Global well-posedness of strong solution to the kinetic Cucker-Smale model with external potential field

1.   Introduction

This paper is concerned with the following kinetic Cucker-Smale model with external potential field:

Here, $f = f(t, x, v)$ is the particle distribution function in space $(x, v)\in\Omega = \mathbb{R}^d\times\mathbb{R}^d$, at time $t\geqslant 0$. The function $U(x) = \frac{ 1 }{2} |x|^{2}$ represents the harmonic potential field. The constant $\sigma > 0$ represents the noise strength. The alignment force $L[f]$ is expressed in the form:

The interaction kernel function $\varphi(|x-y|)$ is a positive nonincreasing $C^2$ function. Without loss of generality, we assume that

The system (1.1) arises as a mean-field kinetic description of the following stochastic Cucker-Smale model with external potential field:

where the deterministic system was studied by ^[1]. Here, $\big(x_{i} (t), v_{i} (t) \big)$ are the position and velocity pair of $i^{th}$-particle, $W_t^{(i)}$ denote independent Wiener processes, and $\sigma$ is the magnitude of the noise. The communication weight function $\varphi : \mathbb{R}^d \to \mathbb{R}_+$ satisfies some symmetry conditions.

The particle Cucker-Smale model was originally proposed to understand the flocking phenomena in bird populations by Cucker and Smale ^[2,3]. Under the "molecular chaos" assumption, Ha and Tadmor ^[4] derived the kinetic Cucker-Smale model formally from the particle Cucker-Smale model using the BBGKY hierarchy method, e.g., ^[5,6,7,8]. For large-scale particle systems, Ha and Liu ^[9] rigorously justified the mean-field limit from the multi-particle Cucker-Smale model to the kinetic Cucker-Smale model, utilizing tools such as measure-valued solutions and the Kantorovich-Rubinstein distance. Furthermore, Carrillo et al. ^[10] proved that the solutions approached exponentially fast in velocity to the mean velocity of the initial condition, while in space they converged to a translational flocking solution.

The Cucker-Smale model has been extended to various complexities, including the presence of different network structures ^[11], communication mechanism ^[12], self-propulsion and friction forces ^[13], and external fields such as fluid field ^[14], temperature field ^[15,16], potential fields ^[1], etc. These extensions significantly influence the dynamics of the system, leading to behaviors that are markedly different from the original model. Moreover, the connection between the kinetic Cucker-Smale model and the Euler-alignment system has been rigorously explored in recent literature. For the Euler-alignment system with pressure effects, Karper et al. ^[17] rigorously justified the hydrodynamic limit of the kinetic Cucker-Smale flocking model. Furthermore, Poyato and Soler ^[18] provided detailed analysis of a compressible Euler-type equation with singular commutator, which is derived from a hyperbolic limit of the kinetic Cucker-Smale model. In the pressure-less case, the derivation from the kinetic Cucker-Smale model to the nonlocal Euler-alignment system was established by ^[19]. Recently, Fabisiak and Peszek ^[20] rigorously derived the macroscopic fractional Euler-alignment system from the kinetic Cucker-Smale equation without performing any hydrodynamic limit.

The well-posedness of solution is a fundamental concept in the theory of partial differential equations. Previous works ^[21,22,23] have established the well-posedness of weak and strong solution to the kinetic Cucker-Smale model without external potentials. Recently, Jin ^[24] developed a unified framework to establish the well-posedness of the model with or without noise. In this paper, the global well-posedness of the noisy version of kinetic Cucker-Smale model with harmonic potential field is studied. We prove the global nonnegativity, existence, and uniqueness of the strong solution for the system (1.1). Our approach is based on a combination of weighted Sobolev spaces and approximation schemes, which have been shown to be effective in dealing with inherent nonlinear and nonlocal problems in ^[25,26,27].

Notation: We denote the usual $L^p$ norms on $\Omega$ by $\left\|f(t)\right\|_{L^p}: = \left\|f(t)\right\|_{L^p(\Omega)}$, and the $i^{th}$ element $L[f]_i$ of the vector $L[f]$ by

Then, we construct three special weighted Sobolev spaces with power $\omega (x, v) = (1+x^2+v^2)^{ \frac{1}{2}}$:

where

In the rest of the paper, we denote $C_d$ and $C_{d, \sigma}$ as positive constants while subscripts are used to indicate specific dependencies of such constants.

Definition 1.1. Let $0 \leqslant f(t, x, v) \in C ([0, \infty); L^1_\omega(\Omega)).$ The function $f(t, x, v)$ is a weak solution to system (1.1) if

We say $f(t, x, v)$ is a strong solution if $f(t, x, v)$ is a weak solution and $f(t, x, v) \in C ([0, \infty); H^1_\omega(\Omega))$.

Now, our main results are stated as follows.

Theorem 1.1. Assume initial datum $f_0(x, v)\in X(\Omega) \cap L^1_\omega(\Omega)$. Then, the system (1.1) admits a unique strong solution in sense of Definition 1.1.

Remark 1.1. In this paper, we consider a special potential function $U = \frac{1}{2}|x|^2$. In fact, similar to the study in ^[1], the potential $U$ can be extended to a more general case: assume that the potential function $U$ satisfies the following conditions:

One can obtain the existence and uniqueness of a strong solution to the system (1.1) with (1.3).

The existence of a strong solution is constructed in the weighted Sobolev space $H^1_\omega(\Omega)$. However, to find the Cauchy sequences in this space, we need to give extra estimates for terms such as $\left\| \omega \nabla_vf_n(t) \right\|_{L^2_\omega}^2$ and $\left\| \omega \nabla_xf_n(t)\right \|_{L^2_\omega}^2$. For this purpose, we construct the weighted Sobolev space $X(\Omega)$ and establish the a priori estimate for the preparation; see Proposition 3.1.

The rest of this paper is organized as follows. In Section 2, we establish a priori estimates of the system (1.1) by taking advantage of three special weighted Sobolev spaces. In Section 3, we first prove the local existence and uniqueness of the strong solution to system (1.1) by an iteration scheme and extend the local existence to the global one.

2.   A priori estimates

This section is devoted to the a priori estimates for the system (1.1). Directly integrating system (1.1) over $[0, t]\times \Omega$ gives that any smooth solution of it satisfies the following conservation law: $\int_\Omega f_0(x, v) \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v = \int_\Omega f(t, x, v) \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v = :\int_\Omega f(t) \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v$. Therefore, w.o.l.g., we assume $\int_\Omega f_0 (x, v) \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v = 1$ in the rest of paper. We first give the following a priori estimates for $\left\| f(t)\right \|_{L^1_\omega}$ and $\left\| f(t)\right \|_{L^2_\omega}$, which are better integrabilities of $f$ with large $v$ and $x$.

Lemma 2.1. Assume the function $f(t, x, v)$ is a smooth solution to system (1.1) with initial datum $0\leqslant f_0\in X(\Omega) \cap L^1_\omega(\Omega)$ and $\int_\Omega f_0 (x, v) \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v = 1$. Then, for $\forall \ t \geqslant 0$, we have

(1) $\int_\Omega f(t) \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v = 1$, $\left\| f(t)\right\|_{L^1} = 1$, and $f(t)\geqslant 0$;

(2) $\left\| f(t)\right \|_{L^1_\omega} \leqslant C(t)$;

(3) $\left \| f(t)\right \|_{L^2_\omega}^2 + 2\sigma \int_{0}^{t} \left \| \nabla_{v} f (\tau) \right \|_{L^2_\omega}^2 \mathop{}\negthinspace\mathrm{d} \tau \leqslant C \mathrm{ exp} \left(\int_{0}^{t} C (\tau) \mathop{}\negthinspace\mathrm{d} \tau \right)$;

where $C$ and $C(t)$ are positive constant and positive continuous functions of $t$ both depending on $\sigma, d$ and the weighted norms of the initial datum $f_0$.

Proof. (1) We multiply system (1.1) by $sgn(f)$ and integrate it over $\Omega$ to obtain

which implies $\left\| f(t)\right \|_{ L^1 } = \left\| f_0\right \|_{ L^1 } = 1, \; \; \forall t\geqslant 0$. Note that

We subtract system (2.1) from (2.2) to obtain $\int_\Omega f(t)\boldsymbol{1}_{[f(t) < 0]} \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v = 0$, which gives $f(t)\geqslant0, \; \forall t \geqslant 0$.

(2) Multiplying system (1.1) by $\omega$ and integrating by parts over $\Omega$ leads to

By applying Grönwall's lemma, we obtain

where $C(t)$ is the function of $t$ depending on $d$, $\sigma$, and $\|f_0\|_{L^1_\omega}$.

(3) We multiply system (1.1) by $2 \omega^{2} f$ to obtain

Then, we integrate system (2.3) by parts over $\Omega$ to obtain

where

Applying Grönwall's lemma to system (2.4), we obtain

where $C(t)$ is the function of $t$ depending on $d$, $\sigma$, and $\|f_0\|_{L^1_\omega}$.

Now we derive the estimates in the weighted Sobolev spaces. The first weighted space $H^1_\omega(\Omega)$ is prepared for the strong solution, while the second space $X(\Omega)$ is constructed to estimate the approximate solutions.

Proposition 2.1. Assume the function $f(t, x, v)$ is a smooth solution to system (1.1) with initial datum satisfying the condition of Lemma 2.1. Then, for $\forall\ t \geqslant 0$, we have

(1) $\left\| f(t)\right\|^2 _{H^1_\omega}+\left\| \omega f(t)\right \|_{L^2_\omega}^2 \leqslant C \mathrm{exp} \left(\int_{0}^{t} C (\tau) \mathop{}\negthinspace\mathrm{d} \tau \right)$;

(2) $\left\|f(t)\right\|^2_X+ \sigma \int_{0}^{t} \left\| \nabla_{v} f (\tau) \right\|^2_X \mathop{}\negthinspace\mathrm{d} \tau \leqslant C \mathrm{exp} \left(\int_{0}^{t} C (\tau) \mathop{}\negthinspace\mathrm{d} \tau\right)$;

where $C$ and $C(t)$ are positive constant and positive continuous functions of $t$ both depending on $\sigma, d$ and the weighted norms of the initial datum $f_0$.

Proof. (1) Applying $\nabla_v$ to system (1.1) gives

Multiplying system (2.5) by $2 \omega^{2} \nabla_vf$ leads to

Then we integrate system (2.6) by parts over $\Omega$ to obtain

By applying $\nabla_x$ to system (1.1), we obtain

Multiplying system (2.8) by $2 \omega^{2} \nabla_xf$ gives

Then, we integrate system (2.9) by parts over $\Omega$ to obtain

Now we estimate the last term $A$ on the right hand side of system (2.10).

where we have used the $\sigma$-Young's inequality and the following facts:

and

Combining system (2.11) with (2.10) yields

Due to the term $\left\| \omega f(t)\right \|_{L^2_\omega}^2$ which appeared in system (2.12), we need to analyze it in detail to close the a priori estimate in $H^1_\omega$. Similarly to the estimate of system (2.4), we multiply system (1.1) by $2\omega^4 f$ and integrate it over $\Omega$ to have

Adding up systems (2.4), (2.7), (2.12), and (2.13), we can get

By Grönwall's lemma and the estimate in Lemma 2.1, we obtain

where $C(t)$ depends on $d$, $\sigma$, and $\|f_0\|_{L^1_\omega}$.

(2) We estimate the first term of $\left\| f(t)\right\| ^2_X$. Multiplying system (2.5) by $2 \omega^{4} \nabla_v f$ gives

Integrating it over $\Omega$, we can get

Adding up the above estimates leads to

Then, we estimate the second term of $\left\| f(t)\right\| ^2_X$. Multiplying system (2.8) by $2 \omega^4 \nabla_x f$ leads to

Similarly, we integrate it over $\Omega$ as follows:

where we use the fact $max\left\{ |\varphi|, |\varphi^{'}|, |\varphi^{''}|\right\}\leqslant 1$;

Similar to the way of estimating the term $A$, we have

Adding up the above estimates for the integrals $\int_\Omega J_i \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v$ $(i = 1, 2, \cdots, 8)$, we obtain

Finally, we consider the last term of $\left\| f(t)\right\| ^2_X$. Similar to the estimate of system (2.4), we multiply system (2.5) by $2 \omega^{6} f$ and integrate over $\Omega$ to have

Adding up systems (2.14)–(2.16), one has

Applying Grönwall's lemma and the estimate $\left\| f(t)\right \|_{ L^1_\omega}$ in Lemma 2.1, we obtain

where $C(t)$ depends on $d$, $\sigma$, and $\|f_0\|_{L^1_\omega}$. Therefore, we finish the proof.

3.   Existence and uniqueness of strong solutions

In this section, we prove the local existence and uniqueness of the strong solution to system (1.1) by constructing a sequence of approximate solutions and extend the local existence to the global. First, let us recall a Grönwall-type lemma in ^[28].

Lemma 3.1. Let $T > 0$ and $(a_n)_{ n \in \mathbb{N} }$ be the sequence of the nonnegative continuous functions on $[0, T]$. Assume that $(a_n)$ satisfies

where $A, B$, and $C$ are nonnegative constants.

If $A = 0,$ there exists a constant $K \geqslant 0$ depending on $B, C$ such that

If $A > 0,$ there exists a constant $K \geqslant 0$ depending on $A, B, C$ such that

3.1.   Uniform bound on approximate solutions.

To begin with, we construct the following iteration scheme on finite time $[0, T]$.

for $n\geqslant1$. We define a sequence of approximate solutions $\{f_{n} \}$ as the solution to the above iterative system (3.1) by induction.

$\diamond$ Initial step ($n = 1$): we set

With this, we solve the initial value problem for the Cucker-Smale model with external potential field and noise:

subject to

$\diamond$ Inductive step: Suppose we have the sequence of smooth approximate solutions $\{f_k \}_{k = 1}^n$. Then, we can solve the following model:

subject to initial datum:

Thus, we can construct the smooth function $\{f_{n+1} \}$ from $\{f_{n} \}$. The solvability of system (3.2) is similar to that in the appendix of reference ^[29]. Therefore, the sequence $\{f_{n} \}$ is well-defined.

Paralleling to the a priori estimate for the solution $f$ in Section 2, we can establish the uniform energy estimates for the approximate sequence $f_{n}$ in the weighted Sobolev spaces.

Lemma 3.2. Let $T_1 > 0$. Assume the function $f_{n} (t, x, v)$ is a smooth solution to system (3.1) with the initial datum satisfying $f_0\in C^\infty(\Omega)\cap X(\Omega) \cap L^1_\omega(\Omega)$ and $\int_\Omega f_0 (x, v) \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v = 1$. Then, for $\forall \ t\leqslant T_1$, we have

(1) $\int_\Omega f_{n} (t) \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v = 1$, $\left\| f_{n}(t)\right \|_{ L^1 } = 1$, and $f_{n} (t) \geqslant 0$;

(2) $\left\| \omega f_{n}(t) \right\|_{L^1} \leqslant C$;

(3) $\left\| f_{n}(t) \right \|_{L^2_\omega } \leqslant C$;

where $C$ denotes the positive constant only depending on $\sigma, d, T_1$, and the weighted norms of the initial datum $f_0$.

Proof. (1) The results are obvious.

(2) We multiply system (3.1) by $\omega$ and integrate it over $\Omega$ to obtain

Then we integrate system (3.3) over $[0, t]$ to obtain

Applying Lemma 3.1 to system (3.4), one gets

where the positive constant $C$ depends on $\left\| f_0 \right \|_{ L^1_{\omega}}$, $d$, and $\sigma$. Then, for any given $T_1 > 0$, there exists a positive $C$ depending $\sigma, d, T_1$, and the weighted norms of the initial datum $f_0$ such that for all $n\geqslant 1$,

(3) Following the way of the computation for equations (2.3) and (2.4), we can conclude that $\left\| f_{n}(t) \right \|_{L^2_\omega } \leqslant C$.

Proposition 3.1. Let $T_1 > 0$. Assume the function $f_{n} (t, x, v)$ is a smooth solution to system (3.1) with initial datum satisfying the condition of Lemma 3.2. Then, for $\forall\ t\leqslant T_1$, we have

(1) $\left(\left\| f_n(t)\right\| ^2_{H^1_\omega(\Omega)} + \left \| \omega f_n (t) \right \|_{L^2_\omega}^2 \right)\leqslant C$;

(2) $\left\|f_n(t)\right\| ^2_X\leqslant C$;

where $C$ denotes positive constant only depending on $\sigma, d, T_1$, and the weighted norms of the initial datum $f_0$.

Proof. Following the similar proof of Proposition 2.1 in Section 2, we can obtain the parallel results.

3.2.   Convergence of approximate solutions.

Next, we show that the approximate solution $f_{n}$ is the Cauchy sequence in $C ([0, T ]; L^1_{\omega} (\Omega) \cap H^1_{\omega} (\Omega))$, where $0 < T < 1$. Setting $h_{n} : = f_{n + 1}- f_{n}$, it follows from system (3.1) that

It is obvious to see that

Proposition 3.2. Assume that initial datum $f_0 (x, v) \in C^\infty(\Omega)\cap X(\Omega) \cap L^1_\omega(\Omega)$, $\left\| f_0\right \|_{ L^1 (\Omega) } = 1$, and $f_0\geqslant 0$. For any given positive small time $T < 1$, we have

where $C$ denotes the positive constant only depending on $\sigma, d$, and the weighted norms of the initial datum $f_0$.

Furthermore, there exists a limit function $f(t, x, v) \in C([0, T ]; L^1_{\omega} (\Omega))$ such that

Proof. For arbitrary $T_1 > 0$ and $t\in[0, T_1]$, multiplying system (3.5) by $\omega sgn(h_n)$ and integrating it over $\Omega$ by parts lead to

where $C$ depends on $\sigma, d$, and the weighed norms of the initial datum $f_0$ and $T_1$, since we used the conclusion of Lemma 3.2. Then, we integrate system (3.6) to obtain

Using Lemma 3.1, we can derive that

Then, for any given small time $0 < T < 1$, we have

where $C$ denotes the positive constant only depending on $\sigma, d$, and the weighted norms of the initial datum $f_0$. This means that $f_n$ is the Cauchy sequence in $C([0, T], L^1_\omega (\Omega))$. Moreover, there exists a limit function $f(t, x, v)\in C ([0, T], L^1_\omega(\Omega))$ such that

Proposition 3.3. Assume that the initial datum $f_0(x, v)$ satisfies the condition of Proposition 3.2. For any given positive small time $T < 1$, we have

where $C$ denotes the positive constant only depending on $\sigma, d$, and the weighted norms of the initial datum $f_0$. Furthermore, there exists a limit function $f(t, x, v) \in C ([0, T ]; H^1_{\omega} (\Omega))$ such that

Proof. (1) Multiplying system (3.5) by $2 \omega^2 h_n$ leads to

Similar to the way to estimate system (2.3), we only need to estimate two extra terms $\int_\Omega K_5 \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v$ and $\int_\Omega K_6 \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v$. Note that

where $\omega^{\ast} = (1+y^2+{v^{\ast}}^2)^{ \frac{1}{2}}$, and we have used the $\sigma$-Young's inequality.

Thus, by integrating system (3.11) over $\Omega$ and using the estimate of Lemma 3.2 and Proposition 3.1, we have

Here, $C$ depends on $d, \sigma$, and weighed norms of initial datum.

By applying Grönwall's lemma to system (3.12) on $0 < t < 1$ and using the conclusion of Proposition 3.2, we obtain

Then for any given small time $T < 1$, we have

Hence, we obtain the estimate system (3.8).

(2) Multiplying system (3.5) by $2 \omega^{4} h_n$ and integrating it over $\Omega$ leads to

which is similar in form to the result of system (3.12). Hence, we obtain estimate system (3.9).

(3) Applying $\nabla_v$ to system (3.5) gives

We multiply system (3.15) by $2 \omega^{2} \nabla_v h_n$ and integrate it over $\Omega$. Comparing to system (2.7), we only need to estimate two extra terms $2 \int_\Omega \omega^{2} \nabla_v h_n \cdot L_1 \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v$ and $2 \int_\Omega \omega^{2 } \nabla_v h_n \cdot L_2 \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v$. Note that

where we have used the $\sigma$-Young's inequality.

Thus, similar to system (2.7), we use systems (3.16) and (3.17) to obtain

Applying $\nabla_x$ to system (3.5) gives

We multiply system (3.19) by $2 \omega^{2} \nabla_x h_n$ and integrate it over $\Omega$. Comparing to system (2.9), we only estimate the extra terms $2 \int_\Omega \omega^{2 } \nabla_x h_n \cdot L_3 \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v$, $2 \int_\Omega \omega^{2 } \nabla_x h_n \cdot L_4 \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v$, and $2 \int_\Omega \omega^{2 } \nabla_x h_n \cdot L_5 \mathop{}\negthinspace\mathrm{d} x \mathop{}\negthinspace\mathrm{d} v$.

Note that

where we have used the $\sigma$-Young's inequality.

Thus, similar to system (2.12), we use systems (3.20)–(3.22) to obtain

Adding up systems (3.12), (3.18), and (3.23), and applying Lemma 3.2-Proposition 3.2 and Proposition 3.3(2), we obtain

Similar to systems (3.13) and (3.14), for any given positive small time $T < 1$, we have

which is system (3.10). This means that the approximate solution $f_n$ is the Cauchy sequence in $C ([0, T ]; H^1_{\omega} (\Omega)).$ Thus, it converges strongly to the limit function $f \in C ([0, T ]; H^1_{\omega} (\Omega))$ as $n \rightarrow \infty.$

With the help of estimates system (3.7) in Proposition 3.2 and (3.24) in Proposition 3.3, we conclude that there exists a constant $C > 0$ depending on $d, \sigma$, and weighed norms of initial datum, such that for any given positive small time $0 < T < 1$,

Therefore, the limit function $f$ of Cauchy sequence $f_n\in C ([0, T ]; L^1_{\omega} (\Omega)\cap H^1_{\omega} (\Omega))$ is a local strong solution to the system (1.1). The uniqueness of the solution can be derived easily. Let $f$ and $\bar{f}$ be the two strong solutions above corresponding to the same initial datum $f_0$. Set

Then, by the same argument as in Lemma 3.2-Proposition 3.3, $\mathcal{E} (t)$ satisfies Grönwall's inequality:

and the standard Grönwall's lemma implies that

which gives the uniqueness of the local solution.

3.3.   Proof of Theorem 1.1.

When initial datum $f_0$ is smooth, the limit function $f$ from Proposition 3.2–Proposition 3.3 is the unique local smooth solution to system (1.1). Combining with Lemma 2.1 and Proposition 2.1, one can extend the local smooth solution to be global-in-time. Hence, we obtain the global smooth solution.

When initial datum $f_0$ is not smooth, we first mollify the initial datum by convolution, i.e.,

where $j_\epsilon(x, v)$ is the standard mollifier. Then, we consider the following modified system

Following the basic idea about the proof of Theorem 3.1 and Theorem 3.2 in ^[24], we can also prove that there exists a sequence $\epsilon_k$, with $\epsilon_k\rightarrow 0$, such that

and $f$ satisfies system (1.1). Hence, the limit function $f(t, x, v)$ is the desired unique strong solution.

Author contributions

All authors contributed equally to the study and the writing of the manuscript.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the Natural Science Foundation of China (No.12001097), Natural Science Foundation of Shanghai Municipality (No. 22ZR1402300), and AI-Enhanced Research Program of Shanghai Municipal Education Commission (No. SMEC-AI-DHUY-01).

Conflict of interest

The authors declare there is no conflict of interest.