# Non-finite type étale sites over fields

Sujeet Dhamore  Department of Mathematics, Indian Institute of Science
Education and Research (IISER) Pune, Dr. Homi Bhabha road, Pashan, Pune 411008
India<EMAIL_ADDRESS>,  Amit Hogadi  Department of
Mathematics, Indian Institute of Science Education and Research (IISER) Pune,
Dr. Homi Bhabha road, Pashan, Pune 411008 India<EMAIL_ADDRESS>and
Rakesh Pawar  UMPA, ENS de Lyon, UMR 5669, CNRS, 46 allée d’Italie 69364 Lyon
Cedex 07, France<EMAIL_ADDRESS>
###### Abstract.

We consider the notion of finite type-ness of a site introduced by Morel and
Voevodsky, for the étale site of a field. For a given field $k$, we conjecture
that the étale site of $Sm/k$ is of finite type if and only if the field $k$
admits a finite extension of finite cohomological dimension. We prove this
conjecture in some cases, e.g. in the case when $k$ is countable, or in the
case when the $p$-cohomological dimension $cd_{p}(k)$ is infinite for
infinitely many primes $p$.

###### 2020 Mathematics Subject Classification:

14F20, 14F35, 14F42 (Primary), 12G05, 12G10

##  1\. Introduction

In modern development of homotopy theory in algebraic geometry e.g., as in
[3], given a site with a Grothendieck topology, one works with the category of
simplicial sheaves (or presheaves) on the site with injective model structure.
One of the important tools to study maps into simplicial sheaf, say
$\mathcal{X}$, is to inductively study maps into its Postnikov truncations
$P^{n}(\mathcal{X})$ for $n\geq 0$. So if one knows that
$\mathcal{X}\to\operatorname{holim}_{n\geq 0}Ex(P^{n}(\mathcal{X}))$ is a weak
equivalence, then the compatible maps $\mathcal{Y}\to P^{n}(\mathcal{X})$ for
all $n$, is equivalent to given a map $\mathcal{Y}\to\mathcal{X}$ (up to
homotopy). This has been important in the applications of
$\mathbb{A}^{1}$-homotopy theory. Moreover, if
$\mathcal{X}\to\operatorname{holim}_{n\geq 0}Ex(P^{n}(\mathcal{X}))$ is a weak
equivalence for all $\mathcal{X}$, then we have an exact fibrant replacement
functor $Ex^{\mathcal{G}}$ for simplicial sheaves which also preserves
fibrations as described in [3, Theorem 1.66, page 69-70]. Unfortunately
however, in general, on an arbitrary site,
$\mathcal{X}\to\operatorname{holim}Ex(P^{n}(\mathcal{X}))$ is not necessarily
a weak equivalence! [3, Example 1.30, page 58]. We recall notion of finite
type site by Morel and Voevodsky.

######  Definition 1.1.

(Finite type site) [3, Definition 1.31, page 58] A site $T$ is called of
finite type, if for each simplicial sheaf $\mathcal{X}$ on $T$, the canonical
map

$\mathcal{X}\to\operatorname{holim}_{n\geq 0}Ex(P^{n}(\mathcal{X}))$

is a weak equivalence.

For a field $k$, we let $\rm{(Sm/k)_{\rm\acute{e}t}}$ denote the étale site of
finite type smooth schemes over a field $k$. The goal of this paper is to give
a criterion for this site to be of finite type. We conjecture the following.

######  Conjecture 1.2.

Let $k$ be any field. Let $\rm{\Delta^{op}Shv}((Sm/k)_{\rm\acute{e}t})$ denote
the category of étale simplicial sheaves on $(Sm/k)_{\rm\acute{e}t}$ with
local injective model structure. Then the site $(Sm/k)_{\rm\acute{e}t}$ is of
finite type if and only if there exists a finite extension $L/k$ such that
$cd(L)<\infty$.

The following is the main result of the paper which proves conjecture for
countable fields.

######  Theorem 1.3.

Conjecture 1.2 holds, if the absolute Galois group $G_{k}=Gal(k^{sep}/k)$ is
first-countable. In particular, if the field $k$ is countable, then $G_{k}$ is
first-countable, hence conjecture 1.2 holds.

For any field $k$, the if part of the Conjecture 1.2 follows from the
following result of Morel and Voevodsky, which gives a sufficient
cohomological criterion for a site to be of finite type.

######  Theorem 1.4.

[3, Theorem 1.37, page 60] Let $T$ be a site and suppose that there exists a
family $(A_{d})_{d\geq 0}$ of classes of objects of $T$ such that the
following conditions hold:

  1. (1)

Any object $U$ in $A_{d}$ has cohomological dimension $\leq d$ i.e. for any
sheaf $F$ on $T/U$ and any $m>d$ one has $H^{m}(U,F)=0$.

  2. (2)

For any object $V$ of $T$ there exists an integer $d_{V}$ such that any
covering of $V$ in $T$ has a refinement of the form $\\{U_{i}\to V\\}$ with
$U_{i}$ being in $A_{d_{V}}$.

Then $T$ is a site of finite type.

######  Remark 1.5.

As a consequence of Theorem 1.4, for a base scheme $S$ of finite Krull
dimension, the sites $(Sm/S)_{Zar}$ and $(Sm/S)_{Nis}$ are of finite type.

Note that the Theorem 1.4 implies that, if $k$ is a field, such that there
exists a finite extension of $k$ which is of finite cohomological dimension,
then $\rm{(Sm/k)_{\rm\acute{e}t}}$ is of finite type. Hence the only if part
of the Conjecture 1.2 is open.

We also show that, certain cohomological boundedness conditions on the field
$k$ are necessary for the site $\rm{(Sm/k)_{\rm\acute{e}t}}$ to be of finite
type. In following results, we show only if part of Conjecture 1.6 for some
cases.

######  Theorem 1.6.

Let $k$ be any field which satisfies the condition that for each
$n\in{\mathbb{N}}$, there exists $G_{k}$-module $A_{n}$ such that

$\limsup_{n\to\infty}exp(H^{n}(G_{k},A_{n}))=\infty$

where for an abelian group $B$, the exponent of $B$ is denoted by $exp(B)$.
Then the site $\rm{(Sm/k)_{\rm\acute{e}t}}$ is not of finite type.

Observe that, a field $k$ which either has $cd_{p}(k)=\infty$ for infinitely
many primes $p$ or has arbitrarily large $cd_{p}(k)$ for arbitrarily large
primes $p$ satisfies hypothesis of Theorem 1.6. This is summarized in the
following result.

######  Proposition 1.7.

Let $k$ be a field such that $\limsup cd_{p}(k)=\infty$ where limit is over
all primes $p$. Then the site $\rm{(Sm/k)_{\rm\acute{e}t}}$ is not of finite
type.

Note that, if a field $k$ satisfies assumptions of either Proposition 1.7 or
Theorem 1.6 then for each finite extension $L/k$, $cd(L)=\infty$.

### Acknowledgments

The first author was supported by Ph.D. fellowship (File No:
09/936(0247)/2019-EMR-I) of Council of Scientific & Industrial Research
(CSIR), India. The third author would like to thank Matthias Wendt for helpful
discussions. The third author is supported by the French ANR project: Motivic
homotopy, quadratic invariants and diagonal classes (ANR-21-CE40-0015).

##  2\. preliminaries

In this section, we fix some notations and recall definitions from [3].

For any Grothendieck site $T$ with enough points, $\rm{\Delta^{op}Shv}(T)$ and
$\rm{\Delta^{op}Shv}_{Ab}(T)$ denote the category of simplicial sheaves and
the category of simplicial abelian sheaves on the site $T$ respectively. We
will consider $\rm{\Delta^{op}Shv}(T)$ and $\rm{\Delta^{op}Shv}_{Ab}(T)$ as a
simplicial model categories with the local injective model structure as
endowed in [3, pages 48-49].

Analogues to the situation for simplicial sets, Morel and Voevodsky introduced
the notion of (simplicial) Postnikov truncations of a given simplicial sheaf.

######  Definition 2.1.

[3, Page 57] Given a simplicial sheaf $\mathcal{X}\in\rm{\Delta^{op}Shv}(T)$
for each $n\geq 0$, $P^{n}(\mathcal{X})$ is the sheafification of the presheaf
given by

$U\longmapsto P^{(n)}(\mathcal{X}(U))={\rm Im\ }(\mathcal{X}(U)\longrightarrow
cosk_{n}(\mathcal{X}(U)))$ for each $U\in T$

where $P^{(n)}(S)$ denotes $n^{th}$ Moore-Postnikov truncation for any
simplicial set $S$.

######  Notations 2.2.

We use the following notations throughout the paper.

  1. (1)

Given a simplicial sheaf $\mathcal{X}$, the homotopy sheaf
$\pi_{0}^{s}(\mathcal{X})$ is defined as the sheafification of the presheaf
$\pi_{0}^{pre}(\mathcal{X})$ given as

$\pi_{0}^{pre}(\mathcal{X})(U)=\pi_{0}(\mathcal{X}(U))$ or each $U\in T$

where $\pi_{0}(S)$ denotes the set of connected components of any simplicial
set $S$.

  2. (2)

Another description of $\pi_{0}^{pre}(\mathcal{X})$ is given as
$U\longmapsto{\rm Hom}_{\mathcal{H}_{s}}(U,\mathcal{X})$, where
$\mathcal{H}_{s}$ is the (simplicial)-homotopy category of
$\rm{\Delta^{op}Shv}(T)$.

  3. (3)

For a given site $T$, let $Ex:\rm{\Delta^{op}Shv}(T)\to\rm{\Delta^{op}Shv}(T)$
denote a resolution functor or a fibrant replacement functor as defined in [3,
Definition 1.6, page 49].

  4. (4)

Given a field $k$ and a fixed separable closure $k^{sep}/k$, for any
intermediate field extension $k^{sep}/L/k$, let $G_{L}:=Gal(k^{sep}/L)$ be
absolute Galois group of $L$.

##  3\. Main Results

The goal of this section is to prove Theorems 1.3, 1.6 and Proposition 1.7.
Lemma 3.2 is at the heart of the proofs of these results. The broad idea of
the proof of this lemma, is borrowed from the following example of Morel and
Voevodsky.

######  Example 3.1.

([3, Example 1.30, page 58]) For $G=\prod_{i>0}\mathbb{Z}/2$, the site
$(G\hbox{-}sets)_{df}$ is not finite type. This relies on the fact that
$H^{i}(\mathbb{Z}/2,\mathbb{Z}/2)\neq 0$ for all $i>0$, on in other words,
$\mathbb{Z}/2$ has infinite (group) cohomological dimension.

######  Lemma 3.2.

Let $k$ be a field. Assume that for each $i\in{\mathbb{N}}$, there is a
continuous $G_{k}$-module $A_{i}$ and classes

$\\{\alpha_{i}\in H^{i}(G_{k},A_{i})\\}_{i\in{\mathbb{N}}}$

such that for each $L\supset k$ finite Galois extension, there is $m$
(possibly depending on $L$) such that $(\alpha_{m})_{|L}\neq 0$ in
$H^{m}(G_{L},A_{m})$. Then the site $(Sm/k)_{\rm\acute{e}t}$ is not of finite
type.

###### Proof.

To prove the lemma we need to construct an étale simplicial sheaf
$\mathcal{X}$ such that $\mathcal{X}\to\operatorname{holim}_{n\geq
0}Ex(P^{n}(\mathcal{X}))$ is not a weak equivalence. We do this in several
steps.
Step 1: In this step we construct the required simplicial sheaf $\mathcal{X}$.

For each $G_{k}$-module $A_{i}$ we have the associated abelian sheaf
$\widetilde{\mathcal{A}_{i}}\in\rm{Shv_{Ab}(({\rm
Spec\,}k)_{\rm\acute{e}t})}$. The inclusion functor $f:({\rm
Spec\,}k)_{\rm\acute{e}t}\rightarrow\rm{(Sm/k)_{\rm\acute{e}t}}$ induces
adjoint pair between respective categories of sheaves by [5, Chapter I, 3.6]
such that,

${\rm{Shv_{Ab}(({\rm
Spec\,}k)_{\rm\acute{e}t})}}$${\rm{Shv_{Ab}((Sm/k)_{\rm\acute{e}t})}}$$\scriptstyle{f^{-1}}$$\scriptstyle{f_{*}}$$\mathrel{\relbar\mkern-4.0mu\relbar\mkern-9.0mu\vbox{\hbox{$\dashv$}}}$

Let $\mathcal{A}_{i}:=f^{-1}(\widetilde{\mathcal{A}_{i}})$ be the sheaves on
$\rm{(Sm/k)_{\rm\acute{e}t}}$ We apply the classical Dold-Kan correspondence
sectionwise as described in [3, Proposition 1.24, page 56] to obtain a model
of Eilenberg-MacLane sheaves
$\mathcal{K}(\mathcal{A}_{i},i)\in\rm{\Delta^{op}Shv}((Sm/k)_{\rm\acute{e}t})$
corresponding to $\mathcal{A}_{i}$ such that

$\mathcal{K}(\mathcal{A}_{i},i)(U)=K(\mathcal{A}_{i}(U),i)$ for any
$U\in\rm{(Sm/k)_{\rm\acute{e}t}}$

where $K(\mathcal{A}_{i}(U),i)$ is the Eilenberg-MacLane simplicial abelian
group corresponding to the abelian group $\mathcal{A}_{i}(U)$. We define
$\mathcal{X}:=\prod_{i>0}\mathcal{K}(\mathcal{A}_{i},i)\in\rm{\Delta^{op}Shv}((Sm/k)_{\rm\acute{e}t})$
as the product of simplicial sheaves.

Step 2: In this step we show that $\mathcal{X}$ is a connected simplicial
sheaf.

For each $U\in\rm{(Sm/k)_{\rm\acute{e}t}}$, the homotopy presheaf
$\pi_{0}^{pre}(\mathcal{X})$ is given as

$\displaystyle\pi_{0}^{pre}(\mathcal{X})(U)=\pi_{0}(\mathcal{X}(U))$
$\displaystyle=\pi_{0}(\prod_{i>0}\mathcal{K}(\mathcal{A}_{i},i)(U))$
$\displaystyle=\pi_{0}(\prod_{i>0}K(\mathcal{A}_{i}(U),i))$
$\displaystyle=\prod_{i>0}\pi_{0}(K(\mathcal{A}_{i}(U),i))$

The last equality holds, since the functor $\pi_{0}(-)$ on simplicial sets
commutes with infinite product of Kan complexes and each
$K(\mathcal{A}_{i}(U),i)$ is a Kan complex. Since $K(\mathcal{A}_{i}(U),i)$ is
connected (for $i>0$), we have that for each $U$,
$\pi_{0}^{pre}(\mathcal{X})(U)=*$. Hence, the homotopy presheaf
$\pi_{0}^{pre}(\mathcal{X})=*$, which implies that the associated sheaf
$\pi_{0}^{s}(\mathcal{X})=*$.

Step 3: We determine Postnikov truncations $P^{n}(\mathcal{X})$ corresponding
to $\mathcal{X}$.

Recall from Definition 2.1 that $P^{n}(\mathcal{X})$ is the sheaf associated
to the presheaf

$U\mapsto P^{(n)}(\mathcal{X}(U))={\rm Im\
}(\mathcal{X}(U)\xrightarrow{\eta}cosk_{n}(\mathcal{X}(U)))$

where $\eta$ for any simplicial set $X$ is the canonical map from $X$ to its
coskeleton $cosk_{n}(X)$ such that for any $m$-simplex $x$ given as a map
$\Delta^{m}\to X$

$\eta(x)=x_{|sk_{n}(\Delta^{m})}$ given as the map $sk_{n}(\Delta^{m})\to X$.

Observe that, since $cosk_{n}$ is right adjoint (hence commutes with limits),
we have that

$cosk_{n}(\mathcal{X}(U))=cosk_{n}(\prod_{i>0}\mathcal{K}(\mathcal{A}_{i},i)(U))=\prod_{i>0}cosk_{n}(K(\mathcal{A}_{i}(U),i)).$

Hence, $\eta$ is given as

$\prod_{i>0}K(\mathcal{A}_{i}(U),i)\xrightarrow{\eta=\prod_{i}{\eta_{i}}}\prod_{i>0}cosk_{n}(K(\mathcal{A}_{i}(U),i))$

where $\eta_{i}:K(\mathcal{A}_{i}(U),i)\to cosk_{n}(K(\mathcal{A}_{i}(U),i))$
is the canonical map to coskeleton.
Also observe that $cosk_{n}(K(\mathcal{A}_{i}(U),i))=*$ for $i>n$. Hence we
have

$P^{(n)}(\mathcal{X}(U))=\prod_{i=1}^{n}{\rm Im\ }(K(\mathcal{A}_{i}(U),i)\to
cosk_{n}(K(\mathcal{A}_{i}(U),i)))=\prod_{i=1}^{n}P^{(n)}(\mathcal{K}(\mathcal{A}_{i}(U),i))$

Since $\mathcal{K}(\mathcal{A}_{i}(U),i)$ is weakly equivalent to
$P^{(n)}(\mathcal{K}(\mathcal{A}_{i}(U),i))$ for $i\leq n$, we get
$\prod_{i=1}^{n}\mathcal{K}(\mathcal{A}_{i},i)(U)$ is weakly equivalent to
$P^{(n)}(\mathcal{X}(U))$ compatible with $n$. Hence, there is weak
equivalence of simplicial sheaves
$\prod_{i=1}^{n}\mathcal{K}(\mathcal{A}_{i},i)\rightarrow P^{n}(\mathcal{X})$.

Step 4: We show that simplicial sheaf
$\prod_{i>0}Ex(\mathcal{K}(\mathcal{A}_{i},i))$ is weakly equivalent to
$\operatorname{holim}_{n}Ex(P^{n}(\mathcal{X}))$. To show that the site
$\rm{(Sm/k)_{\rm\acute{e}t}}$ is not of finite type, it is enough to show that
the homotopy sheaf
$\pi_{0}^{s}(\operatorname{holim}_{n}Ex(P^{n}(\mathcal{X})))\cong\pi_{0}^{s}(\prod_{i>0}Ex(\mathcal{K}(\mathcal{A}_{i},i)))$
is non trivial.

Observe that the map $Ex(\prod_{i=1}^{n}\mathcal{K}(\mathcal{A}_{i},i))\to
Ex(P^{n}(\mathcal{X}))$ is sectionwise weak equivalence of simplicial sets by
[2, Corollary 5.13]. The homotopy limit of simplicial sheaves is defined
sectionwise, then by Homotopy lemma [1, XI, 5.6] the induced map

$\operatorname{holim}_{n}Ex(\prod_{i=1}^{n}\mathcal{K}(\mathcal{A}_{i},i))\longrightarrow\operatorname{holim}_{n}Ex(P^{n}(\mathcal{X}))$

is weak equivalence of simplicial sheaves. Further, we have

$Ex(\prod_{i=1}^{n}\mathcal{K}(\mathcal{A}_{i},i))\longleftarrow\prod_{i=1}^{n}\mathcal{K}(\mathcal{A}_{i},i)\longrightarrow\prod_{i=1}^{n}Ex(\mathcal{K}(\mathcal{A}_{i},i))$

where both maps are weak equivalences, first is part of definition of
resolution functor $Ex$ and second is obtained as finite product of weak
equivalences $\mathcal{K}(\mathcal{A}_{i},i)\rightarrow
Ex(\mathcal{K}(\mathcal{A}_{i},i))$. Then using lifting properties of model
structure, for each $n>0$, we can induce weak equivalences

$Ex(\prod_{i=1}^{n}\mathcal{K}(\mathcal{A}_{i},i))\xrightarrow{f_{n}}\prod_{i=1}^{n}Ex(\mathcal{K}(\mathcal{A}_{i},i))$

which are compatible with $n$. Then by [1, XI, 5.6] we get weak equivalence

$\operatorname{holim}_{n}Ex(\prod_{i=1}^{n}\mathcal{K}(\mathcal{A}_{i},i))\longrightarrow\operatorname{holim}_{n}\prod_{i=1}^{n}Ex(\mathcal{K}(\mathcal{A}_{i},i)).$

Observe that the sequence
$\big{\\{}\prod_{i=1}^{n}Ex(\mathcal{K}(\mathcal{A}_{i},i))\big{\\}}_{n}$ is a
tower of fibrant objects and maps are projections which are fibrations. Then
using [1, XI, 4.1] the natural map

$\lim_{n}\prod_{i=1}^{n}Ex(\mathcal{K}(\mathcal{A}_{i},i))\longrightarrow\operatorname{holim}_{n}\prod_{i=1}^{n}Ex(\mathcal{K}(\mathcal{A}_{i},i))$

is a weak equivalence. Let us denote
$\widetilde{\mathcal{X}}:=\prod_{i>0}Ex(\mathcal{K}(\mathcal{A}_{i},i))=\lim_{n}\prod_{i=1}^{n}Ex(\mathcal{K}(\mathcal{A}_{i},i))$.

Step 5: We show that the homotopy sheaf
$\pi_{0}^{s}(\widetilde{\mathcal{X}})\neq*$.

For each $U\in\rm{(Sm/k)_{\rm\acute{e}t}}$, the homotopy presheaf is given as

$\pi_{0}^{pre}(\widetilde{\mathcal{X}})(U)={\rm
Hom}_{\mathcal{H}_{s}}(U,\prod_{i>0}Ex(\mathcal{K}(\mathcal{A}_{i},i)))=\prod_{i>0}H^{i}(U,\mathcal{A}_{i}).$

where last equality follows from [3, Proposition 1.26, page 57]. Then the
stalk of $\pi_{0}^{s}(\widetilde{\mathcal{X}})$ at geometric point
$\bar{s}={\rm Spec\,}(k^{sep})$ is given by

$\displaystyle\pi_{0}^{s}(\widetilde{\mathcal{X}})_{\bar{s}}=\pi_{0}^{pre}(\widetilde{\mathcal{X}})_{\bar{s}}$
$\displaystyle=\mathop{\varinjlim}\limits_{U}\pi_{0}^{pre}(\widetilde{\mathcal{X}})(U)$
$\displaystyle=\mathop{\varinjlim}\limits_{U}\prod_{i>0}H^{i}(U,\mathcal{A}_{i})$
$\displaystyle=\mathop{\varinjlim}\limits_{k\subset L}(\prod_{i>0}H^{i}({\rm
Spec\,}L,\mathcal{A}_{i}))$

where the colimit runs over étale neighborhoods $U$ of $\bar{s}$ which are
given by finite Galois extensions $L/k$ so that $k\subset L\subset k^{sep}$.
Observe that,

$H^{i}({\rm Spec\,}L,\mathcal{A}_{i})\cong H^{i}({\rm
Spec\,}L,\widetilde{\mathcal{A}_{i}})\cong H^{i}(G_{L},A_{i})$ for all $i>0$

where first isomorphism is by [5, Chatper I, Corollary 3.9.3] and second
isomorphism is by our choice of $\widetilde{\mathcal{A}_{i}}$. Hence we have

$\pi_{0}^{s}(\widetilde{\mathcal{X}})_{\bar{s}}\cong\mathop{\varinjlim}\limits_{k\subset
L}(\prod_{i>0}H^{i}(G_{L},A_{i})).$

Let $\alpha:=\prod_{i}\alpha_{i}\in\prod_{i>0}H^{i}(G_{k},A_{i})$. Then
$\alpha|_{L}\neq 0\ \text{in}\ \prod_{i>0}H^{i}(G_{L},A_{i})$ for each finite
Galois extension $L/k$. Thus, $\alpha$ survives in above colimit, hence
$\pi_{0}^{s}(\widetilde{\mathcal{X}})_{\bar{s}}\neq*$. This implies that the
homotopy sheaf $\pi_{0}^{s}(\widetilde{\mathcal{X}})$ is nontrivial.

Thus, the site $\rm{(Sm/k)_{\rm\acute{e}t}}$ is not of finite type. ∎

######  Remark 3.3.

The hypothesis of Lemma 3.2 also implies that small étale site $({\rm
Spec\,}k)_{\rm\acute{e}t}$ is not of finite type.

###### Proof of 1.3.

By the assumption, the absolute Galois group $G_{k}$ is first-countable, so we
can choose countable nested local basis at identity as open normal subgroups
$\\{G_{L_{i}}\\}_{i\in{\mathbb{N}}}$ which correspond to sequence of finite
Galois extensions

$k\subset L_{1}\subset L_{2}\subset\dots\subset L_{n}\subset\dots\subset
k^{sep}$

such that for any $E/k$, a finite Galois extension, $E\subset L_{m}$ for some
$m$.

We assume that for any finite extension $L/k$, we have $cd(L)=\infty$. For
each $n\in{\mathbb{N}}$, $cd(L_{n})=\infty$, so we can choose nonzero class
$\alpha_{n}\in H^{n}(G_{L_{n}},A_{n})$ for some $G_{L_{n}}$ module $A_{n}$. By
Shapiro’s lemma,

$H^{n}(G_{L_{n}},A_{n})\cong H^{n}(G_{k},M_{n}(A_{n}))$

where $M_{n}(A_{n})\coloneq M^{G_{L_{n}}}_{G_{k}}(A_{n})$ is the induced
$G_{k}$-module. Denote corresponding classes in $H^{n}(G_{k},M_{n}(A_{n}))$ by
$\alpha_{n}$.

By [4, Theorem 6.10.5], for any closed subgroup $H$ of profinite group $G$ and
any discrete $H$-module $A$, the isomorphism of Shapiro’s lemma factors
through restriction map i.e., for each $n\geq 0$ following diagram commutes

$\textstyle{H^{n}(G,M^{H}_{G}(A))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Shapiro}$$\scriptstyle{res}$$\textstyle{H^{n}(H,A)}$$\textstyle{H^{n}(H,M^{H}_{G}(A))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$

where $\nu$ is induced by the counit map $M^{H}_{G}(A)\rightarrow A$.
Thus, the restriction map above is injective and for each $n\in{\mathbb{N}}$,
$(\alpha_{n})_{|L_{n}}\neq 0$ in $H^{n}(G_{L_{n}},M_{n}(A_{n}))$. Furthermore,
given a finite Galois extension $E/k$, there is $m$ such that $E\subset
L_{m}$. Under the composition

$H^{m}(G_{k},M_{m}(A_{m}))\to H^{m}(G_{E},M_{m}(A_{m})|_{E})\to
H^{m}(G_{L_{m}},M_{m}(A_{m}))$

the element $\alpha_{m}\mapsto(\alpha_{m})_{|L_{m}}\neq 0$, hence the image of
$\alpha_{m}$ under

$H^{n}(G_{k},M_{n}(A_{n}))\to H^{n}(G_{E},M_{n}(A_{n})|_{E})$

is non-zero. Thus, the collection $\\{\alpha_{n}\in
H^{n}(G_{k},M_{n}(A_{n}))\\}_{n}$ satisfies the hypothesis of Lemma 3.2, hence
the site $\rm{(Sm/k)_{\rm\acute{e}t}}$ is not of finite type.

Converse follows from Theorem 1.4. ∎

###### Proof of 1.6.

According to the hypothesis, there is a subsequence
$\\{n_{i}\\}_{i\in{\mathbb{N}}}$ such that

$\lim_{i}exp(H^{n_{i}}(G_{k},A_{n_{i}}))=\lim_{i}c_{i}=\infty$

where sequence $\\{c_{i}\\}_{i\in{\mathbb{N}}}$ is either consists of
$\infty^{\prime}s$ only or strictly monotonically increasing. We choose
cohomology classes $\alpha_{n_{i}}\in H^{n_{i}}(G_{k},A_{n_{i}})$ such that
$|\alpha_{n_{i+1}}|>|\alpha_{n_{i}}|$ and $\lim_{i}|\alpha_{n_{i}}|=\infty$
(where for an abelian group $B$ and element $\beta\in B$, the order of $\beta$
is denoted by $|\beta|$ )

Consider any finite Galois extension $L/k$, then the index $[G_{k}:G_{L}]=m$
for some $m\in{\mathbb{N}}$. Consider $n_{i}$ such that $|\alpha_{n_{i}}|>m$.
Under the composition

$H^{n_{i}}(G_{k},A_{i})\xrightarrow{res}H^{n_{i}}(G_{L},A_{i})\xrightarrow{cor}H^{n_{i}}(G_{k},A_{i})$

the element $\alpha_{n_{i}}\mapsto m\alpha_{n_{i}}\neq 0$, hence
$(\alpha_{n_{i}})_{|L}\neq 0$.

Thus, we have a collection $\\{\alpha_{n}\in
H^{n}(G_{k},A_{n})\\}_{n\in{\mathbb{N}}}$, possibly with $\alpha_{n}=0$ and
$A_{n}=0$ whenever $n\neq n_{i}$ for any $i$, which satisfies hypothesis of
Lemma 3.2, hence the site $\rm{(Sm/k)_{\rm\acute{e}t}}$ is not of finite type.
∎

###### Proof of 1.7.

By the hypothesis, there is a subsequence of primes
$\\{p_{i}\\}_{i\in{\mathbb{N}}}$ such that $\lim_{i}cd_{p_{i}}(k)=\infty$ and
sequence {$cd_{p_{i}}(k)\\}_{i\in{\mathbb{N}}}$ either consists of
$\infty^{\prime}s$ only or is strictly monotonically increasing.

For a prime $p_{1}$ there is a $G_{k}$-module $A_{1}$ and a cohomology class
$\alpha_{n_{1}}\in H^{n_{1}}(G_{k},A_{1})$ such that $|\alpha_{n_{1}}|\geq
p_{1}$. For a prime $p_{2}$, since $cd_{p_{2}}(k)=\infty$ or
$cd_{p_{2}}(k)>cd_{p_{1}}(k)$, we can choose $G_{k}$-module $A_{2}$ and class
$\alpha_{n_{2}}\in H^{n_{2}}(G_{k},A_{2})$ such that $|\alpha_{n_{2}}|\geq
p_{2}$ and $n_{2}>n_{1}$. Inductively on $i$, we have classes
$\\{\alpha_{n_{i}}\in H^{n_{i}}(G_{k},A_{i})\\}_{i\in{\mathbb{N}}}$ such that
$|\alpha_{n_{i}}|\geq p_{i}$ and $n_{i+1}>n_{i}$ for each $i$. Thus the
hypothesis of Theorem 1.6 is satisfied and the site
$\rm{(Sm/k)_{\rm\acute{e}t}}$ is not of finite type. ∎

## References

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∎

11institutetext:  H. Joseph, B. Carboni, G. Quaranta, W. Lacarbonara
22institutetext: Department of Structural and Geotechnical Engineering,
Sapienza University of Rome, Via Eudossiana 18,
00184 Rome, Italy

# Deep learning architectures for data-driven damage detection in nonlinear
dynamic systems

Harrish Joseph  Giuseppe Quaranta  Biagio Carboni  Walter Lacarbonara

(Received: date / Accepted: date)

###### Abstract

The primary goal of structural health monitoring is to detect damage at its
onset before it reaches a critical level. The in-depth investigation in the
present work addresses deep learning applied to data-driven damage detection
in nonlinear dynamic systems. In particular, autoencoders (AEs) and generative
adversarial networks (GANs) are implemented leveraging on 1D convolutional
neural networks. The onset of damage is detected in the investigated nonlinear
dynamic systems by exciting random vibrations of varying intensity, without
prior knowledge of the system or the excitation and in unsupervised manner.
The comprehensive numerical study is conducted on dynamic systems exhibiting
different types of nonlinear behavior. An experimental application related to
a magneto-elastic nonlinear system is also presented to corroborate the
conclusions.

###### Keywords:

Autoencoder Convolutional neural network Damage detection Data-driven system
identification Deep learning Generative adversarial network Random vibrations
Structural health monitoring

††journal:

##  1 Introduction

Damage is any unintended alteration in material and/or geometric properties
that may impact a structural system, including its boundary and connectivity
conditions, leading to detrimental effects on both current and future
performance (e.g., Worden et al., 2007). Early damage detection is crucial for
preserving reliability and performance of structural systems throughout their
lifespan by enabling timely proactive maintenance. Vibration-based methods are
particularly attractive for this task because they can detect damage analyzing
the system dynamic response, also when damage is not severe enough to be
visually apparent. The feasibility of vibration-based damage detection methods
has been extensively investigated, yet most efforts so far have focused on
dynamic systems exhibiting linear behavior. Carrying out a complete damage
identification procedure is a multi-level process that encompasses damage
detection, damage localization, damage quantification, and residual life
prediction (Rytter, 1993). By leveraging on sensing, data acquisition and data
transfer systems, structural health monitoring (SHM) techniques are aimed at
identifying the damage by means of suitable processing and analysis methods
(Karakostas et al., 2023). Damage detection can be possibly carried out by
means of pure data-driven methods, and it is especially important to prevent
costly maintenance plans or, even worse, important social, economic and
environmental consequences. Early detection of damage is crucial to optimize
proactive maintenance interventions before the onset of a critical system
state. Vibration-based methods are very common in this context because of the
availability of low-cost sensing solutions that can be deployed with minimal
interference with the construction. Their implementation is based on the
premise that the damage can alter the overall dynamic behavior of the
structure. The main advantage is that damage can be detected from the measured
dynamic response of the structure at the very early stage without the need to
be developed to the level that it is visible.

Several vibration-based methods have been developed either in frequency or
time domain by assuming that the structure exhibits a linear behavior before
and after damage (Hou and Xia, 2021). The widespread use of mode shapes,
natural frequencies and damping ratios for damage detection in linear dynamic
systems is supported by well-established output-only techniques for
operational modal analysis of structures. In particular, modal curvatures have
proven to be especially useful in detecting damages in linear structures
(e.g., Pandey et al., 1991; Quaranta et al., 2016). Conversely, natural
frequencies turn out to be almost insensitive to several damage scenarios
(e.g., Doebling et al., 1998; Fan and Qiao, 2011) while damping ratios do not
always provide consistent results (e.g., Salawu and Williams, 1995; Cao et
al., 2017). Time-domain methods have been also explored for damage detection
in linear structures (Fassois and Sakellariou, 2007), and they usually
leverage on state-space (e.g., Döhler et al., 2014; Greś et al., 2021) or
autoregressive-type models (e.g., Sohn and Farrar, 2001; Lakshmi et al.,
2017).

Damage detection in case of nonlinear dynamics has received considerably less
attention. Initial attempts were based on the assumption that the undamaged
structure behaves linearly while the occurrence of damage causes the
transition to a nonlinear response (Worden et al., 2008). However, the
hypothesis of initial linear behavior does not hold true for a large class of
dynamic systems. Some vibration-based methods have been thus developed for
initially nonlinear dynamic systems. For instance, Adams and Nataraju (2002)
have shown that model reduction near bifurcations can be useful to identify
certain features that can facilitate damage detection in nonlinear dynamic
systems. The extension of the modal-based damage detection approach to
nonlinear dynamic systems has been pursued by Lacarbonara et al. (2016)
through the use of nonlinear normal modes. This analytical study proved that
the sensitivity of the effective nonlinearity coefficients regulating the
nonlinear modal backbones is higher than the sensitivity of the linear
frequencies with respect to damage. Such theoretical evidence was later
confirmed in some experimental results reported by Civera et al. (2019) and
Carboni et al. (2022). Prawin and Rao (2019) addressed the damage detection
for nonlinear dynamic systems in the frequency domain using an improved
describing function. Among the existing time-domain methods, several proposals
deal with the use of Volterra series since such method provides an attractive
way to describe the response of nonlinear dynamic systems. For example, Shiki
et al. (2017) and Villani et al. (2019a, b) applied the discrete-time Volterra
series and a stochastic version of Volterra series, respectively, to detect
damage in dynamic systems exhibiting a nonlinear behavior even in the
undamaged configuration. The output-only version of the Volterra series
proposed by Peng et al. (2021) for nonlinear structural damage detection
employs the structural responses at two different locations in order to
identify the kernel function parameters and to evaluate the contribution of
the nonlinear components.

Recent advances in machine learning methods are paving the way towards
significant advances in damage detection. In particular, there is great
potential in deep learning algorithms for data-driven, vibration-based damage
detection (Avci et al., 2021). Autoencoders (AEs) and generative adversarial
networks (GANs) appear to be among the most promising deep learning
architectures for this task, although their applications are still limited and
primarily focused on linear dynamic systems. Both these architectures employ
an unsupervised learning paradigm, which obviates the need for extensive
labeled datasets for training (Qi and Luo, 2020). For example, Pathirage et
al. (2018) proposed an AE-based damage detection approach where the natural
frequencies and mode shapes of the linear dynamic system are employed as input
data. The AE implemented by Ma et al. (2020) utilizes the measured dynamic
response for detecting damage in a linear system while the application
reported by Shang et al. (2021) is based on the use of the multi-dimensional
cross-correlation functions computed from linear vibrations. Boccagna et al.
(2023) performed damage detection via AE by using univariate statistics
estimated from the measured linear dynamic response after performing
dimensionality reduction through principal component analysis. The AE-based
damage detection approach proposed by Li et al. (2023a) makes use of the power
cepstral coefficients extracted from the acceleration response of a linear
dynamic system to provide a compact representation of its linear modal
properties. Applications of GANs for damage detection are even fewer. The
state-of-the-art review by Avci et al. (2021) highlights the lack of studies
on the use of GANs for damage detection in civil structures until 2019. Some
examples of vibration-based damage detection in linear dynamic systems via
GANs appeared only recently (Luleci et al., 2023; Luo et al., 2023).

Machine learning algorithms applied for data-driven, vibration-based damage
detection in nonlinear dynamic systems are infrequent. A seminal work in this
context is due to Bornn et al. (2010), who reported an experimental
application about the use of the autoregressive support vector machine for
damage detection in a small-scale laboratory shear-type system subject to
random base motion with constant intensity. This system exhibited nonlinear
behavior due to impacts and experienced damage from completely loosened bolts
in a single column. The use of deep learning techniques has only recently
started to emerge. For instance, Jin et al. (2023) successfully applied a deep
multi-kernel extreme learning machine for fault diagnosis of rotating
machineries. Kim et al. (2022) proposed the application of AEs for the
identification of reduced-order model while Zhang et al. (2023) presented a
numerical investigation about the use of AEs for damage detection in
conventional bearings for seismic isolation. In this case, the damage was
simulated through the complete loss of functionality of one or two bearings at
the corners of the isolation level, whereas seismic ground motion records were
selected in such a way to match a target seismic spectrum. Nonlinear dynamical
systems identification through GANs was recently addressed by Yu and Liu
(2024), but no proposals are known to exist about their application in damage
detection problems.

Indeed, output-only, vibration-based damage detection in time-domain of
initially nonlinear systems is very challenging. This is because the features
of nonlinear dynamic responses can change considerably in time, especially
under varying excitation conditions. As a consequence, discerning the effects
induced by damage from those caused by the nonlinear response becomes a truly
challenging task (Bornn et al., 2010). This is a significant concern for
output-only vibration-based damage detection by means of machine learning
methods that deserves in-depth investigations.

This work explores the use of deep learning algorithms for data-driven,
vibration-based damage detection in nonlinear dynamic systems. Within this
framework, the original contribution here lies in a comprehensive
investigation aimed at evaluating the comparative effectiveness and
performance of AEs and GANs. The remaining part of the present work is
organized as follows. Sections 2 and 3 illustrate the adopted deep learning
architectures for damage detection in nonlinear dynamic systems. Section 4
presents the results of an extensive numerical investigation while Section 5
illustrates an experimental application. The most important outcomes of the
present research are finally included in Section 6.

##  2 Autoencoder for damage detection

The autoencoder (AE) originally proposed by Rumelhart et al. (1986) is an
unsupervised algorithm aimed at performing the input reconstruction with the
least possible amount of distortion. Its main components are the encoder, a
latent feature representation, and the decoder. Since comprehensive state-of-
the-art reviews about AEs are available (e.g., Kiranyaz et al., 2021; Yang et
al., 2022; Li et al., 2023b), only details related to the adopted
implementation for damage detection applications in nonlinear dynamic systems
are presented hereafter.

Let $\mathbf{x}\in\mathbb{R}^{n}$ be the normalized time-history response of
the nonlinear dynamic system, the encoder is a function ${\bf
g}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ such that:

$\mathbf{h}={\bf g}(\mathbf{x}),$ (1)

where $\mathbf{h}\in\mathbb{R}^{m}$ is the latent feature representation of
$\mathbf{x}$. The decoder is another function ${\bf
f}:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ such that:

$\tilde{\mathbf{x}}={\bf f}(\mathbf{h})={\bf f}({\bf g}(\mathbf{x})),$ (2)

where $\tilde{\mathbf{x}}\in\mathbb{R}^{n}$ is the input reconstruction. In
most typical AE architectures, the functions $\bf f$ and $\bf g$ representing
the encoding and decoding blocks are artificial neural networks (NNs). In
particular, convolutional NNs (CNNs) are mainly employed in AEs because they
excel at recognizing the most important signal features and at capturing
relevant patterns in the data. 1D CNNs are preferred in the present work
because of their minimum computational complexity. It is noted that a
normalized input is fed into the AE algorithm so as to reduce the variability
among data points from different scales and to speed up the convergence of the
networks during training phase. Scaling the signals to a uniform bounded range
mitigates the dominance of higher magnitude features over smaller ones, which
allows to recognize subtle patterns of damage without bias towards more
prominent signal features.

The components of the implemented CNN-based AE architecture are input layer,
convolutional layer, pooling layer, fully connected layer and output layer.
The input layer of the CNN aims at importing the normalized 1D time-history
response data of the nonlinear dynamic system. Data normalization is
accomplished through min-max scaling within the range $[0,1]$. A deep AE is
here considered, which means that it involves more than one convolutional
layer in the encoding and decoding blocks. Within a convolutional layer, the
convolution operator applies on a given input (i.e., the output from the
previous layer) using a convolution filter to produce a feature map, whereas a
nonlinear activation function is next employed to obtain the output features.
The convolutional operation involves sliding a small filter (also known as
kernel) and computing dot products between the filter weights and the input
values at each location as follows:

${{}^{\ell+1}}y_{i}(j)={{}^{\ell}}\mathbf{K}_{i}*{{}^{\ell}}\hat{\mathbf{x}}(j)+{{}^{\ell}}b_{i},$
(3)

where ${{}^{\ell}}\mathbf{K}_{i}$ and ${{}^{\ell}}b_{i}$ denote the weights
and the bias of the $i$th filter in the $\ell$th layer, respectively,
${{}^{\ell}}\hat{\mathbf{x}}(j)$ is the $j$th receptive field in the $\ell$th
layer (i.e., the local region representing the portion of the input values
that is being processed at a given step of the convolution operation), and
${{}^{\ell+1}}y_{i}(j)$ is the input of the $j$th artificial neuron in the
frame $i$ of the following $(\ell+1)$th layer. One filter corresponds to one
frame in the following layer, and the number of frames is also known as layer
depth. It is noted that the same set of learnable parameters (weights) across
different receptive fields is utilized. Such weight-sharing allows the network
to detect patterns and features regardless of their location. This is because
the same filter is applied across the entire input, reducing the computational
effort. Zero padding is applied to both the beginning and the end of the input
to preserve the original length of the convolution result. This approach
enhances the network’s ability to learn features at the boundaries, which is
particularly important for short inputs. The stride, defined as the number of
positions by which the filter moves across the input during the convolution
operation, is chosen to ensure consistent alignment between the input and
output across the different layers of the network.

The implementation of artificial neurons with a nonlinear activation function
after the convolution operations is of utmost importance. In fact, without a
nonlinear activation function, the output of each convolution would be a
linear transformation of the input, which is unfit to capture inherent complex
patterns from the data. Conversely, introducing nonlinearity allows the
network to learn and represent complex relationships in the input. In the
present study, the Leaky ReLU activation function is utilized. It was proposed
to address one of the main limitations of the traditional ReLU activation
function, which is due to the fact that it sets all negative inputs to $0$,
potentially leading to vanishing gradients (Glorot et al., 2011; Maas et al.,
2013). The Leaky ReLU activation function mitigates this issue by introducing
a small, non-null gradient when the activation function is negative as
follows:

${{}^{\ell+1}}a_{i}(j)=\psi({{}^{\ell+1}}y_{i}(j))=\begin{cases}{{}^{\ell+1}}y_{i}(j)&\text{if
}{{}^{\ell+1}}y_{i}(j)>0\\\ {{}^{\ell+1}}\alpha
y_{i}(j)&\text{otherwise}\end{cases},$ (4)

where ${{}^{\ell+1}}a_{i}(j)$ is the activation value of the convolution
operation result ${{}^{\ell+1}}y_{i}(j)$ in Eq. (3) while $\alpha$ is the
slope coefficient, which controls the angle of the negative slope in case of
negative input values. By using the Leaky ReLU activation function, the
network can learn from negative inputs, enhancing its robustness and ability
to capture complex patterns. Furthermore, using min-max scaling in combination
with Leaky ReLU activation function ensures that the majority of input falls
into an active learning range, reducing the risk of inactive neurons.

A pooling layer after the convolutional layer is meant at lowering both the
size of the features and the parameters of the network. This, in turn, forces
the network to learn only the most important features of the input.
Additionally, it reduces training time and mitigates overfitting. The max-
pooling layer is employed. Let ${{}^{\ell}}q_{i}(s)$ be the value of the
artificial neuron $s$ in the $i$th frame of the $\ell$th layer, the max-
pooling transformation applies as follows:

${{}^{\ell+1}}p_{i}(j)=\max_{(j-1)W+1\leq s\leq jW}{{{}^{\ell}}q_{i}(s)},$ (5)

where ${{}^{\ell+1}}p_{i}(j)$ is the corresponding value of the artificial
neuron in the $(\ell+1)$th layer of the pooling operation and $W$ is the
pooling width.

By adjusting padding and stride, convolutional layers for the encoder are
designed to downscale the input together with pooling layers, while
convolutional layers in the decoder are intended to upscale the input (through
transposed convolution operations, and pooling layers do not take place in
this case). Convolutional layers also serve as output layer of the AE.
Conversely, fully connected layers are implemented in the latent feature
representation to effectively lower the dimensionality of the input, as this
task would be cumbersome with convolutional layers. Controlling the size of
the latent representation is important because it plays an important role in
determining how the machine learning model performs on the assigned data. A
fully connected layer, also known as dense layer, is a standard layer where
each artificial neuron is connected to every artificial neuron in the previous
layer, similar to a traditional NN (e.g., Quaranta et al., 2020). Before the
fully connected layer applies, the output from the last pooling layer in the
encoding block is flattened.

The AE is required to reconstruct the input time-history response of the
nonlinear dynamic system well enough while producing a meaningful latent
representation in such a way to avoid the overfitting phenomenon. This is
accomplished through proper training of the AE, which basically consists of
finding the optimal set of hyperameters $\boldsymbol{\theta}$ ruling the
functions ${\bf f}$ and ${\bf g}$ that represent the encoding and decoding
blocks. To this end, two main strategies leveraging on parsimony are adopted
(Kutz and Brunton, 2022) and are based on creating a bottleneck and adding a
regularization term. A bottleneck involves reducing the dimensionality $m$ of
the latent features compared to that of the data $n$. Including a
regularization term in the training process also prevents the model from
fitting the noise in the training data too closely. In the present work, the
$L_{2}$ technique is implemented, and thus the AE training seeks the solution
of the following optimization problem:

$\min_{\boldsymbol{\theta}}{\left\\{\textsf{E}\left[\Delta\left(\mathbf{x},f(g(\mathbf{x});\boldsymbol{\theta})\right)\right]+\lambda\|\boldsymbol{\theta}\|_{2}^{2}\right\\}},$
(6)

where the first term is a loss function that quantifies how input and output
of the AE differ each other, on average, across all $\mathbf{x}$ (E represents
the expectation operator), while $\lambda$ and $\|\boldsymbol{\theta}\|_{2}$
in the second term are the regularization coefficient and the $L_{2}$-norm of
the hyperameters $\boldsymbol{\theta}$ involved in the functions ${\bf f}$ and
${\bf g}$, respectively. The adopted loss function in the present study is the
mean absolute error (MAE). The set of tuned hyperparameters
$\boldsymbol{\theta}$ includes kernel size, filter numbers, batch size (i.e.,
number of input values processed simultaneously during one training step), and
latent filters. The values of slope coefficient $\alpha$ and regularization
coefficient $\lambda$ are instead established before training (they are set to
$0.2$ and in the range $10^{-4}$ to $10^{-8}$, respectively).

The adaptive moment estimation (ADAM) optimizer is adopted to solve Eq. (6).
It is a gradient-based technique (Kingma and Ba, 2017) that combines the
advantages of the adaptive gradient algorithm and root mean square
propagation. ADAM computes adaptive learning rates for each hyperparameter
based on estimates of the first and second moments of the gradient. The update
rules in ADAM are the following:

$\mathbf{m}_{\tau}=\beta_{1}\mathbf{m}_{\tau-1}+(1-\beta_{1})\nabla_{\tau},$
(7a)
$\mathbf{v}_{\tau}=\beta_{2}\mathbf{v}_{\tau-1}+(1-\beta_{2})\nabla_{\tau}^{2},$
(7b) $\hat{\mathbf{m}}_{\tau}=\frac{\mathbf{m}_{\tau}}{1-\beta_{1}^{\tau}},$
(7c) $\hat{\mathbf{v}}_{\tau}=\frac{\mathbf{v}_{\tau}}{1-\beta_{2}^{\tau}},$
(7d)
$\boldsymbol{\theta}_{\tau+1}=\boldsymbol{\theta}_{\tau}-\frac{\eta}{\sqrt{\hat{\mathbf{v}}_{\tau}}+\epsilon}\hat{\mathbf{m}}_{\tau},$
(7e)

where $\mathbf{m}_{\tau}$ and $\mathbf{v}_{\tau}$ are the updated first (i.e.,
mean) and second (i.e., uncentered variance) moments of the gradients,
respectively, $\beta_{1}$ and $\beta_{2}$ are the exponential decay rates for
these moment estimates (set to 0.9 and 0.999, respectively), $\nabla_{\tau}$
is the objective function gradient at the iteration $\tau$, $\eta$ is the
learning rate (in the range $10^{-2}$ to $10^{-4}$), and $\epsilon$ is a small
scalar added to improve numerical stability (set to $10^{-8}$), while
$\boldsymbol{\theta}_{\tau+1}$ represents the updated hyperparameters for the
next iteration (aka epoch), or the outcome of the optimization if the
convergence criterion is fulfilled. To further strengthen the network against
overfitting phenomenon, early stopping applies by setting a low value of the
patience (i.e., the number of epochs the training process continues without
improvement) between 4 and 6.

The suitability of the AE for data-driven damage detection relies on the
evidence that, once properly trained over time series data collected from an
undamaged dynamic system, it will turn into a large loss function value when
attempting to reconstruct the response of a damaged one. Figure 1 illustrates
the implemented AE architecture based on 1D CNNs for damage detection in
nonlinear dynamic systems. The corresponding algorithm was developed using
TensorFlow (Abadi et al., 2015).

Figure 1: Implemented AE architecture based on 1D CNNs for damage detection in
nonlinear dynamic systems.

##  3 Generative adversarial network for damage detection

A generative adversarial network (GAN) is an unsupervised algorithm originally
proposed by Goodfellow et al. (2014) that consists of two models called
generator and discriminator. The generator tries to create realistic data,
while the discriminator aims to differentiate between real data and the fake
data created by the generator. The details concerning the specific
implementation of the GAN for damage detection applications in nonlinear
dynamic systems will be presented hereafter while more general information can
be found elsewhere (e.g., Gui et al., 2021; Creswell et al., 2018; Xia et al.,
2022).

On the one hand, the generator can be represented as a differentiable function
$G(\mathbf{z};\boldsymbol{\theta}_{G})$ where $\mathbf{z}$ is a random noise
vector from a distribution $\varphi_{\mathbf{z}}(\mathbf{z})$ and
$\boldsymbol{\theta}_{G}$ is a set of hyperparameters. It produces synthetic
data samples, which should resemble real data samples. On the other hand, the
discriminator is another differentiable function
$D(\mathbf{x};\boldsymbol{\theta}_{D})$ where $\mathbf{x}$ is a data sample
from a distribution $\varphi_{data}(\mathbf{x})$ and $\boldsymbol{\theta}_{D}$
is a set of hyperparameters. It produces a scalar output representing the
probability that $\mathbf{x}$ comes from the real data distribution rather
than the generator’s distribution. Data normalization through min-max scaling
is also performed in GAN to enhance the performance of the algorithm as well
as the efficiency of the training phase.

The training of GAN is a non-cooperative game. Hence, GAN optimization is
attained at the Nash equilibrium point (Ratliff et al., 2013) corresponding to
a stable state where neither the generator, nor the discriminator obtains a
beneficial effect (i.e., gain) by deviating from the current configuration. At
this equilibrium point, the generator produces data that are so realistic that
the discriminator cannot distinguish whether they are real or fake, and the
discriminator cannot improve its accuracy by further adjustments. The training
of discriminator $D$ and generator $G$ in the GAN requires the solution of the
following optimization problems:

$\max_{\boldsymbol{\theta}_{D}}\\{\textsf{E}_{\mathbf{x}\sim\varphi_{data}(\mathbf{x})}[\log
D(\mathbf{x};\boldsymbol{\theta}_{D})]+\textsf{E}_{\mathbf{z}\sim\varphi_{\mathbf{x}}(\mathbf{z})}[\log{(1-D(G(\mathbf{z}|\boldsymbol{\theta}_{G});\boldsymbol{\theta}_{D}))}]\\},$
(8)
$\min_{\boldsymbol{\theta}_{G}}\\{\textsf{E}_{\mathbf{z}\sim\varphi_{\mathbf{z}}(\mathbf{z})}[\log{(1-D(G(\mathbf{z};\boldsymbol{\theta}_{G})|\boldsymbol{\theta}_{D}))}]\\}.$
(9)

Altogether, Eqs. (8)-(9) are equivalent to the following min-max optimization
problem:

$\min_{\boldsymbol{\theta}_{G}}{\max_{\boldsymbol{\theta}_{D}}\\{\textsf{E}_{\mathbf{x}\sim\varphi_{data}(\mathbf{x})}[\log
D(\mathbf{x};\boldsymbol{\theta}_{D})]+\textsf{E}_{\mathbf{z}\sim\varphi_{\mathbf{x}}(\mathbf{z})}[\log{(1-D(G(\mathbf{z};\boldsymbol{\theta}_{G});\boldsymbol{\theta}_{D}))}]\\}},$
(10)

wherein the objective function is related to the Jensen-Shannon divergence
between $\varphi_{data}(\mathbf{x})$ and $\varphi_{\mathbf{z}}(\mathbf{z})$
(Gui et al., 2021) (E represents the expectation concerning the distribution
appointed in the subscript). The GAN version based on the optimization problem
in Eq. (10) is commonly referred to as vanilla GAN.

A deep GAN is here implemented by stacking multiple 1D CNNs. In particular, 1D
convolutional layers according to Eqs. (3)-(4) are used for both generator and
discriminator (wherein transposed convolution operations are employed for the
generator). In this regard, a significant novelty deals with the output layer
of the discriminator, which implements a Sigmoid activation function as
follows:

${{}^{\ell+1}}a_{i}(j)=\psi({{}^{\ell+1}}y_{i}(j))=\dfrac{{{}^{\ell+1}}y_{i}(j)}{1+\text{e}^{{{}^{\ell+1}}y_{i}(j)}}.$
(11)

The hyperparameters are tuned alternatively solving Eqs. (8)-(9) throughout
the training stage by means of ADAM update rules in Eq. (7e). During training,
the hyperparameters of one model are updated, while the hyperparameters of the
other are fixed. The set of tuned hyperparameters for GAN includes number
filters, kernel size, stride length, and depth of the neural network (these
hyperparameters are different for the generator and discriminator). The
training process for GAN runs until a number of epochs between $1,000$ and
$2,000$ is achieved (no early stopping applies in this case). The
regularization of GAN discriminator is accomplished via dropout. Accordingly,
throughout the training, an artificial neuron is temporarily dropped at each
iteration (i.e., all the inputs and outputs to this artificial neuron are
disabled at the current iteration) with an assigned probability. The dropped-
out artificial neurons are resampled with the same probability at every
training step, so a dropped out neuron at one step can be active at the next
one.

There are no known proposals regarding the use of GAN in damage detection
problems. Nonetheless, the suitability of GAN for data-driven damage detection
stems on the fact that, upon proper training with time series data collected
from an undamaged system, the discriminator will be no longer able to
differentiate between actual undamaged data and synthetic data produced by the
generator. Conversely, the discriminator will label the data from a damaged
system as fake because they do no longer resemble those of the undamaged
system. In this regard, it is interesting to note that, by implementing the
Sigmoid activation function at the output layer of the discriminator, a
quantitative assessment of the dynamic system state is obtained from the
discriminator. In fact, numerical values of the discriminator output close to
$1$ are associated with an undamaged state while lower values approaching to 0
highlight a deviation from the (trained) undamaged condition, thereby pointing
out the presence of the damage. Hence, the discriminator output can be
exploited to compute a damage index.

This vanilla GAN architecture based on 1D CNNs for damage detection in
nonlinear dynamic systems has been implemented using TensorFlow (Abadi et al.,
2015). Figure 2 shows pictorially the architecture of the GAN.

Figure 2: Implemented GAN architecture based on 1D CNNs for damage detection
in nonlinear dynamic systems.

##  4 Numerical investigation

Numerical simulations are initially performed to evaluate the effectiveness of
both AE and GAN in detecting damage within nonlinear dynamic systems from
their time-domain response only. To replicate the most common and challenging
scenario, the nonlinear dynamic systems under consideration are subjected to
random vibrations of varying intensity, and the response is contaminated by
white Gaussian noise with a noise level of 10%.

The database for training and validating both deep learning architectures
include samples of undamaged system displacements over $2$ s time windows,
assuming a sampling rate of 250 Hz. A total of 2,000 time series was collected
for such tasks: 80% of the generated time-history samples is utilized for
training the algorithms (i.e., training set), while the remaining 20% is
allocated for their validation (i.e., validation set). The validation dataset
was used to check if the training of the network converged properly, avoiding
overfitting or underfitting. The full database thus consists of about 1 h of
recording of the undamaged system response under different magnitudes of the
input random excitation. It is highlighted that neither the applied dynamic
loading conditions, nor the dynamic system characteristics, are ultimately
involved in the implementation of the considered deep learning architectures.
This means that both AE and GAN are meant to perform output-only damage
detection of unknown dynamic systems.

The selected configuration of AE and GAN for each application is first
established through preliminary simulations, and then its trainable parameters
are found. Once both deep learning algorithms have been trained and validated
using the samples of undamaged system response, they are employed to detect
damage. To this end, 2,000 time series of noisy system displacements, each of
2 s duration, were generated for constant damage levels ranging from 5% to
30%. Therefore, both AE and GAN are required to detect damage, if any, from
about 1 h of recording of the system response. While low damage levels are
considered to test the capability of detecting the onset of damage, the
results for increasing damage levels are examined to inspect the proper
working of the implemented deep learning schemes.

###  4.1 Single-degree-of-freedom system with cubic nonlinearity

A large variety of dynamic systems can exhibit a nonlinear behavior for high
levels of excitation due to geometric or material nonlinear effects. The
actual response of such systems can be effectively simulated by means of a
restoring force that is dependent upon the cube of the displacement, known as
a Duffing-type system. Hence, this class of nonlinear dynamic systems provides
a relevant benchmark for evaluating the effectiveness of damage detection
techniques in practical applications. The equation of motion for a single-
degree-of-freedom (1-DOF) system with cubic nonlinearity and linear viscous
damping is the following:

$M\ddot{x}+C\dot{x}+K_{1}x+K_{3}x^{3}=F(t),$ (12)

where $x(t)$ represent the displacement of the system at time $t$ (overdots
denote differentiation with respect to time $t$), whereas $M$ is the mass and
$C$ is the viscous damping coefficient. Moreover, $K_{1}$ and $K_{3}$ are the
linear and nonlinear stiffness terms, respectively, while $F$ is the external
time-varying load. The numerical values of the system parameters in this
numerical application are set to: $M=412$ kg, $C=405.95$ kg/s (corresponding
to a damping ratio of 1%), $K_{1}=1$ kN/mm, $K_{3}=0.001$ kN/mm3. It is
assumed that the presence of damage corresponds to a reduction of both the
linear and nonlinear stiffness by the same amount, without effects on mass and
damping.

The dynamics of such system are first investigated by computing its frequency
response curves (FRCs) considering a harmonic base excitation such that
$F(t)=-AM\cos\Omega t$, where $A$ is the peak acceleration value, $\Omega$ is
the excitation frequency. Results for the undamaged and damaged system are
plotted in Fig. 3 assuming $A$ between $0.003g$ and $0.1g$. They show that the
considered dynamic system exhibits a linear behavior for $A$ less than or
equal to $0.01$ g while a nonlinear behavior is apparent otherwise. In
particular, a moderate nonlinear response is observed for $A$ between $0.01g$
and $0.06g$, whereas a stronger nonlinear behavior occurs for higher values of
the input excitation. For the undamaged system, if $A$ is less than or equal
to $0.01g$, then the resonant frequency is 7.8 Hz, while it shifts to $8.8$ Hz
(increase by 12.8%) when $A$ is $0.1g$.

Figure 3: Frequency response curves (FRCs) of the 1-DOF system for different
excitation amplitudes and damage levels.

Figure 3 is also useful to explain why time-domain, output-only damage
detection in nonlinear dynamic systems subject to varying dynamic excitation
is so challenging as pointed out by Bornn et al. (2010). In fact, Fig. 3 shows
that the shift of the resonant frequency due to the hardening-type nonlinear
response can mislead the fault recognition, thus hindering an accurate early
damage detection.

The effectiveness of both AE and GAN is investigated through numerical
simulations by solving Eq. (12) under random excitation such that
$F=-\ddot{x}_{b}M$, where $\ddot{x}_{b}$ is a white noise. The time-histories
displacement response of the undamaged and damaged system are computed once
time-history samples of $\ddot{x}_{b}$ are generated and the corresponding
peak values are scaled to a uniform random value between $0.01g$ and $0.1g$ so
as to focus on the nonlinear response only, in agreement with Fig. 3. Details
about the configuration of both AE and GAN are provided in Tab. 1.

Table 1: Configuration of the deep learning architectures for damage detection in the 1-DOF system. Type | Component |  Convolution layers |  Trainable parameters |  Kernel size |  Number of filters | Strides
---|---|---|---|---|---|---
AE | Encoder | 3 | 3,767 | (100,50,25) | (10,5,2) | (1,1,1)
Decoder | 3 | 66,056 | (25,50,100) | (5,10,1) | (2,2,1)
GAN | Generator | 3 | 16,018 | (2,3,6) | (32,32,1) | (2,1,2)
Discriminator | 3 | 8,117,729 | (6,6,6) | (64,64,64) | (2,2,2)

The representative results reported in Fig. 4 confirm that the trained AE is
able to reconstruct the time-history displacement of the undamaged nonlinear
system (the system response under the highest excitation is plotted herein).
Conversely, it is no longer able to achieve similar accuracy levels when it
attempts to reconstruct the displacement time-history response of the damaged
system. This mismatch between input and output of the AE highlights a
deviation from the conditions of the training process, and thus points to the
occurrence of the damage. The greater the damage level, the larger the
difference between the input and output of the AE. The GAN operates
differently, requiring the generator to accurately mimic the undamaged system
response. This maximizes the damage sensitivity of the discriminator
throughout the training process. In this regard, a representative pair of
actual and generated displacement time-histories are plotted in Fig. 5 (the
dominant frequency herein is about 8.8 Hz, which corresponds to the highest
excitation level according to Fig. 3). The high similarity between them in the
time-frequency domain demonstrates the fidelity of the GAN in simulating the
dynamics of the undamaged nonlinear system. This, in turn, implies that the
discriminator has been successfully trained and will be thus able to recognize
accurately a deviation from the training conditions, such as the occurrence of
damage.

Figure 4: Reconstruction of the 1-DOF system response through AE for different
damage levels. Figure 5: Time-history and corresponding time-frequency
representation via wavelet transform of the undamaged 1-DOF system response:
actual data and results carried out from the GAN generation block.

Figure 6 compares the performance of both deep learning architectures in
detecting damage. Each dot in the scatter plots represents the reconstruction
loss (in terms of MAE) of the AE and the discriminator output of the GAN for
each time series sample given the damage level, while the trend lines connect
the corresponding average values. Since the discriminator output is bounded
between 0 and 1, a damage index is defined as the complement of the
discriminator output. Figure 6 shows that reconstruction loss values shift
upwards (i.e., the reconstruction loss increases) as the damage severity
grows. It can also be inferred from Fig. 6 that the discriminator outputs are
close to 1 when there is no damage, while a growing number of outcomes
approach 0 as the damage level increases. Therefore, the relative variation of
the normalized reconstruction loss and that of the damage index based on the
discriminator loss properly correlate with the damage level. Notably, a
significant deviation from the reference value corresponding to the undamaged
configuration already occurs at the lowest level of damage (i.e., relative
variation larger than 10% for the lowest damage severity), which demonstrates
a satisfactory sensitivity of both deep learning architectures.

Figure 6: Damage sensitivity of the two deep learning architectures for the
1-DOF system: reconstruction loss values obtained by AE and relative variation
of its normalized average value, together with the discriminator output values
obtained by GAN and relative variation of the associated average damage index.

###  4.2 Two-degree-of-freedom system with cubic nonlinearity

Damage detection in a two-degree-of-freedom (2-DOF) nonlinear dynamic system
with cubic nonlinearity and linear viscous damping is considered next. The
corresponding equations of motion are the following:

$\displaystyle M_{1}\ddot{x}_{1}$
$\displaystyle+C_{1}\dot{x}_{1}+K_{11}x_{1}+K_{13}x_{1}^{3}$ (13a)
$\displaystyle-
C_{2}\left(\dot{x}_{2}-\dot{x}_{1}\right)-K_{21}\left(x_{2}-x_{1}\right)-K_{23}\left(x_{2}-x_{1}\right)^{3}=F_{1}(t),$
$M_{2}\ddot{x}_{2}+C_{2}\left(\dot{x}_{2}-\dot{x}_{1}\right)+K_{21}\left(x_{2}-x_{1}\right)+K_{23}\left(x_{2}-x_{1}\right)^{3}=F_{2}(t),$
(13b)

where $x_{1}(t)$ and $x_{2}(t)$ are the displacements of the two oscillators,
$M_{1}$ and $M_{2}$ are the masses, $K_{11}$ and $K_{21}$ are the linear
stiffness coefficients, $C_{1}$ and $C_{2}$ are the viscous damping
coefficients, $K_{13}$ and $K_{23}$ are the cubic stiffness coefficients,
$F_{1}$ and $F_{2}$ are the excitation forces. The numerical values of the
system parameters in this numerical application are set to: $M_{1}=M_{2}=400$
kg, $C_{1}=C_{2}=$ 400 kg/s (corresponding to a damping ratio of 1%),
$K_{11}=K_{21}=1$ kN/mm, $K_{13}=K_{23}=0.001$ kN/mm3. It is assumed that
linear and nonlinear stiffness coefficients get reduced by the same amount in
both oscillators as a result of damage, while damage does not affect their
masses and damping coefficients. The FRCs of the undamaged and damaged 2-DOF
nonlinear system are shown in Fig. 7 for $F_{1}=-AM\cos\Omega t$ and
$F_{2}=0$, with $A$ between $0.1g$ and $0.6g$.

Figure 7: Frequency response curves (FRCs) of the 2-DOF system for different
excitation amplitudes and damage levels.

A random excitation is employed to detect damage assuming
$F_{1}=-\ddot{x}_{b}M_{1}$ and $F_{2}=0$. The displacement time-histories of
both the undamaged and damaged systems are thus calculated through Eq. (13b)
by generating independent time-history samples of the white noise
$\ddot{x}_{b}$ and adjusting their peaks to match a uniformly random value
between $0.1g$ and $0.6g$ in order to restrict the analysis to a mid-large
nonlinear response according to Fig. 7. The settings adopted to implement the
deep learning architectures are listed in Tab. 2.

Table 2: Configuration of the deep learning architectures for damage detection in the 2-DOF system. Type | Component |  Convolution layers |  Trainable parameters |  Kernel size |  Number of filters | Strides
---|---|---|---|---|---|---
AE | Encoder | 3 | 150,175 | (50,25,12) | (100,50,25) | (1,1,1)
Decoder | 4 | 2,142,902 | (12,25,25,50) | (50,100,100,2) | (2,2,1,1)
GAN | Generator | 3 | 38,151 | (6,6,4) | (64,32,2) | (2,1,2)
Discriminator | 3 | 4,050,770 | (6,6,6) | (32,32,32) | (2,2,2)

Although the interaction of the two nonlinear oscillators complicates the
overall dynamic behavior, both architectures confirm their good performances
as shown in Figs. 8 and 9. As far as the application of the AE is concerned,
Fig. 8 shows that the undamaged response of both oscillators is well
reconstructed after the training process, whereas the accuracy decreases as
expected once the system is damaged. Figure 9 confirms the good performance of
GAN since the real response (i.e., the response obtained as the solution of
the equations of motion) and the generated response of the undamaged
oscillators look similar in the time-frequency domain. This indicates that the
discriminator has been successfully trained to recognize the undamaged system
response as genuine, while the damaged system response is likely to be
classified as fake, with varying degrees of deviation

Figure 8: Reconstruction of the 2-DOF system response through AE for different
damage levels. Figure 9: Time-history and corresponding time-frequency
representation via wavelet transform of the undamaged 2-DOF system response:
actual data and results carried out from the GAN generation block.

The application of both AE and GAN yields the results provided in Fig. 10.
Since both architectures are applied individually to the dynamic response of
each mass, Fig. 10 illustrates the reconstruction loss (in terms of MAE)
obtained by AE and the discriminator output obtained by GAN for all time
series samples of each of the two masses given the damage level. Instead, the
trend lines connect the values obtained by averaging the results calculated
for both masses in order to enable the detection of damage through the
inspection of a single parameter.

Figure 10: Damage sensitivity of the two deep learning architectures for the
2-DOF system: reconstruction loss values obtained by AE and relative variation
of its normalized average value, together with discriminator output values
obtained by GAN and relative variation of the associated average damage index.

While the considered deep learning architectures displayed a similar
performance for the 1-DOF system (see Fig. 6), Fig. 10 shows that AE is more
sensitive to damage than GAN. The lower damage sensitivity of GAN in this
application seems to be associated with the reduced accuracy of its
discriminator in correctly classifying the undamaged response of the 2-DOF
system compared to that of the 1-DOF system. In fact, some discriminator
outputs are somewhat far from 1 for the undamaged system, meaning that GAN
produces a non-negligible number of false positives (i.e., the discriminator
sometimes recognizes as fake the response of the undamaged system, which
implies that it detects damage even when none is present). Anyway, Fig. 10
proves that both algorithms enable the early detection of damage and they
perform as expected when the damage level increases.

###  4.3 Seismic isolator with superelastic hysteresis and negative stiffness

Health monitoring of seismic isolation systems is crucial to ensure that the
isolated structures maintain their target performance levels during
earthquakes. Consequently, several technical codes and guidelines worldwide
require inspections of seismic isolators, both regularly and past seismic
events (CEN/TC167, 2003; Sobanjo and Thompson, 2016). Visual inspections,
whenever possible, can only detect damage or aging effects when they are
visible on the surface, and their reliability heavily depends on the
inspectors’ experience. Therefore, this numerical study investigates the
feasibility of using deep learning architectures to detect damage in seismic
isolators. An innovative concept of seismic isolator with superelastic
hysteresis and negative stiffness proposed by Salvatore et al. (2021) is here
examined. It combines the Bouc-Wen model of hysteresis (Bouc, 1967; Wen,
1976), a negative stiffness mechanism, and a superelastic model proposed by
Charalampakis and Tsiatas (2018). The equation of motion reads:

$\displaystyle M\ddot{x}+f_{r}=F(t),$ (14)

where the total restoring force of the seismic isolation device is given by
$f_{r}=f_{i}+f_{n}+f_{s}$, in which $f_{i}$ corresponds to the traditional
seismic elastomeric isolator restoring force, $f_{s}$ is the superelastic
force and $f_{n}$ reflects the negative stiffness mechanism. The contribution
to the restoring force given by $f_{i}$ is defined as follows:

$f_{i}=C\dot{x}+\alpha K_{i}x+\left(1-\alpha\right)K_{i}z,$ (15a)
$\dot{z}=\dot{x}[1-(\gamma+\beta\text{sign}(z\dot{x}))]|z|^{n},$ (15b)

where $z$ is the hysteretic force governed by $\alpha$, which is the ratio
between the post-elastic and the stiffness $K_{i}$ at the origin. Moreover,
$\gamma$ and $\beta$ control the shape of the hysteresis and $n$ regulates the
smoothness of elastic-to-plastic transition. The contribution to the restoring
force given by $f_{n}$ is defined as follows:

$f_{n}=\left(-K_{n}x+K_{3}x^{3}\right)\dfrac{\left[1+\text{sign}\left(x_{f}-|x|\right)\right]}{2},$
(16)

where $K_{n}$ is the negative linear stiffness, $K_{3}$ is the positive cubic
stiffness, and $x_{f}$ controls the extent of the negative stiffness force.
The last contribution to the restoring force given by $f_{s}$ is defined as
follows (Charalampakis and Tsiatas, 2018):

$\dot{f}_{s}=(1-s)K_{s}\left[\dot{x}-\left|\dot{x}\right|\text{sign}\left(f_{s}-\beta_{s}\right)\left(\dfrac{\left|f_{s}-\beta_{s}\right|}{Y}\right)^{n_{s}}\right]+sK_{m}\dot{x},$
(17a)
$\beta_{s}=K_{s}\alpha_{s}\left\\{x-\dfrac{f_{s}}{K_{s}}+f_{t}\text{tanh}(a_{s}x)\left[\dfrac{1+\text{sign}(-x\dot{x})}{2}\right]\right\\},$
(17b)
$s=\dfrac{\text{tanh}\left[c_{s}\left(\left|x\right|-x_{m}\right)\right]+1}{2},\,f_{t}=\frac{(2Y-y_{s}Y)}{\alpha_{s}K_{s}},\,a_{s}=\frac{\tan^{-1}(\tilde{a}_{s}K_{s})}{Y-y_{s}Y},$
(17c)

where $K_{s}$ and $K_{m}$ are the initial stiffness during the austenitic
phase and the fully martensitic phase, respectively, whereas $Y$ is the
yielding force and $\alpha_{s}$ controls the post-elastic stiffness. The
parameter $n_{s}$ regulates the smoothness of the transition from the initial
elastic to the post-elastic phase while $f_{t}$ and $a_{s}$ control the
twinning hysteresis and super-elasticity and the pinching around the origin
along the cycle, respectively. The complex dynamics resulting from Eqs.
(14)-(17c) is a formidable challenge for testing the ability of AE and GAN to
detect damage in nonlinear systems. The system parameters for this application
are set to: $M=400$ kg, $C=0.4767$ kg/s, $K_{i}=1.1$ kN/mm, $K_{n}=0.5$ kN/mm,
$K_{3}=0.00001$ kN/mm3, $x_{u}=100$ mm, $x_{m}=0.7x_{u}$, $x_{f}=0.7x_{u}$,
$\alpha=0.2$, $\beta=0.0007$, $c=0.01$, $\gamma=0.00001$, $n=1$, and
$n_{s}=3$. Damage is simulated by reducing both $K_{i}$ and $K_{n}$ by the
same amount.

Both AE and GAN are assessed through numerical simulations by solving Eq. (14)
under random excitation such that $F=-\ddot{x}_{b}M$, where the peak value of
the time-history samples of $\ddot{x}_{b}$ are scaled to a uniform random
value between $0.04g$ and $0.80g$. Table 3 shows the details about the
architecture of both AE and GAN for this numerical application.

Table 3: Configuration of the deep learning architectures for damage detection in the seismic isolator. Type | Component |  Convolution layers |  Trainable parameters |  Kernel size |  Number of filters | Strides
---|---|---|---|---|---|---
AE | Encoder | 3 | 38,650 | (20,10,5) | (50,50,50) | (1,1,1)
Decoder | 3 | 791,596 | (5,10,20) | (50,50,1) | (2,2,1)
GAN | Generator | 2 | 898 | (2,8) | (2,1) | (2,2)
Discriminator | 3 | 100,623 | (8,4,2) | (8,4,2) | (2,2,1)

Representative results carried out from the application of AE and GAN are
reported in Figs. 11 and 12, respectively. A close inspection of the results
in Fig. 11 demonstrates that AE continues to reconstruct the undamaged
response of the dynamic system quite satisfactorily while its performance
degrades as expected when the level of damage increases. GAN faces some
difficulties in this application, as it can be inferred from the comparison in
the time-frequency domain between actual and generated response given by Fig.
12 (i.e., some frequency components in the actual response are missing in the
output produced by the generator). Indeed, the discriminator sometimes
mislabels data from the generator as real, even though the comparison with the
reference isolator response in the time-frequency domain points out some
differences. This is due to the very rich dynamics given by Eqs. (14)-(17c)
and anticipates a lower sensitivity of GAN in detecting damage.

Figure 11: Reconstruction of the seismic isolator response through AE for
different damage levels. Figure 12: Time-history and corresponding time-
frequency representation via wavelet transform of the undamaged seismic
isolator response: actual data and results carried out from the GAN generation
block.

Results in Fig. 13 agree with the evidence drawn from Figs. 11 and 12. In
fact, Fig. 13 shows that the AE properly detects the presence of damage. The
sensitivity of AE in this application remains satisfactory for all damage
levels, but it is generally lower compared to Fig. 6 or Fig. 10. The presence
of damage in the seismic isolator is also successfully recognized by GAN,
provided that the damage severity is large enough (the relative variation of
the damage index is around 10% once the damage level approaches 15%) while the
output is almost inconclusive otherwise. This is mainly because the
discriminator tends to produce more false negatives in this application (i.e.,
too many samples of the damaged response of the isolator are incorrectly
classified as true, meaning they are mistakenly considered as representative
of a healthy state).

Figure 13: Damage sensitivity of deep learning architectures for the seismic
isolator: reconstruction loss values obtained by AE and relative variation of
its normalized average value, together with discriminator output values
obtained by GAN and relative variation of the associated average damage index.

##  5 Experimental application

Both AE and GAN are finally tested using real data from a laboratory
experiment reported by Shiki et al. (2017) and Villani et al. (2019b). The
experimental setup is shown in Fig. 14 and is composed by a cantilever
aluminum beam having dimensions of 300 mm $\times$ 19 mm $\times$ 3.2 mm with
a small steel mass attached to its free end that interacts with a neodymium
magnet positioned at a distance of 2 mm. A bolted connection provided with up
to four nuts is positioned 150 mm from the free end to simulate the presence
of damage. Considering the mass of each nut and varying the number of nuts,
the damage levels turn out to be equal to 2%, 4% and 6%. An electrodynamic
shaker is placed 50 mm away from the clamped end of the beam whereas a laser
vibrometer measures the velocity of the free end. A hardening-type
nonlinearity originates in this magneto-elastic dynamic system from the
interaction between the permanent magnet and the steel mass at the beam free
end. While chirp and stepped sine were used as excitation by Shiki et al.
(2017) and Villani et al. (2019b), the present experimental application deals
with output-only damage detection under random vibrations. Accordingly, the
measured system response under band-pass filtered white noise is considered.
The filter bandwidth in the experiment was set to 10-420 Hz whereas the input
excitation levels were equal to 0.01 VRMS, 0.05 VRMS, 0.10 VRMS, 0.11 VRMS,
0.12 VRMS, 0.13 VRMS, 0.14 VRMS and 0.15 VRMS.

Figure 14: Layout of the experimental test consisting of a cantiliver beam and
a magnet close to the tip mass (Villani et al., 2019b, reprinted with
permission).

Once again, the system dynamic response under different (unknown) excitation
levels is taken into account to assess the effectiveness of the deep learning
architectures in detecting damage, regardless of the experienced level of
nonlinearity. The configuration parameters of AE and GAN adopted in such
experimental application are given in Tab. 4.

Table 4: Configuration of the deep learning architectures for damage detection in the magneto-elastic system. Type | Component |  Convolution layers |  Trainable parameters |  Kernel size |  Number of filters | Strides
---|---|---|---|---|---|---
AE | Encoder | 4 | 1,283,525 | (20,100,50,25) | (100,100,50,25) | (1,1,1,1)
Decoder | 3 | 527,679 | (50,50,20) | (50,100,1). | (1,2,2)
GAN | Generator | 3 | 414,629 | (10,20,2) | (100,200,1) | (1,2,2)
Discriminator | 4 | 2,357,201 | (3,50,15,10) | (200,200,100,50) | (1,1,1,1)

Figures 15 and 16 show the performances of both architectures after training
while Fig. 17 demonstrates their effectiveness in detecting damage. On the one
hand, the reconstruction error of AE displays an abrupt increment as soon as a
small damage occurs. On the other hand, a growing number of time series data
is classified as fake by the discriminator of GAN once a small damage occurs
because they do no longer represent the dynamics of a healthy system. The very
good sensitivity of both architectures for the lowest damage level of 2%
confirms their ability in early damage detection. In particular, the damage
index based on the discriminator loss highlights the exceptional performance
of GAN in recognizing the onset of damage in this experimental application.
However, the relative variation of the normalized reconstruction loss and that
of the damage index based on the discriminator loss for damage levels of 4%
and 6% are fairly close to the corresponding values observed for the minimum
damage level equal to 2%. Ultimately, this experimental application further
confirms that both AE and GAN are able to detect damage without a priori
information about the system or the applied dynamic loads, and irrespective of
the degree of nonlinearity.

Figure 15: Reconstruction of the magneto-elastic system response through AE
for different damage levels. Figure 16: Time-history response and
corresponding time-frequency representation via wavelet transform of the
undamaged magneto-elastic system: actual data and results carried out from the
GAN generation block. Figure 17: Damage sensitivity of deep learning
architectures for the magneto-elastic system: reconstruction loss values
obtained by AE and relative variation of its normalized average value,
together with discriminator output values obtained by GAN and relative
variation of the associated average damage index.

##  6 Conclusions

The original contribution of this work lies in the first comprehensive study
on using deep learning architectures for unsupervised, data-driven, vibration-
based damage detection in nonlinear dynamic systems. The performance of both
AE and GAN was deeply examined, considering different nonlinear behaviors. The
excitation scenarios involve random excitation with varying intensity.
Numerical investigations and experimental applications have been conducted
under the assumption that neither the system nor the excitation are known. The
following conclusions can be drawn from the present extensive study on
different systems and datasets.

  * •

Both AE and GAN are able to detect the onset of damage in different nonlinear
dynamic systems. They also show consistent results upon increasing damage
levels.

  * •

On the one hand, it seems that AE displays a more consistent performance than
GAN. GAN did not consistently detect the lowest damage level; however, it can
exhibit exceptional damage sensitivity.

  * •

The AE architecture definition proved easier than configuring GAN. Training AE
was also less time consuming than GAN. Conversely, the main advantage of the
GAN discriminator lies in its output being interpretable as damage index since
it is bounded between two limit values corresponding to healthy and damaged
state. While GAN is applied to detect the occurrence of damage, it can be
possibly used for other tasks, such as synthetic data generation, uncertainty
quantification, and reliability assessment.

In addition to extending both numerical and experimental applications, some
issues should be addressed in future efforts, as listed below.

  * •

It is known that the effects attributable to environmental conditions can
obscure those due to damage. Therefore, understanding whether deep learning
architectures can detect effectively damage under changing environmental
conditions without preliminary data processing is still an open issue.

  * •

Both AE and GAN were employed to establish whether the nonlinear dynamic
system was damaged or not (i.e., novelty detection). Damage localization and
quantification were not addressed. Further studies should aim to understand
whether the output of such deep learning architectures can be eventually
correlated with the damage position and/or magnitude.

## Acknowledgments

The authors would like to thank Americo Cunha (Rio de Janeiro State
University, Brazil) and Samuel Da Silva (São Paulo State University, Brazil)
for sharing the experimental data.

This work was partially carried out within the RETURN Extended Partnership
funded by the European Union Next-GenerationEU (National Recovery and
Resilience Plan – NRRP, Mission 4, Component 2, Investment 1.3 – D.D. 1243
2/8/2022, PE0000005).

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