1. Introduction

Microstructured optical fibers (MOFs) exhibit incomparable properties compared to traditional optical fibers owing to their flexible and adjustable structure, such as no cut-off single mode [1], high nonlinearity [2], high birefringence [3], large mode field [4], and adjustable dispersion [5]. Over more than two decades of rapid development, MOF has been widely used in laser technology [6], sensing [7–9], optical communication [10,11], optoelectronic integration [12], and various other fields. Although the MOF design is greatly mature, only about 20 % can be successfully fabricated, imposing limitations to their practical applications.

The fabrication of MOFs adopts a two-step method: preform and draw. There are various developed mature ways to prepare ideal preforms, such as tube stacking techniques, extrusion, mechanical drilling, three-dimensional (3D) printing, etc. However, hot drawing is the most mature and commonly used technology to fabricate optical fiber from preform. Meanwhile, real-time optimization of the draw parameters during a fiber draw including temperature profile, feed speed, draw speed, and pressure, is an expensive and time-consuming activity that is overly reliant on trial and error. Modeling of fiber draws is advantageous for both the increased efficiency of fiber drawing, i.e., choosing the correct parameters more quickly and wasting a smaller fraction of preform, and for the ability to explore the fabricated parameter space to identify designs that are both feasible and provide high optical performance before performing expensive fabrication trials.

Over the past two decades, studies have provided important insights through simulation analyses of fiber drawing. In 2002, Fitt et al. established a theoretical model of capillary drawing for holey fiber manufacturing, marking a significant milestone in research [13,14]. Their fluid model with a small aspect ratio based on asymptotic analysis quantified the parametric dependence of the drawing process on material parameters (density, viscosity, expansion coefficient, thermal conductivity, specific heat capacity) and draw parameters (pressure, temperature, feed speed, and draw speed). Experimental verification confirmed the feasibility of the model and established a theoretical foundation for the subsequent research on modeling the fiber draw. Subsequently, Voyce et al. from the same team studied the impact of preform rotation on the fiber structure using the Fitt model, thereby reducing or eliminating birefringence. They fabricated two types of fibers to verify the applicability of the model [15–17]. In 2005, Xue et al. reported on the continuous stretching process of MOF fabrication and quantified the influence of material properties and drawing conditions on the fiber’s microstructured holes. They analyzed the effects of surface tension, viscosity, and temperature on the deformation of microhole size by a 3D isothermal drawing model. The results showed that microholes are highly sensitive to viscosity distribution and temperature gradient. Finally, the consistency between the model and the experimental results was verified by experiments [18,19]. In 2008, Voyce et al. proposed a self-pressurization model of optical fiber in a sealed furnace and numerically solved the model. The results represented that increasing the length of the capillary or the temperature at the top of the capillary is beneficial in reducing the fluctuation of the fiber radius [20]. From 2010 to 2012, Luzi et al. focused on the influence of surface tension and hole pressure on the fiber structure during the fiber drawing process based on the asymptotic analysis and correction of the Fitt model and drew a six-hole MOF to verify the numerical results [21–23]. In 2022, the energy conservation equation in the optical fiber drawing process was analyzed using the asymptotic analysis method and coupled with the existing mass and momentum conservation equations. The validity of the model was verified both under pressure and without pressure [24]. In 2014-2015, Jasion et al. introduced a microstructure element method (MSE) for hollow-core photonic bandgap fibers to accurately simulate the structural evolution of MOFs during the drawing process. This method analyzed the force on the nodes in the microstructure, which is an extension of the discrete element method. The advanced nature of the model is verified experimentally [25–28]. In 2019 [29], the research group proposed the hollow-core anti-resonant fiber drawing model. They indicated by simulations that irreversible contact between adjacent capillaries may occur during the drawing process and presented a method to alleviate this phenomenon. It is of great significance to modify the model to guide the actual drawing of 100 km of fiber from a 1-meter preform. In 2016, Chen et al. used the asymptotic analysis method to study the drawing process of six-hole MOF, validating the model through experiments. This study systematically investigated the effects of surface tension, pressure, and draw speed on the air ratio of optical fiber [30–32]. In 2021, the abnormal situation during the drawing process with a large air ratio was observed and discussed both theoretically and experimentally [33]. In 2021, Tafti et al. enhanced the holey capillary model to study the influence of pressure in complex MOFs and verified the correctness of the proposed additional pressure through experiments. However, the model focused on the analysis of the change of the core, and it did not analyze the pressure change of the cladding [34].

The aforementioned studies are all based on the construction of fiber drawing models using the fluid mechanics Navier-Stokes equations and involve a simple hole model, six-hole MOF, anti-resonant hollow core fiber, photonic band gap fiber, and other structures. Nonetheless, the complex partial differential equation is simplified and asymptotically analyzed in the simulation process. Thus, the influence of small parameters is ignored. The validation of the models’ correctness was performed through experimental fabrication. Furthermore, the aforementioned studies relied on traditional numerical methods for solving partial differential equations, such as finite element method (FEM) [35], finite difference method (FDM) [36], and finite volume method (FVM) [37]. However, the aforementioned numerical methods face many challenges in solving complex partial differential equations: rely on grid partitioning, which affects the solution accuracy and stability, and in high-dimensional problems, an increase in grid dimensions leads to exponentially growing computational costs; handling complex boundary conditions can also lead to numerical instability and errors; for certain complex problems, a significant amount of computational resources may be required, limiting the application of traditional numerical methods to large-scale problems.

In recent years, neural network algorithms have been widely applied in various domains, such as image processing [38], natural language processing [39], and pattern recognition [40]. The powerful fitting ability of neural networks has shown excellent performance in numerical prediction and data generation, making neural network-based fluid dynamics simulations a hot research topic in the field of computational fluid dynamics [41–43]. Although deep learning models have a great advantage in computational cost, purely data-driven neural networks cannot learn the hidden physical principle controlled by equations. Therefore, introducing prior knowledge based on physical laws can effectively regulate the minimization process during the training of neural networks, ensuring robust fitting performance with limited samples. Based on this idea, Raissi et al. applied neural networks to approximate solutions of partial differential equations and proposed a physics-informed neural network (PINNs) [44]. Hereafter, they successfully solved both forward and inverse problems of partial differential equations using PINNs and discussed them using fluid control equations as an example. Inspired by the work of PINNs, the concept of physics-informed has sparked popular research interest in deep learning-based fluid flow studies. It has been proved that PINNs can solve the transient/steady state [45], low/high Reynolds [46], and forward/inverse problem [47] in fluid mechanics, offering a new and effective tool for the solution of fluid mechanics.

This article is based on the PINNs and describes the embedding of the complex partial differential equations governing the fiber drawing process into the loss function of a neural network. Leveraging automatic differentiation technology, the solution to complex partial differential equations is transformed into the neural network approximation problem, overcoming the dependence on grid partitioning in traditional numerical methods like FEM. Firstly, the complex partial differential equations are nondimensionalized to facilitate the embedding of the loss function into the neural network. Subsequently, a fully connected layer PINN network and a free boundary network model are constructed, and the optimal network parameters are determined by optimizing the network parameters. Finally, focusing on the rapid development of hollow-core anti-resonant fibers (HC-ARFs), this study investigates the influence of drawing parameters, such as feed speed, draw speed, and temperature, on its structure change. The results indicate that temperature has the most significant influence on the optical fiber drawing, followed by draw speed and feed speed. These insights provide valuable references for practical drawing.

2. Principle and method

The rational construction of the MOF drawing model is significant for describing the actual fiber drawing process. We modeled the drawing process of HC-ARFs, which can explain the change of fiber structure under different drawing parameters. Moreover, the model can also be applied to fibers with more complex structures by simply increasing the number of hole capillary and setting reasonable boundary conditions. By building a PINN for solving the MOF drawing model, the partial differential equation describing the evolution of the fiber is embedded in the loss function of the neural network, resulting in the evolution of the fiber structure.

2.1 Drawing model of HC-ARFs

Figure 1(a) presents the drawing model of the HC-ARFs. The HC-ARF comprises several cladding tubes arranged in a specific pattern, with each cladding tube exhibiting the same variation trend under identical drawing conditions. Therefore, the drawing process of the HC-ARF can be simplified as the drawing of a hole tube.

 

Fig. 1. (a) HC-ARFs drawing model; (b) Furnace temperature and viscosity distributions.

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The silica is in a molten state during the drawing process and follows the Navier-Stokes equation, as presented below:

Table 1. Parameter description

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Throughout the drawing process, the fiber quickly reaches a steady-state condition where the impact of time can be neglected. The primary focus is on the drawing process of the capillary under steady-state conditions. Meanwhile, transient heat transfer is excluded from this study due to the intricate heat transfer mechanisms involved in the capillary drawing process, including conduction, radiation, and convection, and stability of temperature distribution in the capillary under steady-state conditions. Moreover, the temperature distribution is directly specified. The furnace temperature is chosen to follow a Gaussian distribution [28,29], as presented in Eq. (5). Furthermore, the relationship between the viscosity of silica and temperature is described in Eq. (4) [48]. The furnace temperature and viscosity distribution are shown in Fig. 1(b). The density and surface tension of fused silica, the material employed in this work, are given by $\rho$ = 2200 kg/m$^3$ and $\gamma$ = 0.3 N/m.

Since the equation is appropriate for non-dimensionalized to take advantage of the small parameters present in the problem, we set $z = L\bar z$, $r = h\bar r$, $R = h\bar R$, $\mu = {\mu _0}\bar \mu$, $p = ({\mu _0}UL/{h^2})\bar p$, ${u_z} = U{\bar u_z}$, ${u_r} = (hU/L){\bar u_r}$. Here, $h$ denotes a typical drawn capillary size, and $U$ indicates a typical draw speed. Moreover, $\mu _0$ and $L$ represent a typical glass viscosity and length of the capillary, respectively. The equations now become:

The initial and boundary conditions of a partial differential equation are crucial for solving the equation, as they determine the correctness and accuracy of the solution. The initial conditions in the fiber drawing model are as follows:

At the entrance, when z = 0:

The variation of the inner and outer diameters of the fiber is the variable of interest and simultaneously serves as the free boundary conditions for the partial differential equation. These conditions must satisfy both the kinematic boundary conditions and the normal stress boundary conditions [13]:

So far, the mathematical model for fiber drawing has been fully established. By solving the above equations, the variations in the inner and outer diameters of the fiber and parameters, such as the flow velocity and pressure of molten silica, can be obtained. In prior studies, simplifications of the above equations were adopted to facilitate their solution using traditional numerical methods. This approach can be used for a simple simulation of the changes in the fiber’s inner and outer diameters and provides some reference values. However, if the equations are not simplified, the solutions will be closer to the exact solutions. It is worth noting that directly solving the equations using traditional numerical methods can be complex, but PINNs propose a new approach.

In the context of the fiber drawing model, the $air-ratio$ and the wall thickness ($\Delta t$) are considered as criteria for evaluating the quality of fiber drawing in addition to the inner and outer diameters. Therefore, the $air-ratio$ is defined as the ratio of the inner diameter to the outer diameter, and the wall thickness is defined as the difference between the outer and the inner diameters as follows:

2.2 PINNs principle and network architecture

PINNs successfully incorporate prior knowledge about the underlying physics into a loss function. Similar to any other NN model, PINN can minimize the loss and correctly model the underlying phenomenon [44]. L-BFGS(Limited-memory Broyden-Fletcher-Goldfarb-Shanno) is the most commonly used method for solving unconstrained nonlinear programming problems [49], known for its fast convergence speed and low memory overhead. Compared to gradient descent and stochastic gradient descent algorithms, L-BFGS converges faster and is more suitable for large-scale computations. Given the significant involvement of differential operators in our model, we have chosen the L-BFGS algorithm to minimize the loss. The Eqs. (1)–(3) are transformed into the residual form as follows:

The boundary conditions Eqs. (12) and (13) are also transformed into the residual form:

The PINN model is depicted in Fig. 2, consisting of two parts: the PINN network and the free boundary network. The neural network takes spatial coordinates ($r$, $z$) as input and the velocity and pressure ($u_r$, $u_z$, $P$) as outputs. Employing automatic differentiation techniques, the partial differential operators are computed (this can be directly formulated in deep learning frameworks, for example, using the "tf.gradients()" function in TensorFlow) to approximate the complex loss function. It is important to note that an additional neural network needs to be constructed for prediction since the inner and outer radii of the fiber are free boundary conditions. Wang [50] provided a deep learning method for free boundary conditions, which is valuable for this study. First, an additional neural network is constructed with $z$ as input and $R_i$ and $R_o$ as outputs, then they are used as inputs for the $r$ boundary conditions in PINNs. The outputs ($u_r$, $u_z$, $P$) represent the velocity and pressure of the free boundary conditions. The boundary loss is constructed based on kinematic and normal stress boundary conditions, achieving the overall loss. The loss function of the PINNs consists of the residual of the partial differential equations, initial conditions, and boundary conditions:

 

Fig. 2. PINNs neural network architecture.

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3. Results and discussion

By building the MOF drawing model and PINN architecture, the description and solution of the MOF drawing process can be realized. Then, the PINN network parameters will be optimized to more accurately solve complex partial differential equations. We trained the model using TensorFlow deep learning framework and running on the Intel Xeon Platinum 8255C CPU in 12 cores @2.50GHz, 43G memory as well as a RTX2080Ti (11G) GPU. After the network model is trained, the drawing parameters, including the feed speed, draw speed, and temperature will be optimized to explore the evolution trend of the fiber under different drawing parameters and provide guidance for the actual drawing.

3.1 Optimization of neural network parameters

The network size of a neural network plays a pivotal role in determining its performance and directly affects the value of the loss, thus impacting the accuracy of the solution. Therefore, optimizing the parameters of the neural network is necessary to obtain a lower loss. Figure 3(a) illustrates the impact of number of layers and nodes in the PINNs network on the loss. The layer increases from 10 to 30, and the number of nodes starts at 100 and increases by 50 increments up to 500. When the layer is 10, 15, and 20, the loss fluctuates as the number of nodes increases, but overall, it shows a decreasing trend. However, the loss decreases gradually when the layer is 25 and 30, reaching its minimum value at 30 layers and 250 nodes (indicated by the red arrow in Fig. 3(a)). Thus, the optimal number of layers and nodes for the PINNs network are 30 and 250, respectively. Figure 3(b) presents the parameter optimization of the free boundary network, where the number of layers ranges from 5 to 20 with an interval of 5, and the number of nodes is selected among 20, 40, 50, 80, 100, 120, 150, and 200. Results indicate that having more layers does not necessarily lead to better performance, while larger node numbers tend to be more favorable. The loss reaches its minimum value at 5 layers and 200 nodes (indicated by the red arrow in Fig. 3(b)), further confirming the necessity of network parameter optimization.

 

Fig. 3. Parameter optimization of neural network; (a) The impact of the number of layers and nodes in PINN network on $Loss$; (b) The impact of the number of layers and nodes in free boundary network on $Loss$; (c) Loss function weights impact on $Loss$; (d) $air-ratio$ under different loss function weights.

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Since the loss function consists of three components, where $Loss_{IC}$ and $Loss_{BC}$ are at the same level, while $Loss_{res}$ is two orders of magnitude higher than $Loss_{IC}$ and $Loss_{BC}$ (as can be observed in Fig. 3(a) and 3(b)), introduce loss function weights becomes highly necessary to accelerate network convergence and save computational time. As illustrated in Fig. 3(c), with the increase of $\alpha$, both $Loss_{IC}$ and $Loss_{BC}$ gradually decrease, and the $air-ratio$ changes from an upward trend to a downward trend consistent with the actual situation (indicated in Fig. 3(d)). However, when $\alpha$ is too large, $Loss_{res}$ will also gradually increase, resulting in a significant overall change in $Loss$. Therefore, considering both $Loss$ and $air-ratio$, $\alpha$=20 is chosen as the weight of the loss function. This choice can effectively reduce $Loss_{IC}$ and $Loss_{BC}$ without significantly increasing $Loss$, and the change in $air-ratio$ is consistent with the actual situation.

In optimizing the performance of the neural network model, the learning rate and iterations also have an impact on the model’s performance apart from the number of layers and nodes in the network. The learning rate, a crucial parameter in neural networks, affects the optimization speed and convergence of the weights and biases of the neurons during the training process. If the learning rate is too small, the neural network model takes longer to converge, leading to wasted training time and resources. Conversely, the loss experiences significant oscillations when the learning rate is too large, making it difficult for the neural network to converge, as shown in Fig. 4(a). Taking all factors into account, a learning rate of 1e$^{-7}$ is chosen as the optimal learning rate. The number of iterations is another influential factor in the neural network model. When the number of iterations is too small, the loss does not converge. Conversely, the loss starts to fluctuate when the number of iterations is too large, indicating that increasing the number of iterations does not necessarily improve the performance of the neural network. Additionally, the number of iterations also determines the training time of the neural network. Hence, considering all factors, 20,000 iterations are chosen as the optimal number of iterations, as shown in Fig. 4(b). At the same time, when the number of iterations is set to 20,000, the changes in the $air-ratio$ under different learning rates are calculated. As shown in Fig. 4(c), when the learning rates are set to 5e$^{-6}$ and 5e$^{-7}$, the $air-ratio$ exhibits a trend of initially increasing and then decreasing, which is inconsistent with the actual drawing process, where the $air-ratio$ should gradually decrease without applying pressure. Obviously, only a learning rate of 1e$^{-7}$ simultaneously ensures a smooth loss curve and a gradual decrease in the $air-ratio$ trend. Based on the optimized results of the network parameters mentioned above, a PINNs network size of 30 $\times$ 250 and a free boundary network size of 5 $\times$ 200 were used in the subsequent discussions of the drawing parameters. Moreover, the weight of the loss function was set to 20, the learning rate was set to 1e$^{-7}$, and the number of iterations was set to 20,000.

 

Fig. 4. The impact of learning rate and iterations on the loss function and $air-ratio$. (a) $Learning rate$; (b) $Iteration$; (c) $air-ratio$.

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Subsequently, we selected a set of drawing parameters for calculations, where the preform inner and outer diameters were 8 mm and 10 mm respectively, the furnace length was 0.16 m, the internal and external pressures were set to 0, the feed speed was 15 mm/min, the draw speed was 0.5m/min, and the temperature was 2000 $^{\circ }\textrm {C}$. Figure 5(a) shows the process of the fiber’s inner and outer diameters gradually decreasing with the furnace length. In the absence of applied pressure, the $air-ratio$ decreases gradually as depicted in Fig. 5(b). At the same time, as presented in Fig. 5(c), the $z$-velocity $u_z$ increases from the feed speed at the inlet to the draw speed at the outlet, satisfying the given feed speed and draw speed, where the color bar denotes the velocity distribution in the $z$-direction within the fluid. Figure 5(d) presents the pressure distribution inside the fiber, and the color bar represents the pressure distribution within the fluid. It is evident that the pressure experienced by the inner diameter is generally higher than that of the outer diameter. This phenomenon can be explained by the force balance at the inner and outer boundaries. Surface tension produces a pressure proportional to the inverse of the radius of curvature, $P=\gamma /r$ . As the inner diameter is smaller than the outer, the surface tension acting on them will be larger and therefore a higher pressure is required to prevent collapse of the inner diameter than for the outer, which aligns with theoretical expectations. Finally, Fig. 5(e) to 5(g) depict the schematic diagrams of the fiber structure at $z$ = 0 mm, $z$= 80 mm, and $z$ = 160 mm, respectively. The results reveal that the fiber dimensions gradually decrease, which is consistent with the actual fabrication process.

 

Fig. 5. The results of the drawing process with the following parameters: $P_H$=0, $P_a$=0, $u_f$=15 mm/min, $u_d$=0.5 m/min, $L$=0.16 m, $T$=2000 $^{\circ }\textrm {C}$, $R_i$(0)=8 mm, and $R_o$(0)=10 mm. (a) The variation of inner and outer diameters with length; (b) The variation of $air-ratio$; (c) The velocity distribution in the $z$-direction; (d) The pressure distribution; (e)-(g) Schematic diagrams of the fiber structure at different $z$ positions.

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In summary, this work not only predicts the changes in the inner and outer diameters of the fiber with furnace length but also visually represents the velocity and pressure distributions inside the fluid. It provides direct data for the theoretical analysis of fiber collapse and expansion, offering reference value for practical fiber drawing processes.

3.2 Optimization of drawing parameters

The control of drawing parameters during the fiber drawing process has a significant impact on the final quality of the fiber. During the drawing process, the main controlled parameters are temperature, feed speed, draw speed, and pressure. In the stage from preform to cane, due to the relatively large size of the fiber, the collapse phenomenon under the influence of surface tension is not severe. Therefore, it is possible to achieve the desired $air-ratio$ without applying or applying minimal pressure. Hence, in this paper, we set both the core pressure and capillary pressure to 0 and mainly discuss the effects of temperature, feed speed, and draw speed on the fiber drawing process.

In Figs. 6(a) and (b), the influence of feed speed and draw speed on the fiber $air-ratio$ are illustrated, respectively. At a draw speed of 5.0 m/min and a temperature of 2050 $^{\circ }$C, varying the feed speed from 4 mm/min to 24 mm/min reveals a minimal impact on the $air-ratio$, as indicated by the red dashed line in Fig. 6(c), where the final thickness of the fiber remains relatively consistent within this range of feed speeds.

 

Fig. 6. (a) The influence of feed speed on the $air-ratio$; (b) The impact of draw speed on the $air-ratio$; (c) The effect of feed speed and draw speed on the $\Delta t$.

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However, the $air-ratio$ of the fiber exhibits a noticeable change in the draw speed with a feed speed of 8 mm/min and a temperature of 2050 $^{\circ }$C, as shown in Fig. 6(b). With the draw speed increasing from 0.5 m/min to 5.0 m/min, the decrease in $air-ratio$ gradually reduces, indicating that increasing the draw speed can mitigate the fiber collapse phenomenon, which is consistent with the actual drawing process. Figure 6(c) further indicates that as the draw speed increases, the final $\Delta t$ of the fiber gradually decreases, and once the draw speed exceeds 2.5 m/min, the fiber thickness stabilizes. These findings imply that the influence of the feed speed on the inner and outer diameters is not as significant as the effect of the draw speed in the fiber drawing process. To improve the collapse phenomenon during fiber drawing, applying a lower draw speed is necessary.

Finally, the influence of temperature on fiber drawing was analyzed. As shown in Fig. 7(a), the temperature rises from 1900 $^{\circ }$C to 2050 $^{\circ }$C with intervals of 25 $^{\circ }$C. Notably, the temperature acts as a dividing point at 2000 $^{\circ }$C. When the temperature is below 2000 $^{\circ }$C, the $air-ratio$ gradually increases, while the $air-ratio$ gradually decreases when the temperature exceeds 2000 $^{\circ }$C. To investigate the reasons behind this phenomenon, the variations of inner and outer diameters with temperature are provided in Fig. 7(c) and Fig. 7(d), respectively. The maintenance of capillary shape is determined by the combined effects of surface tension, viscous forces, and pressure acting on the capillary wall [28,33,34]. If surface tension and viscous forces dominate, collapse will occur, whereas if pressure dominates, the capillary will expand. When surface tension and pressure are fixed, the relationship between viscosity and temperature as given by Eqs.(4) indicates that higher temperatures result in lower viscosity. As shown in Fig. 5(d), the pressure distribution of the capillary’s inner and outer diameters is not uniform, resulting in different resistances to viscous forces. Therefore, the threshold at which the capillary’s inner and outer walls collapse is also different, leading to inconsistent changes in inner and outer diameters indicated in Fig. 7(c)(d), which in turn result in changes in $air-ratio$.

 

Fig. 7. The influence of different temperatures. (a) The influence of different temperatures on $air-ratio$; (b) The variation of maximum pressure and $\Delta t$ at different temperatures; (c) The change of inner diameter with temperature; (d) The variation of outer diameter with temperature; (e)-(g) The structures at $T_{max}$ = 1900 $^{\circ }$C, $T_{max}$ = 1950 $^{\circ }$C, and $T_{max}$ = 2000 $^{\circ }$C, respectively.

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At the same time, the variations in pressure and $\Delta t$ at different temperatures were calculated, as shown in Fig. 7(b). As the temperature increases leading to fluid flow more fast, resulting in a quicker reduction in capillary size. As the capillary diameter decreases, greater pressure is required to alleviate capillary collapse under the influence of surface tension (the inner pressure is greater than the outer, as shown in Fig. 5(d)). Therefore, in the absence of pressure applied to both the inner and outer sides, the collapse of the inner wall of the fiber intensifies with increasing temperature. Consequently, $\Delta t$ increases with rising temperature, as shown in Figs. 7(e)-(g).

3.3 Discussion

By integrating the above results and leveraging the capability of the PINNs to solve intricate partial differential equations, the original governing equations are directly solved without the need for additional simplifications. This approach allows us to obtain the variations in the inner and outer diameter of each cladding tube in the HC-ARFs under different drawing parameters. Additionally, the velocity and pressure distribution in the fluid region can be obtained, facilitating a better understanding of the impact mechanisms of various parameters on the fiber drawing process. After optimizing the hyperparameters in PINNs, including the number of hidden layers, the number of nodes, the learning rate, and the iterations, the optimal combination of network parameters is obtained. Subsequently, the drawing parameters that affect the fiber structure are discussed, including the feed speed, draw speed, and temperature. The results show that temperature and draw speed have a greater impact on the drawing process compared to the feed speed. Therefore, it is important to focus on these factors during the actual drawing process.

Table 2 provides a summary comparing existing research studies on fiber drawing process simulation with the present study in this paper. The fiber types range from simple single-hole fibers to HC-ARFs and even complex photonic bandgap fibers. However, all these studies used simplified Navier-Stokes equations and employed traditional numerical methods, transitioning from simple single-hole structures to complex ones. However, the proposed approach starts from the original Navier-Stokes equations and directly approximates them using the PINNs to obtain the fiber drawing process. In addition, the previously reported research studies only output the variations in the inner and outer diameters of the fiber, neglecting the velocity and pressure distribution. In contrast, the proposed method can directly obtain the velocity and pressure distribution, which can be used to analyze the mechanisms by which different drawing parameters affect the fiber structure changes. Although this work primarily focuses on the drawing process of single-ring HC-ARFs, the same method can also be extended to complex nested HC-ARFs by incorporating multiple equations.

Table 2. Comparison of existing research work

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Finally, although the PINNs avoid issues such as step size dependence in traditional numerical methods and grid partitioning difficulties in FEM, they still face challenges during the neural network training process. Therefore, further research is required to improve neural network models, such as DeepONet [51], RPINNS [52], and FlowDNN [46], to enhance the speed and accuracy of the solution. Moreover, it becomes necessary to consider the mutual interaction between each layer of air holes when extending from structurally simple HC-ARFs to complex photonic crystal fibers due to the increase in the number of air holes. Consequently, solving the equations becomes more complex, and currently, there is no in-depth research on this aspect. The PINNs have unique advantages in solving high-dimensional and complex partial differential equations, with the potential for acceleration using neural networks. Therefore, a virtual fabrication simulation of complex photonic crystal fibers is expected to be achieved, which will be the focus of our future research.

4. Conclusion

Overall, this paper establishes a virtual fabrication model for MOFs. It employs PINNs for the first time to solve the complex partial differential equations describing the fiber drawing process. Additionally, a specific neural network is constructed to solve the free boundary conditions. The influential parameters in the fiber drawing process, such as temperature, pressure, feed speed, and draw speed, are coupled into the loss function of the neural network through initial and boundary conditions. By optimizing the parameters of the neural network to minimize the loss function value, the virtual fabrication model of microstructure optical fibers is accurately solved. The results indicate that this model not only captures the trends of the inner and outer diameters of the fiber but also provides the velocity and pressure distribution inside the fluid. Furthermore, the pressure distribution results demonstrate that the inner pressure is significantly higher than the outer pressure, providing data to explain capillary collapse in the absence of pressure. Moreover, as temperature increases, the pressure difference gradually amplifies, leading to a more pronounced collapse phenomenon. The paper also investigates key parameters affecting the fabrication of MOFs, such as temperature, feed speed, and draw speed. Through systematic study, it is evident that temperature and draw speed have a more significant impact on the fabrication process than feed speed. Therefore, these two parameters should be given particular attention in practical fabrication processes. The paper provides new reference values for simulating the complex fabrication process of MOFs, enabling preliminary simulations. With improvements in algorithms and computational acceleration, it is anticipated that more precise simulations of the highly complex MOFs fabrication process will be achieved in the future, which holds promise for advancing the development of MOFs.