=$ $\displaystyle 1$ (13) $\displaystyle c(x,0)=$
$\displaystyle 1,\quad 0\leq x\leq R.$

Following Definition 2, our field is $\mathbb{K}=\mathbb{R}$ and
$\Omega=\mathbb{R}_{+}^{3}$. We follow Algorithm 2 to check the
identifiability of the parameters of (10). Details of the computations can be
found in the Maple worksheet nutrient.mpl222https://github.com/rahkooy/PDE-
Identifiability.

Considering the numerator of the rational function (10), i.e., the
differential polynomial

(14)
$-dc(x,t)\partial_{x}^{2}c(x,t)-dc_{0}\partial_{x}^{2}c(x,t)+c(x,t)\partial_{t}c(x,t)+c_{0}\partial_{t}c(x,t)-\lambda
c(x,t),$

we collect four monomials of (14) whose coefficient (with respect to the
elements in $\mathbb{C}(d,c_{0},\lambda)$) is not 1, and compute $W$, the
determinant of their Wronskian to check if it is nonsingular. To do so, we
compute the normal form of $W$ (using Rosenfeld-Gröbner), and check when its
coefficients are zero.

Note that if we obtain nonsingularity of the Wronskian from a subset of the
coefficient, it implies that the Wronskian is nonsingular. The first ten
coefficients are zero if and only if

(15) $\partial_{t}c(x,t)=0\quad\text{or}\quad\partial_{x}c(x,t)=0.$

So we investigate the following two cases.

#### Case 1. $\partial_{t}c(x,t)=0$

In this case, the function $c$ does not depend on $t$ anymore, and therefore,
$c(x,t)=c(x)$. Using the boundary conditions (12) and (13), we obtain
$c(x)=1$, a constant function. On the other hand, considering
$\partial_{t}c(x,t)=0$, the differential polynomial (14) will be simplified to
the following polynomial (which is the LHS of ODE, rather than the original
PDE):

(16) $-dc(x)\partial_{x}^{2}c(x)-dc_{0}\partial_{x}^{2}c(x)-\lambda c(x).$

Substituting $c(x)=1$ into (16), we obtain $-\lambda$, which is not zero, as
the parameters are assumed to be positive. So this case cannot happen.

#### Case 2. $\partial_{x}c(x,t)=0$

This condition means that $c$ does not depend on $x$, hence, $c(x,t)=c(t)$. So
substituting this condition into the PDE (10), we obtain the following ODE:

(17) $\partial_{t}c(t)=\frac{\lambda c(t)}{c_{0}+c(t)}.$

Using boundary condition (12), i.e., $c(R,t)=1$, we obtain
$\partial_{t}c(t)=0$. Therefore ODE (17) is equal to zero if and only if

(18) $\frac{\lambda c(t)}{c_{0}+c(t)}=\frac{\lambda}{c_{0}+1}=0,$

which can happen only if $\lambda=0$. This is impossible according to our
assumption on the positivity of the parameters, i.e.,
$\Omega=\mathbb{R}_{+}^{3}$. So Case 2 neither can happen.

In conclusion, none of the cases considered above can happen. This means that
the Wronskian matrix is non-singular. So, by Proposition 1, the coefficients
of the PDE (14), i.e., $\\{d,dc_{0},c_{0},\lambda\\}$, are all identifiable.

######  Example 2 (Fisher’s Equation).

Next, we consider Fisher’s equation that describes the diffusive spread of a
species which undergoes logistic growth [21]. Fisher’s equation is given by
the following PDE:

(19)
$\partial_{t}n(x,t)=d\partial_{x}^{2}n(x,t)+rn(x,t)\left(1-\frac{n(x,t)}{k}\right),$

where $n(x,t)$ is the input function, $x,t$ are the variables, and $d,r,k$ are
the parameters. The boundary conditions are given by

(20) $n(x,t)\to\begin{cases}k,\quad x\to-\infty\\\ 0,\quad
x\to\infty,\end{cases}$

and the initial condition is given by prescribed $u(x,0)$ as follows, which is
compatible with the boundary conditions (20).

(21) $n(x,0)=n_{0}(x)=\frac{ke^{-\alpha x}}{1+e^{-\alpha x}}.$

Following Definition 2, our field is $\mathbb{K}=\mathbb{R}$ and
$\Omega=\mathbb{R}_{+}^{3}$. Below we follow Algorithm 2 for checking the
identifiability of the parameters. Details of the computations can be found in
the Maple worksheet fisher.mpl333https://github.com/rahkooy/PDE-
Identifiability. Simplifying (19), we obtain

(22)
$\partial_{t}n(x,t)-d\partial_{x}^{2}n(x,t)-rn(x,t)+\frac{rn(x,t)^{2}}{k},$

from which we collect all of the monomials, except for $\partial_{t}n(x,t)$
whose coefficient is $1$, and compute $W$, the determinant of their Wronskian.
Using Rosenfeld-Gröbner, we compute the normal form of the determinant of $W$,
which is a rational function. We would like to check if this rational function
is identically zero. Equivalently, we would like to check if the numerator of
this rational function, say $N$, is zero. For this, we consider the
coefficients of $N$ as a polynomial in terms of parameters. There are four
such coefficients (which are differential polynomials in terms of variables
$x$ and $t$).

Computing the coefficients of the normal form of $W$, one can see that the
coefficients are zero if and only if

(23) $\partial_{x}n(x,t)=0\quad\text{or}\quad\partial_{t}n(x,t)=0,$

which leads to the following cases.

#### Case 1. $\partial_{x}n(x,t)=0$.

In this case, the function $n(x,t)$ does not depend on $x$, hence
$n(x,t)=n(t)$. However, the initial condition (21) implies that $n$ is a
function of $x$ unless $\alpha=0$, which cannot happen as the parameters are
supposed to be positive.

#### Case 2. $\partial_{t}n(x,t)=0$.

This case implies that $n$ does not depend on $t$, hence, $n(x,t)=n(x)$. From
the initial condition (21), we have that

(24) $n(x)=n_{0}(x)=\frac{ke^{-\alpha x}}{1+e^{-\alpha x}}.$

We substitute $\partial_{t}n(x,t)=0$ into Fisher’s Equation (19) to obtain the
following ODE:

(25) $-d\partial_{x}^{2}n(x)-rn(x)+\frac{rn(x)^{2}}{k}=0.$

Equation (24) must satisfy ODE (25). Evaluating the ODE at $n(x)$, we obtain

(26) $\frac{\left((\alpha^{2}d-r)e^{-\alpha x}-\alpha^{2}d-r\right)ke^{-\alpha
x}}{\left(1+e^{-\alpha x}\right)^{3}}.$

Equation (26) is zero if and only if its numerator is zero, which is the case
if and only if either $k=0$ (which is not possible according to our assumption
that parameters are positive), or

(27) $(\alpha^{2}d-r)e^{-\alpha x}-\alpha^{2}d-r=0.$

The above equation can happen only if the coefficient of the exponential term,
as well as the constant term are zero. This leads to the equations

(28) $\displaystyle\alpha^{2}d-r$ $\displaystyle=0$ (29)
$\displaystyle-\alpha^{2}d-r$ $\displaystyle=0,$

which leads to $2\alpha^{2}d=0$. But this cannot happen as $\alpha$ and $d$
are assumed to be non-zero.

So from the above discussions, we conclude that none of the above two cases
can happen, hence, the coefficients of the Fisher equation (19), that is, by
Proposition 1, these functions of parameters are identifiable: $d$,
$\frac{r}{k}$ and $\frac{1}{k}$. By the second part of Proposition 1, the
parameters $d,r,k$ are identifiable.

######  Example 3 (Coupled Reaction-Diffusion Equations).

Next, we consider the following system of two coupled reaction–diffusion
equations. Here species $u$ and $v$ undergo random motion and logistic growth
while competing for resources [21]. The PDEs are:

(30)
$\begin{cases}\partial_{t}u(x,t)=d_{1}\partial_{x}^{2}u(x,t)+u(x,t)\left(a_{1}-b_{1}u(x,t)-c_{1}v(x,t)\right),\\\
\partial_{t}v(x,t)=d_{2}\partial_{x}^{2}v(x,t)+v(x,t)\left(a_{2}-b_{2}u(x,t)-c_{2}v(x,t)\right),\\\
y_{1}(x,t)=u(x,t),\\\ y_{2}(x,t)=v(x,t).\end{cases}$

The model parameters are $a_{1},b_{1},c_{1},d_{1},a_{2},b_{2},c_{2},d_{2}$.
The boundary conditions are given by

(31) $\displaystyle u(x,t)$
$\displaystyle\to\begin{cases}\frac{a_{1}}{b_{1}}\quad x\to-\infty\\\ 0\quad
x\to\infty\end{cases}$ (32) $\displaystyle v(x,t)$
$\displaystyle\to\begin{cases}\frac{a_{2}}{b_{2}}\quad x\to\infty\\\ 0\quad
x\to-\infty\end{cases}$

and the initial conditions are given by prescribed $u(x,0)$ and $v(x,0)$,
e.g.,

(33) $\displaystyle u(x,0)$
$\displaystyle=u_{0}(x)=\frac{\left(\frac{a_{1}}{b_{1}}\right)e^{-\alpha_{1}x}}{1+e^{-\alpha_{1}x}}$
(34) $\displaystyle v(x,0)$
$\displaystyle=v_{0}(x)=\frac{\left(\frac{a_{2}}{b_{2}}\right)e^{-\alpha_{2}x}}{1+e^{-\alpha_{2}x}}\cdot$

We note that for the special case with $a_{1}<0=b_{1}=b_{2}$ equations (30)
reduce to the classical Lotka-Volterra equations with random motion.

Following Definition 2, our field is $\mathbb{K}=\mathbb{R}$ and
$\Omega=\mathbb{R}_{+}^{8}$. Below we follow our method (2) for testing the
identifiability of the parameters. Details of the computations can be found in
the Maple worksheet LV-PDE.mpl444https://github.com/rahkooy/PDE-
Identifiability.

For the first equation, the normal form of the determinant of the Wronskian
yields a polynomial with 34 coefficients. We have considered 22 first
coefficients and obtained that the determinant of the Wronskian is zero if and
only if

(35)
$\partial_{x}u(x,t)=0\quad\text{or}\quad\partial_{t}u(x,t)=\partial_{t}v(x,t)=0\quad\text{or}\quad
v(x,t)=0,$

which leads us to the following cases.

#### Case 1. $\partial_{x}u(x,t)=0$

In this case, the derivatives of $u(x,t)$ with respect to $x$ is zero, which
means that the function $u$ does not depend on $x$, i.e., $u(x,t)=u(t)$.
However, according to the initial condition (33), $u$ depends on $x$, unless
$-\alpha_{1}=0$, in which case $u=\frac{a_{1}}{b_{1}}$ is a constant function.
Using the boundary condition (31) when $x\rightarrow-\infty$, functions $u$
becomes zero. This implies that $\frac{a_{1}}{b_{1}}=0$, which cannot happen
as we have assumed that the parameters are nonzero. So this case does not
happen either.

#### Case 2. $\partial_{t}u(x,t)=\partial_{t}v(x,t)=0$

In this case, $u$ and $v$ do not depend on it, i.e., $u(x,t)=u(x)$ and
$v(x,t)=v(x)$. Since the initial conditions (33) and (34) only depend on $x$,
therefore,

(36) $\displaystyle u(x,t)$
$\displaystyle=u(x)=\frac{\left(\frac{a_{1}}{b_{1}}\right)e^{-\alpha_{1}x}}{1+e^{-\alpha_{1}x}}$
(37) $\displaystyle v(x,t)$
$\displaystyle=v(x)=\frac{\left(\frac{a_{2}}{b_{2}}\right)e^{-\alpha_{2}x}}{1+e^{-\alpha_{2}x}}.$

On the other hand, substituting $\partial_{t}u(x,t)=\partial_{t}v(x,t)=0$ into
the original PDE system (30), one obtains the following two ODEs:

(38) $\displaystyle-
d_{1}\partial_{x}^{2}u(x)-u(x)\left(a_{1}-b_{1}u(x)-c_{1}v(x)\right)=0$ (39)
$\displaystyle-
d_{2}\partial_{x}^{2}v(x)-v(x)\left(a_{2}-b_{2}u(x)-c_{2}v(x)\right)=0.$

Therefore, $u(x)$ and $v(x)$ should satisfy the ODEs (38) and (39). Having
substituted (36) into the ODEs (38), one obtains

(40)
$\frac{1}{b_{1}(1+e^{-ax})^{3}}\left(a_{1}e^{-ax}\left((d_{1}a^{2}+2c_{1}v(x)-a_{1})e^{-ax}-d_{1}a^{2}+v(x)e^{-2ax}c_{1}+c_{1}v(x)-a_{1}\right)\right).$

Finally, substituting (37) into the above, we obtain a rational function of
the form

(41) $\frac{1}{c_{2}(1+e^{bx})b_{1}(1+e^{-ax})^{3}}F(x)a_{1}e^{-ax},$

where

(42) $\displaystyle F(x)=$
$\displaystyle(-a_{2}c_{1}e^{-x(2a-b)}+\left((-d_{1}a^{2}+a_{1})c_{2}-2a_{2}c_{1}\right)e^{x(-a+b)}$
$\displaystyle-c_{2}(d_{1}a^{2}-a_{1})e^{-ax}$
$\displaystyle+\left(c_{2}(d_{1}a^{2}+a_{1})-a_{2}c_{1}\right)e^{bx}$
$\displaystyle+c_{2}(d_{1}a^{2}+a_{1})$

The rational function in (41) is identically zero if and only if either
$a_{1}(e)^{-ax}=0$ or $F(x)=0$. Since $a_{1}\neq 0$, it cannot happen that
$a_{1}(e)^{-ax}=0$. The second case, $F(x)=0$, can happen if the exponential
terms appearing in $F(x)$ are linearly dependent, which only can happen if the
exponents of those terms are equal. It is enough to consider the cases in
which the first exponent is equal to some other exponents. This way, we obtain
below four cases, each of which is discussed and it has been shown that none
can happen.

For convenience, set

(43) $\displaystyle u$ $\displaystyle:=-a_{2}c_{1},$ (44) $\displaystyle w$
$\displaystyle:=-d_{1}a^{2}+a_{1}$ (45) $\displaystyle t$
$\displaystyle:=d_{1}a^{2}+a_{1}.$

Then the coefficients of $F(x)$ will be $u,wc_{2}-2u,c_{2}w,c_{2}t-u$ and
$c_{2}t$. Also, let $y=e^{x}$. The four cases are the following.

#### Case 2.1. $-2a+b=-a+b$

This implies that $a=0$ which cannot happen.

#### Case 2.2. $-2a+b=-a$

This means that $a=b$, and the exponents become
$y^{-a},y^{0},y^{-a},y^{a},y^{0}$. collecting coefficients of equal exponents,
we obtain that

(46) $\displaystyle u+c_{2}w$ $\displaystyle=0,$ (47) $\displaystyle
c_{2}w-2u+c_{2}t$ $\displaystyle=0,$ (48) $\displaystyle c_{2}t-u$
$\displaystyle=0.$

Simplifying the above we have $2u=0$ which is not possible.

#### Case 2.3. $-2a+b=b$

This case implies that $a=0$ which is not possible.

#### Case 2.4. $-2a+b=0$

From the conditions of this case, we obtain that $b=2a$, and the exponents
will be $y^{0},y^{a},y^{-a},y^{2a},y^{0}$. since $y^{a}$ and $y^{2a}$ are
linearly independent, if their coefficients do not kill each other, then we
are done. This is the case as the coefficients are $c_{2}w-2u$ and $c_{2}w$.

Considering coefficients of the normal form of the determinant of the second
Wronskian, we obtain the following conditions:

(49)
$\displaystyle\frac{\partial^{2}}{\partial_{x}\partial_{t}}u(x,t)=\partial_{x}v(x,t)=0\quad\text{or}$
(50) $\displaystyle\partial_{t}u(x,t)=\partial_{t}v(x,t)=0\quad\text{or}$ (51)
$\displaystyle u(x,t)=0\quad\text{or}\quad$ (52) $\displaystyle v(x,t)=0.$

Three out of the four cases in (49), (50), (51), and (52) have been considered
earlier. The only remaining case
$\frac{\partial^{2}}{\partial_{x}\partial_{t}}u(x,t)=\partial_{x}v(x,t)=0$.
Similar to the argument for $\partial_{x}u(x,t)=0$, one can check that
$\partial_{x}v(x,t)=0$ cannot happen. So this case is also impossible.

In conclusion, none of the Wronskians can be identically singular. Hence, by
Proposition 1, the coefficients of the PDEs (30) are identifiable. Since the
coefficients of the system are the parameters, all parameters are
identifiable.

######  Example 4 (Single Output Lotka Volterra).

In this example, we consider the single output version of the Coupled
Reaction-Diffusion Equations studied in Example 3:

(53)
$\begin{cases}\partial_{t}u(x,t)=d_{1}\partial_{x}^{2}u(x,t)+u(x,t)\left(a_{1}-b_{1}u(x,t)-c_{1}v(x,t)\right),\\\
\partial_{t}v(x,t)=d_{2}\partial_{x}^{2}v(x,t)+v(x,t)\left(a_{2}-b_{2}u(x,t)-c_{2}v(x,t)\right),\\\
y(x,t)=u(x,t).\end{cases}$

The initial and boundary conditions are the same as the Coupled Reaction-
Diffusion Equations and are given by (31), (32), (33), and (34). Similarly,
the ground field is $\mathbb{K}=\mathbb{R}$ and $\Omega=\mathbb{R}_{+}^{8}$.

For this example, we initially tried to apply Algorithm 2. We used [1]
applying FieldToIdeal and then FilterGenerators to regroup the monomials and
to simplify the system, and obtained a 13 $\times$ 13 matrix instead of the
original 17 $\times$ 17 Wronskian. However, computing the determinant (and
then normal form) of this simpler matrix was not possible within a reasonable
time and memory. Hence, instead of computing directly the normal form of the
determinant of the Wronskian, we used the following steps to make computations
faster:

  1. (1)

First, we compute the normal form of each entry of the Wronskian

  2. (2)

Secondly, using the equations of the system (53), we eliminate the derivatives
of $u(x,t)$ and $v(x,t)$ with respect to $t$, so that the equations only
depend on $u,v$ and their derivatives with respect to $x$. Then we could
substitute the initial conditions efficiently to the system (substituting
initial conditions into the original system is very time-consuming).

  3. (3)

Third, we substituted random values for parameters as computations with
symbolic parameters were not possible in a reasonable time.

  4. (4)

Lastly, we computed the numeric value of the determinant of the Wronskian,
evaluated in a generic point $x$.

While computing the determinant of the Wronskian and its normal form did not
finish within three days, the above optimizations made the computation finish
in less than four hours, most of which is spent on normal form computations.
Having stored the output of the normal forms, one can carry on the
computations for different values of $x$ and the parameters in the initial
conditions in just less than a second.

As the determinant was not zero in a generic point, we conclude that the
determinant is not a zero function for random nonzero values of the
parameters. Although this does not prove strong identifiability, however, we
succeeded in presenting strong numeric evidence that generically the
parameters are identifiable. Additional studies can check particular values of
the parameters that vanish the Wronskian. We note that due to the lack of
resources for symbolic computations, we chose random values for parameters.
For the details of the computations, we refer to our Maple code LV-Single-
Output.mpl555https://github.com/rahkooy/PDE-Identifiability.

Finally, we also see that the parameters $c_{1}$ and $c_{2}$ are not
identifiable using the following argument. Following Definition 2, let

$\displaystyle\mathbf{k}_{1}=(a_{1},b_{1},c_{1},d_{1},a_{2},b_{2},c_{2},d_{2}),\
\ \mathbf{v}_{1}=(u,v)$ $\displaystyle\mathbf{k}_{2}=(a_{1},b_{1},\lambda
c_{1},d_{1},a_{2},b_{2},\lambda c_{2},d_{2}),\ \
\mathbf{v}_{2}=(u,v/\lambda),$

where $\lambda$ is an arbitrary nonzero number. One sees that, substituting
$\mathbf{k}_{2}$ and $\mathbf{v}_{2}$ into (53), $\lambda$ will be eliminated
and we will obtain (53) again. However, $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$
are not equal. This means that $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ are not
identifiable. As $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ only differ in the
values of $c_{1}$ and $c_{2}$, one can conclude that $c_{1}$ and $c_{2}$ are
not identifiable.

######  Example 5 (A Reaction-Diffusion Model of Cancer Invasion).

Our final example from mathematical biology is a PDE model of cancer invasion
which couples two reaction-diffusion equations for the concentrations of
tumour cells $v(x,t)$ and acid (or pH) $w(x,t)$ with a time-dependent ordinary
differential equation for healthy cells $u(x,t)$. The healthy cells and the
cancer cells undergo logistic growth and compete with each other for space.
The healthy cells do not move but they are killed by acid which is produced by
the tumour cells and undergoes natural decay while it diffuses through the
domain. The diffusive movement of the cancer cells is assumed to be a linearly
decreasing function of the concentration of healthy cells [10]. The model is
given by the following three PDEs:

(54)
$\displaystyle\partial_{t}u(x,t)=r_{1}u(x,t)\left(1-\frac{u(x,t)}{k_{1}}-\frac{v(x,t)}{k_{2}}a_{12}\right)-d_{1}w(x,t)u(x,t),$
(55)
$\displaystyle\partial_{t}v(x,t)=r_{2}v(x,t)\left(1-\frac{v(x,t)}{k_{2}}-\frac{u(x,t)}{k_{1}}a_{21}\right)+d_{2}\partial_{x}\left((1-u(x,t))\partial_{x}v(x,t)\right),$
(56)
$\displaystyle\partial_{t}w(x,t)=d_{4}\partial_{x}^{2}w(x,t)+r_{3}v(x,t)-d_{3}w(x,t),$

along with the output equations

(57) $\displaystyle y_{1}(x,t)$ $\displaystyle=u(x,t),$ (58) $\displaystyle
y_{2}(x,t)$ $\displaystyle=v(x,t),$ (59) $\displaystyle y_{3}(x,t)$
$\displaystyle=w(x,t).$

The parameters of the system are
$r_{1},k_{1},a_{12},k_{2},d_{1},r_{2},a_{21},d_{2},d_{3},r_{3},d_{4}$, the
boundary conditions are given by

(60) $\displaystyle u(x,t)$ $\displaystyle\to\begin{cases}k_{1},\quad
x\to+\infty\\\ 0,\quad x\to-\infty\end{cases}$ (61) $\displaystyle v(x,t)$
$\displaystyle\to\begin{cases}0,\quad x\to+\infty\\\ k_{2},\quad
x\to-\infty\end{cases}$ (62) $\displaystyle w(x,t)$
$\displaystyle\to\begin{cases}0,\quad x\to+\infty\\\
\frac{k_{2}r_{3}}{d_{3}},\quad x\to-\infty,\end{cases}$

and the initial conditions are given by prescribed $u(x,0)$ and $v(x,0)$,
e.g.,

(63) $\displaystyle w(x,0)$ $\displaystyle=0$ (64) $\displaystyle u(x,0)$
$\displaystyle=\frac{k_{1}e^{\gamma_{1}x}}{1+e^{\gamma_{1}x}}$ (65)
$\displaystyle v(x,0)$
$\displaystyle=\frac{k_{2}e^{-\gamma_{2}x}}{1+e^{-\gamma_{2}x}}\cdot$

Our ground field is $\mathbb{K}=\mathbb{R}$ and $\Omega=\mathbb{R}_{+}^{11}$,
corresponding to the number of parameters.

Details of the computations can be found in the Maple worksheet reaction-
diffusion.mpl666https://github.com/rahkooy/PDE-Identifiability.

We consider each of the three equations separately, computing three Wronskians
and checking if they are nonsingular. We note that for the third equation, as
there are no monomials with coefficient 1, we divide the polynomial by
$k_{1}k_{2}$ so that it has a monomial with coefficient one. Below is the
summary of our computations.

  * •

The first Wronskian yields 40 coefficients. Considering ten of the
coefficients, Rosenfeld-Gröbner results in the following conditions:

(66) $\partial_{x}u(x,t)=0\quad\text{or}\quad v(x,t)=0.$

The above conditions can easily be refuted by looking at the boundary and
initial conditions, using arguments similar to the Coupled Reaction-Diffusion
equations.

  * •

For the second equation, the normal form contains 8 coefficients. Our
procedure results in the following conditions for Wronskian to be singular:

(67) $w(x,t)\frac{\partial w(x,t)}{\partial t\partial
x}=\partialderivative{w(x,t)}{t}\partialderivative{w(x,t)}{x}\quad\text{or}\quad
w(x,t)=0\quad\text{or}\quad v(x,t)=0.$

The second and third conditions, i.e., $w(x,t)=0$ and $v(x,t)=0$ can be easily
refuted using boundary and initial conditions. For the first condition, we
show that one can solve using the specified initial condition and obtain only
zero solution. More precisely, substituting (63) in the first condition, one
obtains that either $\partialderivative{w(x,t)}{x}=0$ at $t=0$ or
$\partialderivative{w(x,t)}{t}=0$ at $t=0$. One can easily check that both of
these contradict the initial and boundary conditions, as the parameters are
not allowed to be zero.

  * •

For the third equation, the first ten coefficients result in the following
conditions

(68) $\partialderivative{u(x,t)}{x}=0\quad\text{or}\quad v(x,t)=0$

The second condition ($v(x,t)=0$) can be trivially refuted. For the first
conditions, adding $\partialderivative{u(x,t)}{x}$ to the system and computing
Rosenfeld-Gröbner, we obtain 9 equations, three of which are the input-output
equations, three are the partial derivatives of the output equations with
respect to $x$, and the remaining three equations involve the parameters. The
latter three are of interest to us. They show that the states are not constant
with respect to $x$. One can check that the boundary and initial conditions
would imply that several parameters are zero, which is not permitted according
to our assumptions. This can simply be seen as the normal forms of $u,v$, and
$w$ with respect to the Rosenfeld-Gröbner are zero.

###  5.2. PDEs from applied mathematics

Finally, in this section, we consider two well-known PDEs in applied
mathematics that model natural phenomena.

######  Example 6.

The following PDE is an example of an elliptic PDE.

(69) $\partial_{x}^{2}u(x,y)+\partial_{y}^{2}u(x,y)=\theta,$

where $0\leq x\leq L$ and $0\leq y\leq H$. The boundary conditions are

(70) $\displaystyle u(x,H)$ $\displaystyle=\frac{\alpha x}{L}$ (71)
$\displaystyle u(L,y)$ $\displaystyle=\frac{\alpha y}{H},$

and the initial conditions are

(72) $\displaystyle u(x,0)$ $\displaystyle=0$ (73) $\displaystyle u(0,y)$
$\displaystyle=0.$

The parameters of the system are $\theta$ and $\alpha$, and the domain for
parameters is $\Omega=\mathbb{R}_{+}^{2}$.

In order to remove the parameter $\alpha$ from the boundary condition, one can
define a new variable $v:=u/\alpha$. Then the new equation and boundary
conditions will be

(74) $\alpha\partial_{x}^{2}v(x,y)+\alpha\partial_{y}^{2}v(x,y)=\theta,$

the boundary conditions will be

(75) $\displaystyle v(x,H)$ $\displaystyle=\frac{x}{L}$ (76) $\displaystyle
v(L,y)$ $\displaystyle=\frac{y}{H},$

and the initial conditions will be

(77) $\displaystyle\alpha v(x,0)$ $\displaystyle=0$ (78) $\displaystyle\alpha
v(0,y)$ $\displaystyle=0.$

Since $\alpha$ is assumed to be non-zero, the initial conditions become

(79) $\displaystyle v(x,0)$ $\displaystyle=0$ (80) $\displaystyle v(0,y)$
$\displaystyle=0.$

Dividing both sides of (74) by $\alpha$, we obtain
$\partial_{x}^{2}v(x,y)+\partial_{y}^{2}v(x,y)=\frac{\theta}{\alpha}$. Hence,
$\frac{\theta}{\alpha}$ is identifiable, as it is in terms of derivatives of
$v(x,y)$, however, the parameters $\alpha$ and $\theta$ are not identifiable.
For the special case of $\alpha=1$, the parameter $\theta$ will be
identifiable.

######  Remark 4.

Note that the current work does not address the general case of systems with
parameters in the boundary conditions. This is a potential future work using
prolongations as in SIAN [12].

######  Example 7.

The following PDE is the well-known _wave equation_ and is an example of a
hyperbolic PDE.

(81) $\partial_{t}^{2}u(x,t)-\sigma^{2}\partial_{x}^{2}u(x,t)=0,$

where $\sigma$, the parameter, is the wave speed. The initial conditions are
given by

(82) $\displaystyle u(x,0)=L-x,\quad\partial_{t}u(x,0)=\beta.$

The output function is $y(x,t)=u(x,t)$. By d’Alembert’s formula [22, Theorem
2.15], (81) and (82) define a unique solution

(83) $y(x,t)=-x+\beta t+L.$

Following Definition 2, our ground field is $\mathbb{K}=\mathbb{R}$ and
$\Omega=\mathbb{R}_{+}$. Let us now try to apply Algorithm 2 (Proposition 1)
to see that computing Wronskian is essential. The IO-equation is

(84) $\partial_{t}^{2}y(x,t)-\sigma^{2}\partial_{x}^{2}y(x,t)=0.$

Of its two monomials, the coefficient of $\sigma^{2}\partial_{x}^{2}y(x,t)$ is
not one. The Wronskian of this monomial (with respect to both $x$ and $t$) is
itself. Furthermore, this monomial $\partial_{x}^{2}y(x,t)$ vanishes on every
solution (83), so Algorithm 2 is not applicable here. Finanlly, since the
solution (83) depends only on the initial conditions and not on $\sigma$,
$\sigma$ is nonidentifiable by Definition 2.

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