Decoy simulation flow field modeling method based on PINN-U-Net and local-global feature fusion

Abstract

The application discloses a decoy bomb simulation flow field modeling method based on PINN-U-Net and local-global feature fusion, which comprises the following steps: constructing a simulation data set of a decoy bomb flow field through numerical simulation, and establishing a physical loss function meeting a decoy bomb temperature field control equation and a boundary condition; designing a global feature extraction module based on a Transformer, splicing a continuous multi-frame time sequence physical quantity sequence in the data set and adding spatial coordinate information to form an input vector, inputting the input vector into the global feature extraction module, extracting and outputting a feature vector corresponding to each physical point; replacing a main network of PINN with a U-Net network structure to construct a PINN-U-Net network, splicing frame data and a feature vector corresponding to each point of the corresponding frame to form an input tensor of the PINN-U-Net network; and outputting physical quantities in the decoy bomb flow field under the constraint of the physical loss function. The modeling speed and the calculation accuracy for a decoy bomb combustion scene are significantly improved, various limitations suffered by traditional numerical calculation are avoided, and a high-speed and accurate scene modeling function can be provided.

Description

Technical Field

[0001] The present invention relates to the fields of computational fluid dynamics and deep learning technology, and specifically to a decoy bomb simulation flow field modeling method based on PINN-U-Net and local-global feature fusion. Background Art

[0002] With the continuous evolution of infrared guidance technology, infrared decoy flares, as a crucial defense against infrared-guided weapons, have a direct impact on the survivability of battlefield platforms. Decoy flares simulate infrared radiation characteristics similar to those of target aircraft, interfering with enemy infrared seekers and achieving target misdirection and interception evasion. To enhance the effectiveness of decoy flares in complex combat environments, high-precision and efficient modeling and simulation of their temperature field evolution and radiation distribution characteristics are required, particularly dynamic modeling of the convective field while ensuring real-time responsiveness.

[0003] Traditional numerical simulation methods, such as computational fluid dynamics (CFD) models based on finite differences or finite volumes, while offering strong physical consistency and accuracy guarantees, suffer from high computational overhead and long simulation cycles, making them incapable of meeting the application requirements of highly dynamic and real-time combat missions. On the other hand, deep neural network models, which rely solely on large-scale sample data for training, while offering significant advantages in inference speed, lack physical mechanism constraints and are prone to modeling results that violate fundamental physical laws. This leads to poor robustness and reliability under extreme conditions.

[0004] Physics-Informed Neural Networks (PINNs), which have emerged in recent years, effectively enhance the neural network's ability to follow physical laws by introducing governing equations (such as the Navier-Stokes equations and the convection-diffusion equation) into the loss function as physical constraints. This improves modeling accuracy and generalization. However, traditional PINNs often use a feedforward, fully connected structure, which has limited ability to model the large-scale nonlinear structures, spatiotemporal coupling, and global correlations present in complex temperature fields. Summary of the Invention

[0005] The purpose of the present invention is to provide a decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion to solve the problems existing in the above background technology.

[0006] To solve the above technical problems, the present invention adopts the following technical solutions: a decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion, comprising:

[0007] The simulation data set of the decoy bullet flow field is constructed through numerical simulation. The data set includes the spatial temperature field distribution, density field distribution, three-dimensional velocity field and pressure field at each moment.

[0008] Design a flow field modeling framework under physical constraints, including:

[0009] Establish a physical loss function that satisfies the temperature field control equations and boundary conditions of the decoy missile;

[0010] A Transformer-based global feature extraction module is designed. This module concatenates multiple frames of continuous temporal physical quantity sequences in the dataset and appends spatial coordinate information to form an input vector. This vector is then fed into the global feature extraction module to extract and output the feature vector corresponding to each physical point.

[0011] The U-Net network structure replaces the main network of PINN to construct a PINN-U-Net network. The frame data is spliced ​​with the corresponding feature vector of each point in the corresponding frame to form the input tensor of the PINN-U-Net network. The encoder part of the U-Net network performs feature extraction, and the decoder part gradually restores the spatial resolution. Under the constraint of the physical loss function, the velocity field, pressure field, temperature field and density field in the flow field of the decoy bullet are output.

[0012] Preferably, the calculation results at each moment in the data set are saved in the form of a complete three-dimensional scene data cube, and the data clearly records the spatial temperature field distribution, density field distribution, and three-dimensional velocity field and pressure field at each moment.

[0013] Preferably, the maximum and minimum value normalization is used to process the data of spatial temperature field distribution and density field distribution, and the quartile normalization method is used to process the data of three-dimensional velocity field and pressure field. After normalization, the data are divided into training set and test set.

[0014] Preferably, the automatic differentiation method of pytorch is used for the simulation data set, and the differential equation loss function is calculated based on the Navier-Stokes equation ; Calculate the initial state loss of each physical quantity that meets the preset initial conditions at the initial moment ; Joint differential equation loss function and initial state loss , and the boundary condition loss function , establish the physical loss function as:

[0015] ;

[0016] in, is the weight coefficient.

[0017] Preferably, four groups of Transformer encoding blocks are stacked in the global feature extraction module, each Transformer encoding block contains eight attention heads, and the embedding dimensions of the input and output are both 128 dimensions; and

[0018] In each Transformer encoding block, the multi-head attention sublayer is followed by a feed-forward neural network consisting of two fully connected layers:

[0019] The first fully connected layer maps 128-dimensional features to a 256-dimensional latent space and uses the GELU activation function to enhance nonlinear modeling capabilities;

[0020] The second fully connected layer maps it back to 128 dimensions;

[0021] Each sub-layer adopts a residual connection mechanism;

[0022] Further preferably, the Transformer module models the input space-time physical point sequence based on the self-attention mechanism, takes a three-frame three-dimensional physical quantity sequence as input, and combines position encoding to capture the long-range dependency and spatiotemporal correlation characteristics in the flow field data; it can effectively integrate the global information between different regions, construct a deep feature representation that is expressive of complex flow patterns, provide rich prior knowledge and structural guidance for the subsequent PINN-U-Net model, and enhance its ability to understand high-dimensional nonlinear physical processes.

[0023] Preferably, the distribution information of the three-dimensional physical field on the spatial grid is extracted for each frame of data, and the size of the grid is constructed. The input tensor is Represents six original physical quantity channels, are the spatial resolution dimensions of the current frame, t represents the normalized time information corresponding to each frame, and is used to explicitly introduce the time state; and

[0024] The frame data and the corresponding feature vector of each point in the corresponding frame are concatenated to form the final input tensor. The input dimension is .

[0025] Preferably, the encoder part of the U-Net network gradually extracts local spatial features through multi-layer three-dimensional convolution operations and downsampling structures, effectively capturing the local change trend of physical quantities in the decoy missile flow field. The decoder part jumps and fuses feature information of different levels to achieve high-precision spatial field reconstruction, and outputs the velocity field vector in the decoy missile temperature field. , pressure field p, temperature field T and density field , realizing the joint prediction of multiple physical quantities.

[0026] Preferably, the flow field modeling framework is optimized, including:

[0027] Combine physical constraint loss and data loss to form a comprehensive loss function;

[0028] Based on the data in the simulation dataset, the flow field modeling framework is trained and optimized by minimizing the loss function through the gradient descent method.

[0029] Furthermore, after the model training is completed, the initial state of the decoy missile evolution process to be modeled (i.e., initial spatial coordinates, initial physical parameters, and initial temperature and density distribution) is input into the network, and the future time of modeling is specified. The network will automatically generate the numerical calculation results of each physical field at the specified modeling time. The calculation results are compared with the actual numerical solution for a detailed error analysis, and the root mean square error (RMSE) and relative error are used to calculate the error. Error indicators quantitatively evaluate modeling accuracy and model reliability, providing an accurate reference for further improving model performance.

[0030] Beneficial effects: By designing a flow field modeling framework under a physical constraint mechanism and integrating the partial differential equations reflecting physical laws into the loss function in the form of residuals, the present invention circumvents the iteration and CFL restrictions required for traditional numerical solutions through deep learning network training, effectively improving the speed of modeling from the initial scene to the steady-state scene, and the network fitting results conform to physical laws.

[0031] In addition, by integrating the Transformer global feature extraction module and the U-Net structure, the collaborative modeling of the global evolution trend and local spatial features in complex temperature field data is achieved, significantly enhancing the model's expressive ability in the decoy bomb temperature field modeling task; the Transformer global feature extraction module is based on the self-attention mechanism, which can effectively capture the long-distance spatiotemporal dependencies existing in the physical field, and enhance the model's global perception of the overall flow structure, heat transfer process and evolution law; the extracted global context features are fused with the original physical quantity input data and passed to the U-Net network as multi-channel input; the U-Net encoder-decoder structure has excellent local feature extraction and multi-scale restoration capabilities, and on the basis of maintaining the consistency of the physical partial differential equation constraints, it realizes high-precision joint modeling of multiple physical quantities such as velocity, temperature, pressure and density in the decoy bomb temperature field; this fusion strategy effectively overcomes the problems of limited field of view and local overfitting of traditional neural networks when dealing with nonlinear and strongly coupled temperature fields, and provides a new modeling path for achieving high-speed and high-fidelity decoy bomb flow field simulation.

[0032] The technical solution of the present invention significantly improves the modeling speed and calculation accuracy of the decoy bomb combustion scene, circumvents various limitations of traditional numerical calculations, and can provide high-speed and accurate scene modeling functions. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0034] In the attached figure:

[0035] Figure 1 Schematic diagram of the overall network structure of the method of the present invention.

[0036] Figure 2 This is a schematic diagram of the Transformer feature extraction module structure in the present invention.

[0037] Figure 3 Schematic diagram of the PINN-U-Net network structure.

[0038] Figure 4 This is a comparison diagram of the network results and the real data. DETAILED DESCRIPTION

[0039] The following describes the embodiments of the present invention in conjunction with the accompanying drawings. The terms used in the embodiments of the present invention are only used to explain the specific embodiments of the present invention and are not intended to limit the present invention. The following describes the embodiments of the present application in conjunction with the accompanying drawings.

[0040] refer to Figure 1-Figure 3 As shown in FIG, the decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion includes:

[0041] The simulation data set of the decoy bullet flow field is constructed through numerical simulation. The data set includes the spatial temperature field distribution, density field distribution, three-dimensional velocity field and pressure field at each moment.

[0042] The temperature field model of the decoy is described by three equations: the Navier-Stokes equation, the temperature field convection-diffusion equation, and the density field equation. The Navier-Stokes equation describes the velocity field and pressure of the gas during the combustion of the decoy:

[0043] Momentum equation:

[0044] ;

[0045] Continuity equation:

[0046] ;

[0047] in The velocity contains velocity components in three directions, For time, is the density, is the pressure, is the viscosity coefficient, For wind power, is the buoyancy term.

[0048] The Navier-Stokes equations can be broken down into four terms for solution: convection term, external force term, viscous force term, and pressure term. The convection term is solved using the semi-Lagrangian method, which tracks the current velocity field backward, and the final velocity is obtained by interpolation of the grid points where it is located, so it will not exceed the maximum and minimum values ​​of the original two fields. The external force term includes buoyancy term and eddy current suppression force. The buoyancy is controlled by the buoyancy adjustment factor, and the eddy current suppression force is obtained by solving the velocity field curl, calculating the gradient of the curl field modulus and normalizing it. The two are combined and the temperature field is updated through a forward Euler integral. The viscous force term is solved by constructing the Poisson equation and using iterative relaxation technology. The pressure field and viscous force terms are solved in the same way.

[0049] Temperature field equation:

[0050] ;

[0051] in The velocity contains velocity components in three directions, For time, is the temperature, is the thermal diffusivity, is the Stefan-Boltzmann constant, is the ambient air temperature, is the thermal efficiency factor, As heat source.

[0052] Density field equation:

[0053] ;

[0054] is the density, The velocity contains velocity components in three directions, is the density diffusion coefficient, Field source density.

[0055] Since the temperature field and density field equations are similar, the solution methods are also the same. The convection term is solved by the semi-Lagrangian method, and the diffusion term constructs the Poisson equation and is solved by the iterative relaxation technique.

[0056] In a specific case, the simulation parameters are set to space size: 80X250X80, grid size 0.1, time resolution 0.001, air temperature 300K, buoyancy adjustment coefficient 0.6, thermal efficiency factor 0.7, wind speed 3 , thermal diffusion coefficient is 0.3, density value is 1.293 , the viscosity coefficient is , with a density diffusion coefficient of 0.3. Initially, a spherical heat source with a radius of 0.4m is placed in the center of the scene. The initial temperature of the heat source is 3500K, the initial value of the velocity field is 0, the initial value of the background temperature field is 300K, the density at the heat source is 50, and the background is 0. Boundary condition settings: Velocity field: No-slip boundary condition, meaning the velocity value at the boundary is 0; Temperature field: Adiabatic boundary condition, meaning the temperature field at the boundary is 0.

[0057] The numerical solution results are normalized in a targeted manner, where the temperature field and density field are normalized using the maximum and minimum values:

[0058] ;

[0059] Quartile normalization is used for velocity and pressure fields:

[0060] ;

[0061] Design a flow field modeling framework under physical constraints, including:

[0062] Step 1: Establish a physical loss function that satisfies the temperature field control equations and boundary conditions of the decoy bomb;

[0063] Temperature field loss function:

[0064] ;

[0065] NS momentum equation loss function:

[0066] ;

[0067] ;

[0068] ;

[0069] NS continuous equation loss function:

[0070] ;

[0071] Density field equation loss function:

[0072] ;

[0073] Therefore, the total differential equation loss function is:

[0074] ;

[0075] Initial state loss, that is, when t is 0, each physical quantity needs to meet the preset initial conditions. Assume that the initial conditions are , , , , but:

[0076] Boundary condition loss function:

[0077] ;

[0078] After calculating all the above losses, the physical loss function of the PINN model is obtained:

[0079] ;

[0080] in is the weight coefficient, which is used to balance the contribution of different losses to the total physical loss;

[0081] like Figure 2 As shown in the figure, a global feature extraction module based on Transformer is designed. The continuous multi-frame time series physical quantity sequences in the data set are spliced ​​and the spatial coordinate information is added to form an input vector, which is input into the global feature extraction module to extract and output the feature vector corresponding to each physical point.

[0082] Specifically: the time series frame dataset is input into the Transformer feature extraction module, and the features of each time frame can be expressed as: , by concatenating three consecutive frames of time points into a complete input vector and appending spatial coordinate information: This feature vector serves as the input to the Transformer module; each Transformer encoding block contains eight attention heads (Multi-head Attention), with both the input and output embedding dimensions set to 128, resulting in a dimension of 16 per head. The Multi-head Attention mechanism computes multiple attention subspaces in parallel, enabling the network to simultaneously capture multiple correlations between different regions in the input sequence.

[0083] In each Transformer encoding block, the multi-head attention sub-layer is followed by a feed-forward neural network (FFN), which consists of two fully connected layers. The first fully connected layer maps 128-dimensional features to a 256-dimensional latent space, using the GELU activation function to enhance nonlinear modeling capabilities. The second fully connected layer maps them back to 128 dimensions to ensure consistent module input and output dimensions. Each sub-layer (including the attention layer and the FFN) uses a residual connection mechanism, adding the input to the sub-layer output to improve gradient propagation stability. The network is also normalized in conjunction with the normalization layer to accelerate convergence and improve training stability.

[0084] The entire Transformer feature extraction module is composed of four stacked groups of the above structures. It can globally encode the input sequence data consisting of three-dimensional spatial position, timestamp and initial physical quantity, and extract the context-related deep feature representation corresponding to each physical point;

[0085] refer to Figure 2 As shown in the figure, the U-Net network structure replaces the main network of PINN to construct a PINN-U-Net network, which is used to model the spatiotemporal evolution relationship of multiple physical quantities in the three-dimensional flow field of the decoy missile;

[0086] Specifically, for each frame of data, the distribution information of its three-dimensional physical field (including three-dimensional velocity components, pressure, temperature and density) on the spatial grid is extracted, and a grid of size The input tensor is Represents six original physical quantity channels, are the spatial resolution dimensions of the current frame respectively. Represents the normalized time information corresponding to each frame, which is used to explicitly introduce the time state.

[0087] Combined with the 128-dimensional context feature tensor generated by the Transformer module for each spatial grid point, it is further spliced ​​to the above physical channel to form the final input tensor. The input dimension is , that is, a total of 135 channels, including original physical quantities, time channels, and global spatiotemporal context features.

[0088] The input tensor is fed into the U-Net network with a three-dimensional convolutional structure. The U-Net network adopts a typical encoder-decoder architecture, which consists of a five-layer encoder and a four-layer decoder. It has the ability to model complex three-dimensional physical fields at multiple scales. In the encoder part, the network stacks five convolutional modules in sequence, each module contains two layers. The three-dimensional convolution layer is used with batch normalization (BatchNorm3D) and LeakyReLU activation function for nonlinear feature extraction, and a kernel size of The maximum pooling layer is used for spatial downsampling.

[0089] The encoder channel settings are: 134 (input channels, including 5 physical channels, 1 time channel and 128-dimensional Transformer features) → 64 → 128 → 256 → 512 → 1024. The decoder part consists of four groups of deconvolution and convolution modules. Each group is first upsampled by transposed convolution, then jump-connected to the corresponding encoding layer feature map, and then twice Convolution, normalization and activation operations, the channel settings are: 1024 → 512 → 256 → 128 → 64. Finally, through a layer The convolution of the feature map compresses the number of channels to 6, corresponding to the three-dimensional velocity components ,pressure( ),temperature( ) and density ( )’s predicted value.

[0090] During the forward propagation process, the input data tensor passes through the encoder in sequence for feature extraction, and is then restored to the complete spatial resolution layer by layer by the decoder, outputting a continuous six-dimensional physical quantity prediction map. During the training process, the output result is first subjected to point-by-point error calculation with the corresponding real data to form a data supervision loss (such as MSE loss). At the same time, the network output is denormalized and then participates in the calculation of the residual of the physical partial differential equation, including the residual loss of the control equation (such as Navier-Stokes momentum conservation, mass conservation), the residual loss of the heat conduction equation, and the boundary condition constraint loss. These physical losses are calculated through automatic differentiation of high-order derivative residual terms, and together with the supervision loss, they constitute the total loss function, and the network parameters are optimized through the backpropagation mechanism;

[0091] Optimize the flow field modeling framework, including:

[0092] Combine physical constraint loss and data loss to form a comprehensive loss function;

[0093] Based on the data in the simulation dataset, the flow field modeling framework is trained and optimized by minimizing the loss function through the gradient descent method.

[0094] In one specific case, the Adam optimizer was used for training, with the maximum learning rate set to 0.0001, the minimum learning rate set to 0.000001, and the batch size set to 3, meaning three frames of the scene were input each time, and the number of training rounds was 30,000. The Adam optimization algorithm was used to train the PINN-Transformer fusion model to obtain an optimized model for modeling the temperature and density fields.

[0095] Using the trained modeling network, the initial scenario, i.e. the initial value and initial distribution of each physical quantity and the time to be predicted, is input, and the network outputs all the numerical results of the physical field at the corresponding time, such as Figure 4 Combined with the numerical solution, the error analysis of the modeling results is carried out. Here we show the temperature field calculation results. Figure 4 The above figure is the network calculation result. Figure 4 The following figure shows the theoretical calculation results. The evaluation criteria include mean square error MSE, relative Error. The smaller the value of both, the higher the modeling accuracy. The value is It means that the prediction results are credible. When the MSE value is less than 1 when the data set range is [300-3500], it means that the error is less than 1°C and the modeling accuracy is high.

[0096] Based on the detailed introduction above, the present invention proposes a decoy missile temperature field modeling method based on the fusion of Transformer and PINN. This method uses the Transformer network to extract features from the three-dimensional spatiotemporal distribution of the decoy missile's physical field, forming prior prediction information with global perception capabilities. This information is introduced as a data-driven supervision term into the PINN loss function, and the network parameters are optimized together with the physical constraint residual term constructed by automatic differentiation. While maintaining physical consistency, the global features provided by the Transformer module significantly improve the modeling capability of nonlinear temperature field structures. This method not only enhances the model's prediction accuracy and generalization ability for complex temperature fields, but also effectively shortens the simulation time, showing good potential for actual combat deployment and engineering application prospects.

[0097] The above describes the embodiments of the present invention in detail with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. After knowing the contents described in the present invention, ordinary technicians in this technical field can make several equivalent changes and substitutions without departing from the principles of the present invention. These equivalent changes and substitutions should also be regarded as falling within the scope of protection of the present invention.

Claims

1. A decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion, characterized by: include: The simulation data set of the decoy bullet flow field is constructed through numerical simulation. The data set includes the spatial temperature field distribution, density field distribution, three-dimensional velocity field and pressure field at each moment. Design a flow field modeling framework under physical constraints, including: Establish a physical loss function that satisfies the temperature field control equations and boundary conditions of the decoy missile; A Transformer-based global feature extraction module is designed. This module concatenates multiple frames of continuous temporal physical quantity sequences in the dataset and appends spatial coordinate information to form an input vector. This vector is then fed into the global feature extraction module to extract and output the feature vector corresponding to each physical point. The global feature extraction module consists of four stacked Transformer encoding blocks, each containing eight attention heads, with both input and output embedding dimensions of 128 dimensions; and In each Transformer encoding block, the multi-head attention sublayer is followed by a feed-forward neural network consisting of two fully connected layers: The first fully connected layer maps 128-dimensional features to a 256-dimensional latent space and uses the GELU activation function to enhance nonlinear modeling capabilities; The second fully connected layer maps it back to 128 dimensions; Each sub-layer adopts a residual connection mechanism; The U-Net network structure replaces the main network of PINN to construct a PINN-U-Net network. The frame data is spliced ​​with the corresponding feature vector of each point in the corresponding frame to form the input tensor of the PINN-U-Net network. The encoder part of the U-Net network performs feature extraction, and the decoder part gradually restores the spatial resolution. Under the constraint of the physical loss function, the velocity field, pressure field, temperature field and density field in the flow field of the decoy bullet are output.

2. The decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion according to claim 1 is characterized by: The calculation results at each moment in the dataset are saved in the form of a complete three-dimensional scene data cube.

3. The decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion according to claim 2 is characterized by: The maximum and minimum value normalization method is used to process the spatial temperature field distribution and density field distribution data, and the quartile normalization method is used to process the three-dimensional velocity field and pressure field data.

4. The decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion according to claim 1 is characterized in that: For the simulation data set, the automatic differentiation method of pytorch is used to calculate the differential equation loss function based on the Navier-Stokes equation. ; Calculate the initial state loss of each physical quantity that meets the preset initial conditions at the initial moment ; Joint differential equation loss function and initial state loss , and the boundary condition loss function , establish the physical loss function as: ； in, is the weight coefficient.

5. The decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion according to claim 1 is characterized in that: For each frame of data, the distribution information of the three-dimensional physical field on the spatial grid is extracted, and the size of the grid is constructed. The input tensor is Represents six original physical quantity channels, are the spatial resolution dimensions of the current frame, and t represents the normalized time information corresponding to each frame, which is used to explicitly introduce the time state; as well as The frame data and the corresponding feature vector of each point in the corresponding frame are concatenated to form the final input tensor. The input dimension is .

6. The decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion according to claim 5 is characterized by: The encoder part of the U-Net network gradually extracts local spatial features through multi-layer three-dimensional convolution operations and downsampling structures, and the decoder part uses jump connections to fuse feature information at different levels to achieve high-precision spatial field reconstruction. The output includes the velocity field, pressure field, temperature field and density field in the decoy bomb temperature field.

7. The decoy missile simulation flow field modeling method based on PINN-U-Net and local-global feature fusion according to claim 1 is characterized by: It also includes optimization of the flow field modeling framework, including: Combine physical constraint loss and data loss to form a comprehensive loss function; Based on the data in the simulation dataset, the flow field modeling framework is trained and optimized by minimizing the loss function through the gradient descent method.

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