A probabilistic neural network has been implemented to predict the malignancy of breast cancer cells, based on a data set, the features of which are used for the formulation and training of a model for a binary classification problem. The focus is placed on considerations when building the model, in order to achieve not only accuracy but also a safe quantification of the expected uncertainty of the calculated network parameters and the medical prognosis. The source code is included to make the results reproducible, also in accordance with the latest trending in machine learning research, named Papers with Code. The various steps taken for the code development are introduced in detail but also the results are visually displayed and critically analyzed also in the sense of explainable artificial intelligence. In statistical-classification problems, the decision boundary is the region of the problem space in which the classification label of the classifier is ambiguous. Problem aspects and model parameters which influence the decision boundary are a special aspect of practical investigation considered in this work. Classification results issued by technically transparent machine learning software can inspire more confidence, as regards their trustworthiness which is very important, especially in the case of medical prognosis. Furthermore, transparency allows the user to adapt models and learning processes to the specific needs of a problem and has a boosting influence on the development of new methods in relevant machine learning fields (transfer learning).
Citation: Anastasia-Maria Leventi-Peetz, Kai Weber. Probabilistic machine learning for breast cancer classification[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 624-655. doi: 10.3934/mbe.2023029
[1] | Benedict C. Posadas . Socioeconomic determinants of the level of mechanization of nurseries and greenhouses in the southern United States. AIMS Agriculture and Food, 2018, 3(3): 229-245. doi: 10.3934/agrfood.2018.3.229 |
[2] | Germaine Ibro, Ibro Madougou Abdoulaye, Gry Synnevåg, Jens B. Aune . Food security and child malnutrition in the regions of Maradi, Tahoua and Tillabéri in Niger: The status, the causes, and transformative change. AIMS Agriculture and Food, 2022, 7(3): 704-720. doi: 10.3934/agrfood.2022043 |
[3] | Radhwane Derraz, Farrah Melissa Muharam, Noraini Ahmad Jaafar . Uncertainty sources affecting operational efficiency of ML algorithms in UAV-based precision agriculture: A 2013–2020 systematic review. AIMS Agriculture and Food, 2023, 8(2): 687-719. doi: 10.3934/agrfood.2023038 |
[4] | Priscilla Charmaine Kwade, Benjamin Kweku Lugu, Sadia Lukman, Carl Edem Quist, Jianxun Chu . Farmers’ attitude towards the use of genetically modified crop technology in Southern Ghana: The mediating role of risk perception. AIMS Agriculture and Food, 2019, 4(4): 833-853. doi: 10.3934/agrfood.2019.4.833 |
[5] | Celina Gómez, Luigi Gennaro Izzo . Increasing efficiency of crop production with LEDs. AIMS Agriculture and Food, 2018, 3(2): 135-153. doi: 10.3934/agrfood.2018.2.135 |
[6] | José M Peña, Ana I de Castro, Jorge Torres-Sánchez, Dionisio Andújar, Carolina San Martín, José Dorado, César Fernández-Quintanilla, Francisca López-Granados . Estimating tree height and biomass of a poplar plantation with image-based UAV technology. AIMS Agriculture and Food, 2018, 3(3): 313-326. doi: 10.3934/agrfood.2018.3.313 |
[7] | Mohamad Harisudin, Kusnandar, Erlyna W. Riptanti, Nuning Setyowati, Isti Khomah . Determinants of the Internet of Things adoption by millennial farmers. AIMS Agriculture and Food, 2023, 8(2): 329-342. doi: 10.3934/agrfood.2023018 |
[8] | Fernando Arias, Maytee Zambrano, Kathia Broce, Carlos Medina, Hazel Pacheco, Yerenis Nunez . Hyperspectral imaging for rice cultivation: Applications, methods and challenges. AIMS Agriculture and Food, 2021, 6(1): 273-307. doi: 10.3934/agrfood.2021018 |
[9] | Marcello Mastrorilli, Raffaella Zucaro . Towards sustainable use of water in rainfed and irrigated cropping systems: review of some technical and policy issues. AIMS Agriculture and Food, 2016, 1(3): 294-314. doi: 10.3934/agrfood.2016.3.294 |
[10] | Ogunniyi Adebayo, Omonona Bolarin, Abioye Oyewale, Olagunju Kehinde . Impact of irrigation technology use on crop yield, crop income and household food security in Nigeria: A treatment effect approach. AIMS Agriculture and Food, 2018, 3(2): 154-171. doi: 10.3934/agrfood.2018.2.154 |
A probabilistic neural network has been implemented to predict the malignancy of breast cancer cells, based on a data set, the features of which are used for the formulation and training of a model for a binary classification problem. The focus is placed on considerations when building the model, in order to achieve not only accuracy but also a safe quantification of the expected uncertainty of the calculated network parameters and the medical prognosis. The source code is included to make the results reproducible, also in accordance with the latest trending in machine learning research, named Papers with Code. The various steps taken for the code development are introduced in detail but also the results are visually displayed and critically analyzed also in the sense of explainable artificial intelligence. In statistical-classification problems, the decision boundary is the region of the problem space in which the classification label of the classifier is ambiguous. Problem aspects and model parameters which influence the decision boundary are a special aspect of practical investigation considered in this work. Classification results issued by technically transparent machine learning software can inspire more confidence, as regards their trustworthiness which is very important, especially in the case of medical prognosis. Furthermore, transparency allows the user to adapt models and learning processes to the specific needs of a problem and has a boosting influence on the development of new methods in relevant machine learning fields (transfer learning).
Let Ω⊆RN be a bounded domain with a C2-boundary ∂Ω. In this paper, we study the following quasilinear Robin problem
{−Δu−Δ(u2)u+a(x)u=λf(x,u)inΩ,∂u∂n+β(x)u=0on∂Ω, | (1.1) |
where ∂u∂n=∇u⋅n with n(x) being the outward unit normed vector to ∂Ω at its point x, a(x)∈L∞(Ω) satisfying ess infx∈Ωa(x)>0, β(x)∈C0,τ(∂Ω,R+0) for some τ∈(0,1) and β(x)≠0. λ>0 is a real parameter.
The Robin boundary problems with Laplacian operator have been widely investigated in recent years ([1,2,3,4,5,6]). For example, Papageorgiou et al. in [4] considered the Robin problems driven by negative Laplacian with a superlinear reaction and proved the existence and multiplicity theorems by variational methods. However, the Robin boundary problems with quasilinear Schrödinger operator have not been dealt with yet. Based on the above issues, this paper aims to study the existence and multiplicity of solutions to problem (1.1). In our work, one of the main difficulties is that there is no suitable space on which the energy functional is well defined. To the best of our knowledge, the first existence result for the equation of the following form
−Δu−Δ(u2)u+a(x)u=λf(x,u),x∈RN | (1.2) |
is due to Poppenberg, Schmitt and Wang([7]), and their approach to the problem is the constrained minimization argument. Since then, some ideas and approaches were developed to overcome these difficulties. Liu and Wang, in [8], reduced the quasilinear equation to a semilinear one by the change of variable. In [9], using the same methods in [8], Colin and Jeanjean considered the problem (1.2) on the usual Sobolev space. In this paper, we will use the change of variable to overcome the main difficulties.
Let f:Ω×R→R be a Carathéodory function such that f(x,0)=0 for a.e. x∈Ω and let
F(x,t):=∫t0f(x,ξ)dξ,(x,t)∈Ω×R. |
We first posit the following assumptions on f.
(f1)|f(x,t)|≤c0(1+|t|r−1) for a.e. x∈Ω, where c0>0, 4<r<2⋅2∗;
(f2)limt→+∞F(x,t)t4=0 and limt→+∞F(x,t)=−∞ uniformly for x∈Ω;
(f3) There exists ˜u∈Lr(Ω) such that ∫ΩF(x,˜u)dx>0;
(f4) There is a constant δ>0 such that f(x,t)≤0 for a.e. x∈Ω and t∈(0,δ);
(f5) For every ρ>0, there exists μρ>0 such that t↦f(x,t)√1+2t2+μρt is nondecreasing on [0,ρ];
Remark 1.1. The following function satisfies hypotheses (f1)-(f5), for simplicity we drop the x-dependence,
f(t)={0,if t≤0,−tk1−1+2tk2−1,if t∈(0,1],2tk3−1−tk4−1,if t>1, |
where 3<k1<k2<2k1<+∞ and 1<k3<k4<4.
The theorem below is the first result of our paper.
Theorem 1.1. If hypotheses (f1)-(f5) hold, then there exists a critical parameter value λ∗>0 such that for all λ>λ∗, problem (1.1) has at least two positive solutions v0,v1∈int(C+) with v0≤v1 in Ω.
To study further the multiplicity of solutions for problem (1.1) under the assumption (f1), we give some other conditions on f as follows:
(f6)lim|t|→+∞F(x,t)t4=+∞ uniformly with respect to x∈Ω, and there exist α∈(max{1,(r−4)N4},2∗), ˆc0>0 such that
0<ˆc0≤lim inf|t|→+∞f(x,t)t−4F(x,t)|t|2α |
uniformly for x∈Ω.
(f7)f(x,−t)=−f(x,t).
Remark 1.2. The following function satisfies hypotheses (f1), (f6) and (f7):
f(t)=t3ln(1+|t|) for all t∈R. |
Theorem 1.2. If hypotheses (f1), (f6), (f7) hold and λ>1, then problem (1.1) has infinitely many high energy solutions in H1(Ω).
The paper is organized as follows. In Section 2, we give some preliminary results, which will be used in this paper. We prove Theorem 1.1 and Theorem 1.2 in Section 3 and Section 4 respectively.
In this paper, the main working spaces are H1(Ω), C1(ˉΩ) and Lq(∂Ω), 1≤q≤+∞. We denote the norm of Lq(Ω) and H1(Ω) by
‖u‖q=(∫Ω|u|qdx)1/q,‖u‖=(∫Ω|∇u|2dx+∫Ω|u|2dx)1/2. |
The Banach space C1(ˉΩ) is an ordered Banach space with positive (order) cone
C+:={u∈C1(ˉΩ):u(x)≥0, ∀ x∈ˉΩ}. |
This cone has a nonempty interior given by
int(C+):={u∈C+:u(x)>0, ∀ x∈ˉΩ}. |
On ∂Ω we will employ the (N−1)-dimensional Hausdorff measure σ. By applying this measure we can define the Lebesgue spaces Lq(∂Ω) (1≤q≤+∞). The Trace Theorem says that there exists a unique continuous linear map γ0:H1(Ω)→L2(∂Ω), known as "trace map", such that
γ0(u)=u|∂Ω,∀ u∈H1(Ω)∩C(ˉΩ). |
Consequently, the trace map extends the notion of boundary values to any Sobolev function, and
imγ0=H12,2(∂Ω), kerγ0=H10(Ω). |
Moreover, the trace map γ0 is compact from H1(Ω) into Lq(∂Ω) with q∈[1,2(N−1)N−2) if N≥3 and q∈[1,+∞) if N=1, 2. In the sequel, for the sake of notational simplicity, the use of the trace map γ0 will be dropped. The restrictions of all Sobolev functions on the boundary ∂Ω are understood in the sense of traces.
We know that (1.1) is the Euler-Lagrange equation associated to the energy functional
I(u)=12(∫Ω|∇u|2(1+2u2)dx+∫Ωa(x)u2dx+∫∂Ωβ(x)u2(1+u2)dσ)−λ∫ΩF(x,u)dx. |
Unfortunately, the functional I could not be well defined in H1(Ω) for N≥3. To overcome this difficulty, we use the argument developed in [9]. More precisely, we make the change of variable u=g(v), which is defined by
g′(t)=1√1+2g2(t)on[0,+∞),g(t)=−g(−t)on(−∞,0]. |
Now we present some important results about the change of variable u=g(v).
Lemma 2.1. ([10]) The function g and its derivative satisfy the following properties:
(ⅰ) g is uniquely defined, C2 and invertible;
(ⅱ) |g′(t)|≤1 for all t∈R;
(ⅲ) |g(t)|≤t for all t∈R;
(ⅳ) g(t)t→1, as t→0;
(ⅴ) |g(t)|≤21/4|t|1/2 for all t∈R;
(ⅵ) g(t)/2<tg′(t)<g(t) for all t>0, and the reverse inequalities hold for t<0;
(ⅶ) g(t)√t→a1>0, as t→+∞;
(ⅷ) |g(t)g′(t)|≤1/√2 for all t∈R;
(ⅸ) There exists a positive constant C1 such that
|g(t)|≥{C1|t|,|t|≤1,C1|t|1/2,|t|≥1. |
Therefore, after the change of variable, the functional I(u) can be rewritten in the following way
J(v)=12∫Ω|∇v|2dx+12∫Ωa(x)g2(v)dx+12∫∂Ωβ(x)g2(v)(1+g2(v))dσ−λ∫ΩF(x,g(v))dx. | (2.1) |
From Lemma 2.1, by a standard argument, we see that J is well defined in H1(Ω) and J∈C1. In addition,
J′(v)φ=∫Ω∇v∇φdx+∫Ωa(x)g(v)g′(v)φdx+∫∂Ωβ(x)g(v)g′(v)(1+2g2(v))φdσ−λ∫Ωf(x,g(v))g′(v)φdx | (2.2) |
for all v,φ∈H1(Ω).
It is easy to see that the critical points of J correspond exactly to the weak solutions of the semilinear problem
{−Δv+a(x)g(v)g′(v)=λf(x,g(v))g′(v)inΩ,∂v∂n+β(x)g(v)g′(v)(1+2g2(v))=0on∂Ω. | (2.3) |
Hence, to prove our main results, we shall look for solutions of problem (2.3), i.e., the critical points of the functional J.
Let X be a Banach space and X∗ be its topological dual. By ⟨⋅,⋅⟩ we denote the duality brackets for the pair (X,X∗). We now give the definitions of the (PS)c condition and Cerami condition in X as follows.
Definition 2.2. Let ψ∈C1(X,R) and c∈R. We say that ψ satisfies the (PS)c condition, if every sequence {un}⊆X such that
ψ(un)→c, ψ′(un)→0 in X∗ as n→∞ |
admits a strongly convergent subsequence.
Definition 2.3. Let ψ∈C1(X,R) and c∈R. We say that ψ satisfies the Cerami condition, if every sequence {un}⊆X such that
ψ(un)→c, (1+‖un‖X)ψ′(un)→0 in X∗ as n→∞ |
admits a strongly convergent subsequence.
The following two lemmas are known as Mountain Pass Theorem and Fountain Theorem.
Lemma 2.4. (Mountain Pass Theorem, [11]) If ψ∈C1(X,R) satisfies the (PS)c-condition, there are u0, u1∈X with ‖u1−u0‖X>ϱ>0,
max{ψ(u0), ψ(u1)}<inf{ψ(u):‖u−u0‖X=ϱ}=mϱ |
and c=infγ∈Γmax0≤τ≤1ψ(Γ(τ)) where Γ={γ∈C([0,1], X):γ(0)=u0, γ(1)=u1}, then c≥mϱ and c is a critical value of ψ.
If X is a reflexive and separable Banach space, then there are ej∈X and e∗j∈X∗ such that
X=¯span{ej|j=1, 2, ⋯}, X∗=¯span{e∗j|j=1, 2, ⋯}, |
⟨ei,e∗j⟩={1,i=j,0,i≠j. |
For convenience, we write
Xj:=span{ej}, Yk:=⊕kj=0Xj, Zk:=¯⊕∞j=kXj. | (2.4) |
Lemma 2.5. (Fountain Theorem, [12]) Let ψ∈C1(X,R) be an even functional. If, for every k∈N, there exists ρk>γk>0 such that
(A1)ak:=supu∈Yk,‖u‖X=ρkψ(u)≤0,
(A2)bk:=infu∈Zk,‖u‖X=γkψ(u)→∞ as k→∞,
(A3)ψ satisfies the Cerami condition for every c>0.
Then ψ has an unbounded sequence of critical values.
In this section we prove the existence of positive solutions of problem (1.1). For simplicity, we may take f(x,t)=0 for a.e. x∈Ω, all t≤0.
Proposition 1. If (f1)–(f3) and (f5) hold, then problem (2.3) admits a solution v0∈int(C+).
Proof. We consider the C1-functional J+:H1(Ω)→R defined by
J+(v)=12(‖∇v‖22+‖v−‖22+∫Ωa(x)g2(v)dx+∫∂Ωβ(x)g2(v)(1+g2(v))dσ)−λ∫ΩF(x,g(v+))dx. | (3.1) |
We claim that J+ is coercive. Arguing by contradiction, assume that we can find a sequence {vn}n≥1⊆H1(Ω) and a constant M such that
‖vn‖→+∞ as n→∞ and J+(vn)≤M for all n≥1. | (3.2) |
Then from (3.2) we have
M≥12(‖∇vn‖22+‖v−n‖22+∫Ωa(x)g2(vn)dx+∫∂Ωβ(x)g2(vn)(1+g2(vn))dσ)−λ∫ΩF(x,g(v+n))dx≥12(‖∇vn‖22+‖v−n‖22)−λ∫ΩF(x,g(v+n))dx | (3.3) |
for all n≥1. Hypotheses (f1) and (ⅴ) of Lemma 2.1 imply that there exist c2, c3>0 such that
|F(x,g(v+n))|≤c2(1+|g(v+n)|r)≤c3(1+|v+n|r2). |
Combining this with (3.2), (3.3), we get
‖v+n‖→+∞ as n→∞. |
Let wn=v+n‖v+n‖, n≥1. Then ‖wn‖=1 for all n≥1, and so we may assume that
wn⇀w in H1(Ω), wn→w in Lθ(Ω) and in Lθ(∂Ω) for each θ∈(1,2∗). | (3.4) |
From (3.3) we have
12‖∇wn‖22−λ∫ΩF(x,g(v+n))‖v+n‖2dx≤M‖v+n‖2for all n≥1. | (3.5) |
Invoking (f1), (f2) and (ⅴ) of Lemma 2.1, we can find cϵ>0 such that
|F(x,g(t))|≤ϵ2t2+cϵfor a.e. x∈Ω, all t≥0, |
which yields, for n≥1 large enough,
∫Ω|F(x,g(v+n))|‖v+n‖2dx≤ϵ2+cϵ|Ω|N‖v+n‖2≤ϵ. | (3.6) |
Passing to the limsup as n→∞ in (3.5), and using (3.6), we have
lim supn→+∞‖∇wn‖22≤0. | (3.7) |
In addition, from (3.4) we have ‖∇w‖22≤lim infn→+∞‖∇wn‖22. Combining this with (3.7), we obtain that ∇wn→∇w=0 in L2(Ω). Hence,
wn→w in H1(Ω), |
and so ‖w‖=1, w≥0. Since ∇w=0, we have w=1/√|Ω|N, and v+n(x)→+∞ for a.e. x∈Ω.
On the other hand, combining with (f1), (f2), (3.3), (ⅶ) of Lemma 2.1 and Fatou's lemma, we get
M≥J+(vn)≥−λ∫ΩF(x,g(v+n))dx→+∞ as n→∞, |
which is impossible. Thus, J+ is coercive.
Furthermore, from Lemma 2.1, Sobolev embedding theorem and trace theorem, we see that J+ is sequentially weak lower semicontinuous. By Weierstrass-Tonelli theorem we can find v0∈H1(Ω) such that
J+(v0)=inf{J+(v):v∈H1(Ω)}. | (3.8) |
Consider the integral functional JF:Lr(Ω)→R defined by
JF(v)=∫ΩF(x,g(v))dx. |
From hypothesis (f1), (ⅰ) of Lemma 2.1 and the dominated convergence theorem, we see that JF is continuous. By hypothesis (f3) and (ⅸ) of Lemma 2.1, there exists ˜v∈Lr(Ω) such that
JF(˜v)>0. |
Exploiting the density of H1(Ω) in Lr(Ω), we can find ˉv∈H1(Ω) such that
JF(ˉv)>0. | (3.9) |
Therefore, from (3.1) and (3.9), there exists λ∗>0 such that
J+(v0)<J+(ˉv)<0=J+(0) |
for all λ>λ∗. This ensure that v0≠0. From (3.8), we have J′+(v0)=0, that is,
∫Ω∇v0∇φdx−∫Ωv−0φdx+∫Ωa(x)g(v0)g′(v0)φdx |
+∫∂Ωβ(x)g(v0)g′(v0)(1+2g2(v0))φdσ=λ∫Ωf(x,g(v+0))g′(v+0)φdx | (3.10) |
for all φ∈H1(Ω). In (3.10) choosing φ=−v−0, we obtain
‖∇v−0‖22+‖v−0‖22≤0, |
which implies v0≥0 a.e. in Ω. Hence,
∫Ω∇v0∇φdx+∫Ωa(x)g(v0)g′(v0)φdx+∫∂Ωβ(x)g(v0)g′(v0)(1+2g2(v0))φdσ=λ∫Ωf(x,g(v0))g′(v0)φdx | (3.11) |
for all φ∈H1(Ω). That is to say v0 is a weak solution of problem (2.3). By Theorem 4.1 in [13], we know that v0∈L∞(Ω). Furthermore, applying Theorem 2 in [14], we have v0∈C+∖{0}. Taking ρ=‖g(v0)‖∞, from (f5), there exists μρ>0 such that
f(x,g(v0))g′(v0)+μρg(v0)≥0for a. e. x∈Ω. | (3.12) |
From (2.3), (3.12) and (ⅲ) of Lemma 2.1 we obtain
−Δv0+(λμρ+‖a(x)‖∞)v0≥0for a. e. x∈Ω. |
Therefore, v0∈int(C+) by the strong maximum principle(see [15]).
To look for another positive solution to problem (2.3), we consider the functional ˆJ+∈C1(H1(Ω),R) defined by
ˆJ+(v)=12(‖∇v‖22+‖v−‖22+‖(v−v0)+‖22+∫∂ΩˆK(x,v)dσ)−∫ΩˆF(x,v)dx, | (3.13) |
where v0 is given in Proposition 1, ˆF(x,t)=∫t0ˆf(x,s)ds, ˆK(x,t)=∫t0ˆk(x,s)ds, and
ˆf(x,s)={0if s≤0,λf(x,g(s))g′(s)−a(x)g(s)g′(s)if 0<s<v0(x),λf(x,g(v0))g′(v0)−a(x)g(v0)g′(v0)if s≥v0(x), | (3.14) |
for all (x,s)∈Ω×R.
ˆk(x,s)={0if s≤0,β(x)g(s)g′(s)(1+2g2(s))if 0<s<v0(x),β(x)g(v0)g′(v0)(1+2g2(v0))if s≥v0(x), | (3.15) |
for all (x,s)∈∂Ω×R. It is easy to see that ˆf(x,s) and ˆk(x,s) are Carathéodory functions.
Proposition 2. If (f1), (f5) hold and v is a critical point of ˆJ+(v), then v is a solution of problem (2.3), and v∈[0,v0]∩int(C+).
Proof. The assumption yields the following equality:
∫Ω∇v∇φdx−∫Ωv−φdx+∫Ω(v−v0)+φdx+∫∂Ωˆk(x,v)φdσ=∫Ωˆf(x,v)φdx | (3.16) |
for all φ∈H1(Ω). Taking φ=−v− in (3.16), we obtain (arguing as in Proposition 1) v≥0 a.e. in Ω.
On the one hand, letting φ=(v−v0)+ in (3.16) again, we have
∫Ω∇v∇(v−v0)+dx+‖(v−v0)+‖22+∫Ωa(x)g(v0)g′(v0)(v−v0)+dx+∫∂Ωβ(x)g(v0)g′(v0)(1+2g2(v0))(v−v0)+dσ=λ∫Ωf(x,g(v0))g′(v0)(v−v0)+dx. | (3.17) |
On the other hand, the fact that v0>0 is a critical point of J+ yields
∫Ω∇v0∇(v−v0)+dx+∫Ωa(x)g(v0)g′(v0)(v−v0)+dx+∫∂Ωβ(x)g(v0)g′(v0)(1+2g2(v0))(v−v0)+dσ=λ∫Ωf(x,g(v0))g′(v0)(v−v0)+dx. | (3.18) |
Then from (3.17) and (3.18), we obtain
‖∇(v−v0)+‖22+‖(v−v0)+‖22=0, |
and (v−v0)+=0. Therefore v∈[0,v0] a.e. in Ω. Now, (3.16) becomes
∫Ω∇v∇φdx+∫Ωa(x)g(v)g′(v)φdx+∫∂Ωβ(x)g(v)g′(v)(1+2g2(v))φdσ=λ∫Ωf(x,g(v))g′(v)φdx |
for all φ∈H1(Ω). So v is a weak solution of problem (2.3). Reasoning as in the proof of Proposition 1, we have v∈int(C+).
Lemma 3.1. If hypotheses (f1)–(f5) hold, then ˆJ+ satisfies the (PS)c condition.
Proof. We first check that ˆJ+ is coercive. Since v0∈C1¯(Ω), using (f1), (ⅲ) of Lemma 2.1 and (3.14), we can find a constant c4>0 such that
|∫ΩˆF(x,v)dx|≤λc4‖v‖2for all v∈H1(Ω). | (3.19) |
Because of (3.15) we have
∫∂ΩˆK(x,v)dσ≥0for all v∈L2(∂Ω). | (3.20) |
Moreover, for every v∈H1(Ω) we see that
‖(v−v0)+‖22=‖(v+−v0)+‖22=‖v−v0‖22−∫{v+≤v0}(v0−v)2dx≥12‖v+‖22−12‖v0‖22−∫{v+≤v0}v20dx≥12‖v+‖22−2‖v0‖22. | (3.21) |
From (3.13), (3.19), (3.20) and (3.21) we derive
ˆJ+(v)≥14‖v‖2−‖v0‖22−λc4‖v‖2, ∀ v∈H1(Ω). |
It is easy to see that ˆJ+ is coercive.
Now let {vn}n≥1 be a sequence such that
ˆJ+(vn)→c, ˆJ′+(vn)→0 in H1(Ω) as n→∞. | (3.22) |
The coercivity of ˆJ+ and (3.22) imply that {vn}n≥1 is bounded in H1(Ω). Hence, there is v∈H1(Ω) such that, along a relabeled subsequence {vn}n≥1,
vn⇀v in H1(Ω), vn→v in Lθ(Ω) and in Lθ(∂Ω) for each θ∈(1,2∗). | (3.23) |
The second part of (3.22) yields
∫Ω∇vn∇φdx−∫Ωv−nφdx+∫Ω(vn−v0)+φdx+∫∂Ωˆk(x,vn)φdσ−∫Ωˆf(x,vn)φdx→0 | (3.24) |
as n→∞, for all φ∈H1(Ω). In (3.24) choosing φ=vn−v, using (f1), (3.23) and Lemma 2.1, we find
limn→∞∫Ω∇vn∇(vn−v)dx=0. |
Therefore ‖vn‖2→‖v‖2 as n→∞. Recalling that the Hilbert space H1(Ω) is locally uniformly convex, we obtain vn→v in H1(Ω). This shows that ˆJ+ satisfies the (PS)c condition.
In the next lemma, we state a consequence which depends on the regularity results in [13] and Theorem 2 in [14]. The proof is similar to Proposition 3 in [2].
Lemma 3.2. If (f1) holds and v∈H1(Ω) is a local C1(ˉΩ)-minimizer of ˆJ+, then v∈C1,τ(ˉΩ) and it is a local H1(Ω)-minimizer of ˆJ+.
Proposition 3. If (f1)-(f5) hold, then ˆJ+ admits a critical point v1∈H1(Ω) different from 0 and v0.
Proof. Let v∈C1(ˉΩ) with 0<‖v‖C1(ˉΩ)≤δ. From (f4) and (ⅳ) of Lemma 2.1, we get ˆF(x,g(v))≤0 for a.e. x∈Ω which implies
ˆJ+(v)≥−λ∫ΩˆF(x,g(v))dx>0. |
This shows that 0 is a local C1(ˉΩ)-minimizer of ˆJ+. Applying Lemma 3.2, we derive that 0 is a local H1(Ω)-minimizer of ˆJ+.
If 0 is not a strict local minimizer of ˆJ+, the result is obvious because any neighborhood of 0 in H1(Ω) contains another critical point of ˆJ+.
We next only need to discuss that 0 is a strict local minimizer of ˆJ+. In such a condition, we can find a sufficiently small ϱ∈(0,1) such that
ˆJ+(0)<inf{ˆJ+(v):‖v‖=ϱ}=ˆmϱ. | (3.25) |
Note that
ˆJ+(v0)=J+(v0)<J+(0)=ˆJ+(0)=0. |
This fact, together with (3.25) and Lemma 3.1 permit the use of the Mountain Pass Theorem, which yields a critical point v1 of ˆJ+ different from 0 and v0.
Theorem 1.1 follows immediately from Proposition 1, Proposition 2 and Proposition 3.
Lemma 4.1. If (f1) and (f6) hold, then, for any c>0, the functional J satisfies the Cerami condition.
Proof. Let {vn}n≥1⊆H1(Ω) be a Cerami sequence of J, that is,
J(vn)→c, (1+‖vn‖)J′(vn)→0 in H1(Ω) as n→∞. | (4.1) |
First we prove the boundedness of {vn}n≥1. By (4.1) we have
c+1≥12(∫Ω|∇vn|2dx+∫Ωa(x)g2(vn)dx+∫∂Ωβ(x)g2(vn)(1+g2(vn))dσ)−λ∫ΩF(x,g(vn))dx | (4.2) |
and
ϵn‖φ‖1+‖vn‖≥J′(vn)φ=∫Ω∇vn∇φdx+∫Ωa(x)g(vn)g′(vn)φdx+∫∂Ωβ(x)g(vn)g′(vn)(1+2g2(vn))φdσ−λ∫Ωf(x,g(vn))g′(vn)φdx | (4.3) |
for all φ∈H1(Ω) with ϵn≪1. Choosing φ=φn=−g(vn)g′(vn), we obtain
|∇φn|=|∇vn|(1+2(g(vn)g′(vn))2). |
Because of Lemma 2.1(ⅵ, ⅷ), φn∈H1(Ω). Therefore,
2ϵn≥J′(vn)φn=−∫Ω|∇vn|2((1+2(g(vn)g′(vn))2)dx−∫Ωa(x)g2(vn)dx−∫∂Ωβ(x)g2(vn)(1+2g2(vn))dσ+λ∫Ωf(x,g(vn))g(vn)dx. | (4.4) |
Now using (f6) and (ⅷ) of Lemma 2.1, it follows from (4.2) and (4.4) that
c+1+ϵn2≥∫Ω(12−14(1+2(g(vn)g′(vn))2)|∇vn|2dx+14∫Ωa(x)g2(vn)dx+14∫∂Ωβ(x)g2(vn)dσ+λ4∫Ω(f(x,g(vn))g(vn)−4F(x,g(vn)))dx≥λ4∫Ω(f(x,g(vn))g(vn)−4F(x,g(vn)))dx. | (4.5) |
On the other hand, from (f1) and (f6) we can find ˜c0∈(0,ˆc0) and c5>0 such that
f(x,t)−4F(x,t)≥˜c0|t|2α−c5a.e. x∈Ω. | (4.6) |
By (4.5) and (4.6), we conclude that {g(vn)}n≥1 is bounded in L2α(Ω). Furthermore, due to (ix) of Lemma 2.1,
{vn}n≥1 is bounded in Lα(Ω). | (4.7) |
Suppose that N≠2. By (f6), without loss of generality, we may assume that α<r2<2∗. So, we can find z∈(0,1) such that
2r=1−zα+z2∗. | (4.8) |
Invoking the interpolation inequality, Sobolev embedding theorem and (4.7) we have
‖vn‖r2≤‖vn‖1−zα‖vn‖z2∗≤c6‖vn‖z | (4.9) |
for some c6>0. Taking φ=−φn=g(vn)g′(vn) in (4.3), we have
∫Ω|∇vn|2((1+2(g(vn)g′(vn))2)dx+∫Ωa(x)g2(vn)dx+∫∂Ωβ(x)g2(vn)(1+2g2(vn))dσ≤2ϵn+λ∫Ωf(x,g(vn))g(vn)dx. | (4.10) |
By use of (f1) and (ⅴ) of Lemma 2.1, we can obtain that
f(x,g(vn))g(vn)≤c4(1+|g(vn)|r)≤c5(1+|vn|r2). | (4.11) |
From (f1), we assume that r is close to 2⋅2∗, hence α≥2. Then {vn}n≥1 is bounded in L2(Ω). Combining this with (4.9), (4.10) and (4.11), we have
‖vn‖2≤c6(1+‖vn‖zr2). | (4.12) |
Hypothesis (f6) and (4.8) lead to zr<4, and consequently, {vn}n≥1 is bounded in H1(Ω).
If N=2, then 2∗=+∞ and the Sobolev embedding theorem says that H1(Ω)↪Lη(Ω) for all η∈[1,+∞). Let η>r2>α and z∈(0,1) such that
2r=1−zα+zη, | (4.13) |
or zr=η(r−2α)η−α. Note that
limη→+∞η(r−2α)η−α=r−2α and r−2α<4. |
Repeating the previous proof method, for η>1 large enough, we again obtain that
{vn}n≥1 is bounded in H1(Ω). |
Next, we show that {vn}n≥1 is strongly convergent in H1(Ω). Since {vn}n≥1 is bounded, up to a subsequence(which we still denote by {vn}n≥1), we assume that there exists v∈H1(Ω) such that
vn⇀v in H1(Ω), vn→v in Lθ(Ω) and Lθ(∂Ω) for each θ∈(1,2∗). | (4.14) |
Choosing φ=vn−v∈H1(Ω) in (4.3) and combining (f1) with Lemma 2.1(ⅱ, ⅲ, ⅸ), we obtain limn→∞∫Ω∇vn∇(vn−v)dx=0. Hence ‖vn‖2→‖v‖2 as n→∞. Recalling that the Hilbert space H1(Ω) is locally uniformly convex, we get vn→v in H1(Ω).
Lemma 4.2. ([16]) Let X=H1(Ω) and define Yk, Zk as in (2.4). If 1≤q<2∗, then
dk:=supu∈Zk,‖u‖=1‖u‖q→0 as k→∞. |
Proof of Theorem 1.2. Since g(t) and f(x,t) are odd respect to t, functional J∈C1(H1(Ω),R) is even obviously. Further, from Lemma 4.1, we know that J satisfies the Cerami condition. So we need only to verify J satisfying the conditions (A1) and (A2) in Lemma 2.5.
Since Yk is finite-dimensional and all norms are equivalent on Yk, for v∈Yk, we have
‖v‖2≤C2‖v‖22. |
From the assumption (f6) and (ⅸ) of Lemma 2.1, there exists R>0 such that
F(x,g(t))≥2C2|t|2−C3for all x∈Ω, |t|≥R. |
For v∈Yk, λ>1, we have
J(v)≤‖v‖2−∫ΩF(x,g(v))dx≤−C2‖v‖22+C3|Ω|N≤−‖v‖2+C3|Ω|N. |
So (A1) is satisfied for sufficiently large ‖v‖.
From (f1) and (ⅴ) of Lemma 2.1, there exists constant C4>0 such that
F(x,g(t))≤C4(1+|t|r2),∀ t∈R. | (4.15) |
Let us define
dk:=supv∈Zk,‖v‖=1‖v‖r2. |
For v∈Zk, using (4.15), we get
J(v)≥12‖∇v‖22+12∫Ωa(x)g2(v)dx+12∫∂Ωβ(x)g2(v)(1+g2(v))dσ−C4λ‖v‖r2r2−C4λ|Ω|N≥12‖v‖2−C4λ‖v‖r2r2−12‖v‖22−C4λ|Ω|N≥12‖v‖2−C5dr2k‖v‖r2−C6|Ω|N. |
Choosing γk=(12C5rdr2k)24−r, if ‖v‖=γk, we obtain
J(v)≥(12−2r)(12C5rdr2k)44−r−C6|Ω|N. |
By Lemma 4.2, dk→0 and γk→∞ as k→∞, the condition (A2) is verified. In summary, all conditions of Fountain Theorem are satisfied. Thus J has an unbounded sequence of critical values. Theorem 1.2 is proved.
In this paper, we have studied a class of quasilinear Schrödinger equation in a bounded domain with Robin boundary. By giving different conditions on the reaction, we obtained two existence results of solutions to the equation. The Mountain Pass Theorem and Fountain Theorem were also employed in this study.
The authors wish to thank the referees and the editor for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11171220).
All authors declare no conflicts of interest in this paper.
[1] |
Q. Xia, Y. Cheng, J. Hu, J. Huang, Y. Yu, H. Xie, J. Wang, Differential diagnosis of breast cancer assisted by S-Detect artificial intelligence system, Math. Biosci. Eng., 18 (2021), 3680–3689. https://doi.org/10.3934/mbe.2021184 doi: 10.3934/mbe.2021184
![]() |
[2] |
X. Zhang, Z. Yu, Pathological analysis of hesperetin-derived small cell lung cancer by artificial intelligence technology under fiberoptic bronchoscopy, Math. Biosci. Eng., 18 (2021), 8538–8558. https://doi.org/10.3934/mbe.2021423 doi: 10.3934/mbe.2021423
![]() |
[3] |
S. Biswas, D. Sen, D. Bhatia, P. Phukan, M. Mukherjee, Chest X-Ray image and pathological data based artificial intelligence enabled dual diagnostic method for multi-stage classification of COVID-19 patients, AIMS Biophys., 8 (2021), 346–371. https://doi.org/10.3934/biophy.2021028 doi: 10.3934/biophy.2021028
![]() |
[4] |
T. Ayer, J. Chhatwal, O. Alagoz, C. E. Kahn, R. W. Woods, E. S. Burnside, Comparison of logistic regression and artificial neural network models in breast cancer risk estimation, Radiographics, 30 (2010), 13–21. https://doi.org/10.1148/rg.301095057 doi: 10.1148/rg.301095057
![]() |
[5] | T. van der Ploeg, M. Smits, D. W. Dippel, M. Hunink, E. W. Steyerberg, Prediction of intracranial findings on CT-scans by alternative modelling techniques, BMC Med. Res. Methodol., 11 (2011). https://doi.org/10.1186/1471-2288-11-143 |
[6] |
L. V. Jospin, W. L. Buntine, F. Boussaïd, H. Laga, M. Bennamoun, Hands-On Bayesian neural networks – A tutorial for deep learning users, IEEE Comput. Intell. Mag., 17 (2022), 29–48. https://doi.org/10.1109/MCI.2022.3155327 doi: 10.1109/MCI.2022.3155327
![]() |
[7] |
A. A. Abdullah, M. M. Hassan, Y. T. Mustafa, A review on Bayesian deep learning in healthcare: Applications and challenges, IEEE Access, 10 (2022), 36538–36562. https://doi.org/10.1109/ACCESS.2022.3163384 doi: 10.1109/ACCESS.2022.3163384
![]() |
[8] |
E. Hüllermeier, W. Waegeman, Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods, Mach. Learn., 110 (2021), 457–506. https://doi.org/10.1007/s10994-021-05946-3 doi: 10.1007/s10994-021-05946-3
![]() |
[9] | G. Deodato, Uncertainty modeling in deep learning, Variational inference for Bayesian neural networks, 2019. Available from: http://webthesis.biblio.polito.it/id/eprint/10920. |
[10] | H. Huang, Y. Wang, C. Rudin, E. P. Browne, Towards a comprehensive evaluation of dimension reduction methods for transcriptomic data visualization, Commun. Biol., 5 (2022). https://doi.org/10.1038/s42003-022-03628-x |
[11] | L. McInnes, J. Healy, J. Melville, UMAP: Uniform manifold approximation and projection for dimension reduction, preprint, arXiv: 1802.03426. https://doi.org/10.48550/arXiv.1802.03426 |
[12] |
A. Diaz-Papkovich, L. Anderson-Trocmé, S. Gravel, A review of UMAP in population genetics, J. Hum. Genet., 66 (2021), 85–91. https://doi.org/10.1038/s10038-020-00851-4 doi: 10.1038/s10038-020-00851-4
![]() |
[13] |
J. Salvatier, T. V. Wiecki, C. Fonnesbeck, Probabilistic programming in Python using PyMC3, PeerJ Comput. Sci., 2 (2016), e55. https://doi.org/10.7717/peerj-cs.55 doi: 10.7717/peerj-cs.55
![]() |
[14] |
A. Kucukelbir, D. Tran, R. Ranganath, A. Gelman, D. M. Blei, David M., Automatic differentiation variational inference, J. Mach. Learn. Res., 18 (2017), 1–45. https://doi.org/10.48550/arXiv.1603.00788 doi: 10.48550/arXiv.1603.00788
![]() |
[15] | M. Krasser, Variational inference in Bayesian neural networks. Available from: http://krasserm.github.io/2019/03/14/bayesian-neural-networks/. |
[16] | L. van der Maaten, G. Hinton, Visualizing data using t-SNE, J. Mach. Learn. Res., 9 (2008), 2579–2605. |
[17] | F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, et al., Scikit-learn: Machine learning in Python, J. Mach. Learn. Res., 12 (2011), 2825–2830. |
[18] | Heaton Research, The number of hidden layers. Available from: https://www.heatonresearch.com/2017/06/01/hidden-layers.html. |
[19] | T. Wood, What is the Sigmoid Function?, 2020. Available from: https://deepai.org/machine-learning-glossary-and-terms/sigmoid-function. |
[20] | A. K. Dhaka, A. Catalina, M. R. Andersen, M. Magnusson, J. H. Huggins, A. Vehtari, Robust, accurate stochastic optimization for variational inference, in Advances in Neural Information Processing Systems (eds. H. Larochelle et al.), Curran Associates, Inc., 33 (2020), 10961–10973. |
[21] |
A. Gelman, B. Carpenter, Bayesian analysis of tests with unknown specificity and sensitivity, J. R. Stat. Soc. Ser. C, 69 (2020), 1269–1283. https://doi.org/10.1111/rssc.12435 doi: 10.1111/rssc.12435
![]() |
[22] | D. Lunn, C. Jackson, N. Best, A. Thomas, D. Spiegelhalter, The BUGS book: A practical introduction to Bayesian analysis, Chapman and Hall/CRC, Boca Raton, USA, 2012. |
[23] | D. B. Rizzo, M. R. Blackburn, Harnessing expert knowledge: Defining Bayesian network model priors from expert knowledge only–prior elicitation for the vibration qualification problem IEEE Syst. J., 13 (2019), 1895–1905. https://doi.org/10.1109/JSYST.2019.2892942 |
[24] | A. M. Stefan, D. Katsimpokis, Q. F. Gronau, E. J. Wagenmakers, Expert agreement in prior elicitation and its effects on Bayesian inference, Psychon. Bull. Rev., 2022. https://doi.org/10.3758/s13423-022-02074-4 |
[25] | V. Fortuin, Priors in Bayesian deep learning: A review, Int. Stat. Rev., 2022. https://doi.org/10.1111/insr.12502 |
[26] |
C. Esposito, G. A. Landrum, N. Schneider, N. Stiefl, S. Riniker, GHOST: Adjusting the decision threshold to handle imbalanced data in machine learning, J. Chem. Inf. Model., 61 (2021), 2623–2640. https://doi.org/10.1021/acs.jcim.1c00160 doi: 10.1021/acs.jcim.1c00160
![]() |
[27] |
Q. Kim, J.-H. Ko, S. Kim, N. Park, W. Jhe, Bayesian neural network with pretrained protein embedding enhances prediction accuracy of drug-protein interaction, Bioinformatics, 37 (2021), 3428–3435. https://doi.org/10.1093/bioinformatics/btab346 doi: 10.1093/bioinformatics/btab346
![]() |
[28] |
S. Budd, E. C. Robinson, B. Kainz, A Survey on active learning and human-in-the-Loop deep learning for medical image analysis, Med. Image Anal., 71 (2021), 102062. https://doi.org/10.1016/j.media.2021.102062 doi: 10.1016/j.media.2021.102062
![]() |
[29] | V. S. Bisen, What is Human in the Loop Machine Learning: Why & How Used in AI?, 2020. Available from: https://medium.com/vsinghbisen/what-is-human-in-the-loop-machine-learning-why-how-used-in-ai-60c7b44eb2c0. |
[30] | K. Bykov, M. M. C. Höhne, A. Creosteanu, K. R. Müller, F. Klauschen, S. Nakajima, et al., Explaining Bayesian neural networks, preprint, arXiv: 2108.10346. https://doi.org/10.48550/arXiv.2108.10346 |