Solving method for micro-vibration transmission model of flexible cable of spatial magnetic suspension vibration isolator

Abstract

The invention discloses a solving method for a micro-vibration transmission model of a flexible cable of a spatial magnetic suspension vibration isolator, and relates to the technical field of spatial vibration isolation. The method comprises the following steps: establishing a spatial magnetic suspension vibration isolator flexible cable micro-vibration transmission model based on central line telescopic deformation and cross section shear deformation; constructing a partial differential equation solving model based on the physical information neural network; constructing a loss function of a partial differential equation solving model by using the residual error, the residual error gradient, the initial condition and the boundary condition of the partial differential equation so as to train the partial differential equation solving model, and performing offline storage on the trained model; and then solving the micro-vibration transmission model of the flexible cable of the spatial magnetic suspension vibration isolator to obtain a solving result. According to the method, the accuracy of the flexible cable micro-vibration transmission model in the construction and solving process can be improved, support is provided for the design of vibration isolation compensation control, and then the problem of dynamic disturbance introduced by the flexible cable in a space magnetic suspension vibration isolation system is solved.

Description

Technical Field

[0001] The invention relates to the technical field of space vibration isolation, and in particular to a method for solving a micro-vibration transmission model of a flexible cable of a space magnetic suspension vibration isolator. Background Art

[0002] With the development of space science and remote sensing technology, it is particularly important for high-precision optical payloads to operate stably in a micro-vibration environment. Space magnetic levitation isolators provide a very effective solution. Through non-contact connection design, the physical transmission path of micro-vibration is cut off, so that the upper platform can achieve vibration isolation effect from zero frequency. This design not only improves the vibration isolation effect, but also reduces the input requirements for micro-vibration sources on the satellite, reducing risks and costs. However, the current magnetic levitation isolators face the problem of flexible cables transmitting micro-vibrations in the process of achieving high-precision vibration isolation. These cables are used to provide power and data transmission. The presence of flexible cables may introduce unnecessary platform micro-vibrations and affect the vibration isolation effect. Therefore, how to establish a flexible cable micro-vibration model and suppress the induced micro-vibrations accordingly has become an important issue that needs to be solved urgently.

[0003] In the existing flexible cable modeling technology, although the spring-damper model can capture some of the dynamic characteristics of the cable, it cannot give a full description of the disturbance characteristics on the rotational degree of freedom. The Kirchhoff model can describe the behavior of the cable under certain conditions, but in practical applications, factors such as the assumed centerline expansion and contraction deformation and cross-sectional shear deformation are often not fully considered. This makes the cable simulation results based on the Kirchhoff model significantly different from the actual situation, resulting in the limited effectiveness of the model and the inability to accurately reflect the actual dynamic behavior of the cable. In addition, the flexible cable model is a set of complex partial differential equations. The analytical solution of the partial differential equation is usually difficult to obtain, so it is necessary to use numerical methods to find approximate solutions. Traditional numerical methods such as the finite element method require a large amount of data (such as geometric models, material properties, boundary conditions, etc.) to be prepared in advance. The preparation process is cumbersome and error-prone, especially when dealing with complex or irregular geometric shapes, the amount of calculation increases significantly. The finite difference rule replaces the differential by the difference quotient, which has poor adaptability and stability. For complex partial differential equations, it is difficult to converge and find approximate solutions. The application of PINN (Physical Information Neural Network) does not require grid division and can better solve nonlinear and high-dimensional partial differential equations. However, when dealing with large and complex equations, PINN requires a large amount of data for training, which will significantly increase the time for network training and learning, which will limit the design of the controller and prevent real-time control. Since the cable dynamics model has 18 equations, in order to ensure the accuracy of the solution, it is difficult to control the time to meet the requirements by simply relying on the optimization control algorithm to shorten the solution time. If the method of offline calculation and online calling can be adopted, not only can the problem of solution time be avoided, but also the controller can be designed for real-time control. Therefore, this field urgently needs a model that can accurately reflect the dynamic characteristics of the cable, as well as an effective method for solving dynamic models based on partial differential equations. Summary of the invention

[0004] In view of the above-mentioned deficiencies in the prior art, the present invention provides a method for solving the micro-vibration transmission model of the flexible cable of a spatial magnetic levitation isolator.

[0005] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is: The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension vibration isolator includes the following steps: S1. Based on the centerline expansion deformation and cross-sectional shear deformation, a micro-vibration transmission model of the flexible cable of the spatial magnetic suspension isolator is established; S2. Construct a partial differential equation solving model based on physical information neural network; S3, constructing a loss function of a partial differential equation solution model using the residual, residual gradient, initial conditions and boundary conditions of the partial differential equation, training the partial differential equation solution model based on the loss function of the partial differential equation solution model, and saving the trained partial differential equation solution model offline; S4. Based on the trained partial differential equation solving model saved offline, the micro-vibration transmission model of the flexible cable of the spatial magnetic levitation isolator is solved to obtain the solution result.

[0006] Furthermore, in step S1, a micro-vibration transmission model of the flexible cable of the space magnetic levitation isolator is established based on the centerline expansion and contraction deformation and the cross-sectional shear deformation. The specific process is: an inertial coordinate system is set, and the cross-sectional attitude angle of the flexible cable is established based on the inertial coordinate system. Based on the cross-sectional attitude angle, the centerline expansion and contraction deformation and the cross-sectional shear deformation of the flexible cable, the arc coordinate component of the flexible cable is determined, and the forward strain model, shear strain model, velocity model and internal force model of the flexible cable are constructed. The cross-sectional attitude angle, arc coordinate component, forward strain model, shear strain model, velocity model and internal force model of the flexible cable are combined to establish the micro-vibration transmission model of the flexible cable of the space magnetic levitation isolator.

[0007] Furthermore, the forward strain model of the flexible cable is expressed as:

[0008] in: , , They are the positive strain of the flexible cable in the inertial coordinate system x , y , z The components in three directions, , , They are the lower edge of the flexible cable in the inertial coordinate system. x , y , z Displacement in three directions, , , They are respectively used to determine the attitude angles of the three directions of the cross section of the flexible cable. are the arc coordinates of the flexible cable.

[0009] Furthermore, the shear strain model of the flexible cable is expressed as:

[0010] in: , , They are the shear strain of the flexible cable in the inertial coordinate system. flat, Plane and The weight of the plane, , , They are the lower edge of the flexible cable in the inertial coordinate system. x , y , z Displacement in three directions, , , They are respectively used to determine the attitude angles of the three directions of the cross section of the flexible cable. are the arc coordinates of the flexible cable.

[0011] Furthermore, the velocity model of the flexible cable is expressed as:

[0012] in: , , They are the flexible cable in the inertial coordinate system. x , y , z The speed in three directions, , , They are the lower edge of the flexible cable in the inertial coordinate system. x , y , z Displacement in three directions, , , They are respectively used to determine the attitude angles of the three directions of the cross section of the flexible cable. For time.

[0013] Furthermore, the internal force model of the flexible cable is expressed as:

[0014] in: , , They are the flexible cable in the inertial coordinate system. x , y , z Internal forces in three directions, , , They are the flexible cable in the inertial coordinate system. x , y , z The normal strain in three directions, , , The flexible cable is in the non-drag state. x ,y , z The normal strain in three directions, is the shear stiffness coefficient, is the tensile stiffness coefficient.

[0015] Further, in step S2, the partial differential equation solving model includes an input layer, a hidden layer and an output layer which are connected in sequence and take arc coordinates and time as inputs, and take cross-sectional attitude angles, arc coordinate components, forward strain, shear strain, velocity and internal force as outputs.

[0016] Further, step S3 includes the following steps: S31. Using the residual and residual gradient of the partial differential equation, construct a residual function and residual gradient function of the partial differential equation solving model; S32, constructing a loss function of a partial differential equation solving model using initial conditions and boundary conditions of the partial differential equation, and constructing an initial condition loss function and a boundary condition loss function of the partial differential equation solving model; S33. Based on the residual function, residual gradient function, initial condition loss function and boundary condition loss function of the partial differential equation solution model, a loss function of the partial differential equation solution model is constructed, which is expressed as:

[0017] in: is the loss function of the partial differential equation solving model, is the weight of the residual term, Find the total residual value for the partial differential equation solution model, is the weight of the residual gradient term, Find the total residual gradient value for the partial differential equation solution model, is the weight of the initial condition term, Initial condition loss for solving partial differential equation models, is the weight of the boundary condition term, Solve boundary condition losses for partial differential equation models.

[0018] Further, in step S31, the residual function and residual gradient function of the partial differential equation solving model are expressed as:

[0019]

[0020]

[0021] in: Find the total residual value for the partial differential equation solution model, Find the first sample residual value for the partial differential equation solution model, The second sample residual value of the partial differential equation solution model is: Solve the model for the partial differential equation Sample residual values, is the number of samples used for residual calculation, is the residual function, , the dynamic equations of the flexible cable include 18 partial differential equations, corresponding to 18 residual functions, among which is the solution predicted by the neural network, and are the partial derivatives of the solution with respect to space and time, respectively. Find the total residual gradient value for the partial differential equation solution model, is the gradient symbol, and the gradient of the residual is the rate of change of the residual with respect to the input variable.

[0022] Further, in step S32, the initial condition loss function and boundary condition loss function of the partial differential equation solving model are expressed as:

[0023]

[0024] in: Initial condition loss for solving partial differential equation models, The boundary condition loss for solving the partial differential equation model is, is the number of samples used for initial condition calculation, are the values ​​of the initial conditions predicted by solving the model through the partial differential equations, are known initial conditions, is the number of samples used for boundary condition calculation, is the value of the boundary condition predicted by solving the model through the partial differential equations, The boundary conditions are known.

[0025] The present invention has the following beneficial effects: (1) The present invention constructs a micro-vibration transmission model of a flexible cable of a spatial magnetic levitation isolator based on the centerline expansion deformation and the cross-sectional shear deformation, so that the constructed micro-vibration transmission model of a flexible cable of a spatial magnetic levitation isolator can more accurately reflect the real dynamic characteristics of the cable, thereby providing more reliable prediction results in complex environments; (2) The present invention constructs a partial differential equation solution model based on a physical information neural network, and constructs a loss function of the partial differential equation solution model by using the residual, residual gradient, initial conditions and boundary conditions of the partial differential equation. The partial differential equation solution model is trained based on the loss function of the partial differential equation solution model, which can improve the accuracy of the partial differential equation solution model. Then, by saving the trained partial differential equation solution model offline, the trained partial differential equation solution model can be repeatedly called for prediction, which provides a model reference and technical guarantee for the design of the subsequent vibration isolation compensation controller, thereby ensuring the effectiveness and reliability of the controller in cable disturbance suppression. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 This is a flow chart of the solution method for the micro-vibration transmission model of the flexible cable of the space magnetic levitation isolator; Figure 2 A model diagram for solving the partial differential equation in the present invention; Figure 3 This is a flowchart of the flexible cable solution process in the present invention. DETAILED DESCRIPTION

[0027] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0028] like Figure 1 As shown, the method for solving the micro-vibration transmission model of the flexible cable of the spatial magnetic suspension vibration isolator includes steps S1-S4, which are specifically as follows: S1. Based on the centerline expansion deformation and cross-sectional shear deformation, a micro-vibration transmission model of the flexible cable of the spatial magnetic levitation isolator is established.

[0029] In an optional embodiment of the present invention, the present invention establishes a micro-vibration transmission model of a flexible cable of a space magnetic levitation isolator based on the centerline expansion and contraction deformation and the cross-sectional shear deformation. The specific process is: setting an inertial coordinate system, establishing a cross-sectional attitude angle of the flexible cable based on the inertial coordinate system, determining the arc coordinate component of the flexible cable based on the cross-sectional attitude angle, the centerline expansion and contraction deformation and the cross-sectional shear deformation of the flexible cable, and constructing a forward strain model, a shear strain model, a velocity model and an internal force model of the flexible cable, and combining the cross-sectional attitude angle, the arc coordinate component, the forward strain model, the shear strain model, the velocity model and the internal force model of the flexible cable to establish the micro-vibration transmission model of the flexible cable of the space magnetic levitation isolator.

[0030] The present invention establishes a coordinate system , and any point on the cable centerline P The location and O The points coincide. Around Axis rotation Corner , then go around Axis rotation is the principal axis coordinate system of the cable cross section . Around y Axis rotation Angle is the cable cross-section joint coordinate system, denoted as . , , To determine the attitude angle of the cable section.

[0031] Cable before shear deformation and expansion deformation P Point relative to fixed point O The radius vector is , the radius vector after shear deformation and expansion deformation is recorded as , is the initial geometry of the cable, then P The displacement of the point under the influence of the rigid body rotation, shear deformation and expansion deformation of the cable section is: .

[0032] In the present invention, considering the expansion and contraction deformation of the cable centerline and the shear deformation of the cross section, the strain of the cable comes not only from the rotation and shear deformation of the cross section, but also from the tension or compression of the centerline. The strain will include the positive strain caused by the linear displacement. , , and the shear strain due to cross-sectional rotation , , The present invention thus obtains the cable cross-section attitude angle , , and displacement Projection The positive strain and cut silhouettes , , kinematic relationship.

[0033] The present invention compares the time on both sides of the equal sign t Derivative, and then find the cable cross-section attitude angle , , and deformation speed of each point on the cable centerline and displacement Projection The kinematic relationship between them.

[0034] Then the present invention considers the arc coordinates and of and When the micro-arc segment between points is Point in time Internal force and The impact of Point in time Force and Under the influence of distribution force and Under the action of, according to the momentum theorem and momentum theorem of Newtonian mechanics combined with the kinematic relationship established above, we can get 18 unknown variables, including the cross-sectional attitude angle , , , the components of the arc coordinates of each point , , , Internal Force , , ,speed , , , positive strain , , and shear strain , , The closed system of equations.

[0035] Before the cable undergoes centerline expansion and shear deformation, the centerline is represented by arc coordinates (s) and time (t) in three-dimensional space, and its loss of diameter can be expressed as: .

[0036] The expansion and contraction deformation of the center line can be expressed as displacement: .

[0037] Cable diameter loss after expansion and contraction It can be expressed as: .

[0038] In the present invention, considering the expansion and contraction deformation of the cable centerline and the shear deformation of the cross section, the strain of the cable comes not only from the rotation and shear deformation of the cross section, but also from the tension or compression of the centerline. The strain will include the positive strain caused by the linear displacement. , , and the shear strain due to cross-sectional rotation , , These strain components will together describe the deformation state of the cable.

[0039] The forward strain model of the flexible cable is expressed as:

[0040] in: , , They are the positive strain of the flexible cable in the inertial coordinate system x , y , z The components in three directions, , , They are the lower edge of the flexible cable in the inertial coordinate system. x , y , z Displacement in three directions, , , They are respectively used to determine the attitude angles of the three directions of the cross section of the flexible cable. are the arc coordinates of the flexible cable.

[0041] The shear strain model of the flexible cable is expressed as:

[0042] in: , , They are the shear strain of the flexible cable in the inertial coordinate system. flat, Plane and The weight of the plane, , , They are the lower edge of the flexible cable in the inertial coordinate system. x , y , z Displacement in three directions, , , They are respectively used to determine the attitude angles of the three directions of the cross section of the flexible cable. are the arc coordinates of the flexible cable.

[0043] The velocity model of the flexible cable is expressed as:

[0044] in: , , They are the flexible cable in the inertial coordinate system. x , y , z The speed in three directions, , , They are the lower edge of the flexible cable in the inertial coordinate system. x , y , z Displacement in three directions, , , They are respectively used to determine the attitude angles of the three directions of the cross section of the flexible cable. For time.

[0045] The present invention will Time on both sides of the equal sign Seeking guidance, for P Deformation speed of a point , the initial velocity of the cable when it is in equilibrium , and there is .

[0046] Then, a velocity model of the flexible cable is constructed.

[0047] The internal force model of the flexible cable is expressed as:

[0048] in: , , They are the flexible cable in the inertial coordinate system. x , y , z Internal forces in three directions, , , They are the flexible cable in the inertial coordinate system. x , y , z The normal strain in three directions, , , The flexible cable is in the non-drag state. x , y , z The normal strain in three directions, is the shear stiffness coefficient, is the tensile stiffness coefficient.

[0049] In the present invention, the initial values ​​of the bending and torsion and the initial values ​​of the strain of the cable in the non-drag state are respectively , , and , , According to the linear constitutive relationship, the internal moment of the cable is The scalar form after projection is:

[0050] in: and The cross-sectional area is Axis and Bending stiffness of the shaft, For cross section The torsional stiffness of the shaft, and are the shear and tensile stiffness coefficients, and are the elastic modulus and shear modulus of the cable, is the cable cross-sectional shape factor, and Respectively, the cross-section Axis and Moment of inertia of the axis, Relative cross section Polar moment of inertia about the axis.

[0051] The present invention considers arc coordinates and of and When the micro-arc segment between points is Point in time Internal force and The impact of Point in time Force and Under the influence of distribution force and Under the action of, according to the momentum theorem of Newtonian mechanics and the momentum theorem about the center of mass, the balance equation of the cable can be derived:

[0052] in, is the density of the cable; is the cross-sectional area of ​​the cable; is the inertia tensor per unit length of the flexible cable, along Axis and The components of the axis have the same principal moments of inertia ,along The principal moment of inertia of the shaft is , and is the moment of inertia and polar moment of inertia of the cross section. Relative to the arc length Taking the derivative, we get:

[0053] Then we get:

[0054] .

[0055] The present invention can finally obtain according to the above model , , , , , , , , , , , , , , , , , The closed system of equations with 18 unknown variables is formed. If the boundary conditions at both ends are given and The above model can be solved by numerical integration if the constraints at the locations and the initial conditions of the variables are taken into account, as well as the initial shape of the cable before deformation.

[0056] S2. Construct a partial differential equation solving model based on physical information neural network.

[0057] In an optional embodiment of the present invention, the partial differential equation solving model includes an input layer, a hidden layer and an output layer which are connected in sequence and take arc coordinates and time as inputs, and take cross-sectional attitude angles, arc coordinate components, forward strain, shear strain, velocity and internal force as outputs.

[0058] like Figure 2As shown, the neural network architecture designed by the present invention is suitable for the partial differential equation problem. The input layer is designed to be 2 (arc coordinates s and time t), the hidden layer is designed to be H according to simulation and experimental results, the hidden layer neurons are designed to be K, the output layer is designed to be 18, and the activation function is selected as ReLU to improve the running speed.

[0059] S3. Construct a loss function of a partial differential equation solving model using the control equations and boundary conditions of the partial differential equation, train the partial differential equation solving model based on the loss function of the partial differential equation solving model, and save the trained partial differential equation solving model offline.

[0060] In an optional embodiment of the present invention, the present invention uses the control equation and boundary conditions of the partial differential equation to construct a loss function. The loss function includes various related residuals, initial conditions and boundary conditions to evaluate the difference between the output of the neural network and the solution of the equation. In order to solve the gradient weakening problem that PINN may encounter during the solution process, the present invention embeds the gradient information of the residual of the partial differential equation into the loss function. In this way, the neural network is not only constrained by the PDE residual during the training process, but also by its gradient residual, thereby improving the accuracy and stability of the solution. And in order to balance the influence of boundary conditions, initial conditions and PDE residuals on the solution results, the present invention proposes a weighted loss function. In this function, the present invention assigns different weights to boundary conditions, initial conditions, PDE residuals and residual gradients, respectively, to better control the solution process and improve the accuracy of the solution results. In order to simplify the manual derivation and programming workload in the PDE solution process, the present invention utilizes the automatic differentiation function of the deep learning framework. Through automatic differentiation, the present invention can directly handle the continuity and boundary conditions of PDE without manually deriving gradient formulas and writing complex codes. This greatly reduces the complexity and workload of the solution process and improves the solution efficiency. In the automatic differentiation process, the present invention uses a back propagation algorithm to calculate the gradient and update the weights of the neural network. In this way, the present invention can solve PDE more efficiently and obtain more accurate results.

[0061] The present invention initializes the weights and biases of the neural network. In each training step, randomly selected training points are loaded in batches. The loss function value is calculated and back-propagated to obtain the gradient. The network parameters are updated using the L-BFGS optimizer, and the neural network is trained iteratively. The network parameters are optimized by minimizing the loss function, so that the neural network learns an accurate partial differential equation solution. In order to improve the solution speed of PINN, the present invention uses a large-scale GPU for parallel computing. By distributing the training tasks of the neural network to multiple GPUs, the present invention can accelerate the training process of the neural network and improve the computing efficiency. In the parallel computing process, the present invention adopts the strategies of data parallelism and model parallelism to make full use of the computing resources of the GPU. In this way, the present invention can obtain more accurate solution results in a shorter time. Then the present invention converts the learning results of the neural network into data that can be recorded offline for subsequent application. Download offline data, repeatedly call and predict the cable model in real time, thereby optimizing the control strategy of micro-vibration.

[0062] Step S3 includes the following steps: S31. Using the residual and residual gradient of the partial differential equation, the residual function and residual gradient function of the partial differential equation solving model are constructed, which can be expressed as:

[0063]

[0064]

[0065] in: Find the total residual value for the partial differential equation solution model, Find the first sample residual value for the partial differential equation solution model, The second sample residual value of the partial differential equation solution model is: Solve the model for the partial differential equation Sample residual values, is the number of samples used for residual calculation, is the residual function, , the dynamic equations of the flexible cable include 18 partial differential equations, corresponding to 18 residual functions, among which is the solution predicted by the neural network, and are the partial derivatives of the solution with respect to space and time, respectively. Find the total residual gradient value for the partial differential equation solution model, is the gradient symbol, and the gradient of the residual is the rate of change of the residual with respect to the input variable.

[0066] S32. Using the initial conditions and boundary conditions of the partial differential equation to construct the loss function of the partial differential equation solution model, construct the initial condition loss function and boundary condition loss function of the partial differential equation solution model, expressed as:

[0067]

[0068] in: Initial condition loss for solving partial differential equation models, The boundary condition loss for solving the partial differential equation model is, is the number of samples used for initial condition calculation, are the values ​​of the initial conditions predicted by solving the model through the partial differential equations, are known initial conditions, is the number of samples used for boundary condition calculation, is the value of the boundary condition predicted by solving the model through the partial differential equations, The boundary conditions are known.

[0069] S33. Based on the residual function, residual gradient function, initial condition loss function and boundary condition loss function of the partial differential equation solution model, a loss function of the partial differential equation solution model is constructed, which is expressed as:

[0070] in: is the loss function of the partial differential equation solving model, is the weight of the residual term, Find the total residual value for the partial differential equation solution model, is the weight of the residual gradient term, Find the total residual gradient value for the partial differential equation solution model, is the weight of the initial condition term, Initial condition loss for solving partial differential equation models, is the weight of the boundary condition term, Solve boundary condition losses for partial differential equation models.

[0071] In the process of constructing the loss function, in order to simplify the manual derivation and programming workload in the PDE solution process, the present invention utilizes the automatic differentiation function of the deep learning framework. Through automatic differentiation, the present invention can directly handle the continuity and boundary conditions of the PDE without manually deriving the gradient formula and writing complex codes. This greatly reduces the complexity and workload of the solution process and improves the solution efficiency. In the automatic differentiation process, the present invention uses a backpropagation algorithm to calculate the gradient and update the weights of the neural network. In this way, the PDE can be solved more efficiently and more accurate results can be obtained.

[0072] Next, the L-BFGS optimizer is used to train the network. During the training process, the L-BFGS optimizer performs optimization through the following steps: (1) Calculate the loss: According to the parameters of the current network, forward propagation is used to calculate the activation value of the hidden layer neurons through weighted sum from the input layer and the hidden layer, and finally output , and calculate the loss function .

[0073] (2) Calculate the gradient: Use automatic differentiation techniques to compute the gradient of the loss function with respect to each network parameter: .

[0074] (3) L-BFGS update steps: The core of L-BFGS lies in the process of updating parameters. The L-BFGS optimization algorithm updates parameters according to the following rules:

[0075] in: is the search direction calculated in the L-BFGS algorithm, is the step size, determined by line search.

[0076] (4) Approximate Hessian matrix: By maintaining information from previous iterations (step size and gradient changes), L-BFGS uses this information to construct an approximation to the Hessian matrix without calculating the Hessian matrix itself. This approach greatly reduces memory usage and is suitable for large-scale problems.

[0077] (5) Iteration: Repeatedly calculate the loss, gradient and update the parameters until the convergence criterion is reached (the convergence criterion is set to the change in the loss function below a certain threshold or the maximum number of iterations is reached) N max ).

[0078] In order to improve the solution speed of PINN, the present invention uses large-scale GPUs for parallel computing. By distributing the training tasks of the neural network to multiple GPUs, the present invention can accelerate the training process of the neural network and improve the computing efficiency. In the parallel computing process, the present invention adopts the strategies of data parallelism and model parallelism to make full use of the computing resources of the GPU. In this way, the present invention can obtain more accurate solution results in a shorter time.

[0079] Finally, the trained PINN model is saved offline and used to solve the cable. A set of independent test points can be used as the input of the cable, offline data can be downloaded, and the cable model can be repeatedly called and predicted in real time to obtain the output response of the cable online and evaluate the prediction ability of the model. The PINN method is helpful for the design of the controller and can effectively compensate for the disturbance introduced by the cable through the real-time controller. The flexible cable solution process is as follows Figure 3 shown.

[0080] S4. Based on the trained partial differential equation solving model saved offline, the micro-vibration transmission model of the flexible cable of the spatial magnetic levitation isolator is solved to obtain the solution result.

[0081] Simulation experiment: At present, PINN can only be used for simple conditions such as periodic boundary conditions and fixed boundary conditions, and cannot be applied to complex conditions such as random boundary conditions. First, the complexity and diversity of random boundary conditions make it difficult for neural networks to learn effectively, and the need for data sparsity may lead to inefficient training. In addition, changes in random boundaries may affect the convergence and numerical stability of the model, especially in nonlinear problems, which may lead to an unstable learning process. At the same time, the complexity of boundary conditions also makes the expression of physical laws inaccurate, thus affecting model performance. Therefore, although PINN provides new ideas for solving partial differential equations, its application under random boundary conditions still needs in-depth research and improvement.

[0082] In order to test the effectiveness of the PINN solution method in the present invention, a two-dimensional wave equation numerical example is taken as an example to verify the method.

[0083] Solve the wave equation in two dimensions:

[0084]

[0085] The initial conditions are:

[0086]

[0087] The analytical solution to the partial differential equations is:

[0088]

[0089] The present invention uses PINN to solve the problem. The simulation parameters are: 2 input layers, 2 output layers, 50 hidden layers, 1500 maximum iterations, and loss threshold of loss<1e-4. The number of initial values ​​is 50, 0< u <1,0< v <1,0< t <0.1, and compare the results with the analytical solution, as shown in the following table: Table 1 Comparison of solution results of examples

[0090] It can be seen from the simulation results that the root mean square errors of the numerical solution and the analytical solution of u and v obtained by the finite difference method are 0.06 and 0.30 respectively. The root mean square errors of the numerical solution and the analytical solution of u and v obtained by the method of the present invention are 0.01 and 0.04 respectively. It can be seen that the method of the present invention has higher solution accuracy than the traditional finite difference method, thereby verifying the effectiveness of the method.

[0091] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0092] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0093] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0094] The present invention uses specific embodiments to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as a limitation on the present invention.

[0095] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A method for solving the micro-vibration transmission model of the flexible cable of a space magnetic suspension isolator, characterized in that: The following steps are involved: S1. Based on the centerline expansion deformation and cross-sectional shear deformation, a micro-vibration transmission model of the flexible cable of the spatial magnetic suspension isolator is established; S2. Construct a partial differential equation solving model based on physical information neural network; S3, constructing a loss function of a partial differential equation solution model using the residual, residual gradient, initial conditions and boundary conditions of the partial differential equation, training the partial differential equation solution model based on the loss function of the partial differential equation solution model, and saving the trained partial differential equation solution model offline; S4. Based on the trained partial differential equation solving model saved offline, the micro-vibration transmission model of the flexible cable of the spatial magnetic levitation isolator is solved to obtain the solution result.

2. The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension isolator according to claim 1 is characterized in that: In step S1, a micro-vibration transmission model of a flexible cable of a space magnetic levitation isolator is established based on the centerline expansion and contraction deformation and the cross-sectional shear deformation. The specific process is as follows: an inertial coordinate system is set, and a cross-sectional attitude angle of the flexible cable is established based on the inertial coordinate system. Based on the cross-sectional attitude angle, the centerline expansion and contraction deformation and the cross-sectional shear deformation of the flexible cable, the arc coordinate component of the flexible cable is determined, and a forward strain model, a shear strain model, a velocity model and an internal force model of the flexible cable are constructed. The cross-sectional attitude angle, the arc coordinate component, the forward strain model, the shear strain model, the velocity model and the internal force model of the flexible cable are combined to establish the micro-vibration transmission model of the flexible cable of the space magnetic levitation isolator.

3. The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension isolator according to claim 2 is characterized in that: The forward strain model of the flexible cable is expressed as: in: , , They are the positive strain of the flexible cable in the inertial coordinate system x , y , z The components in three directions, , , They are the lower edge of the flexible cable in the inertial coordinate system. x , y , z Displacement in three directions, , , They are respectively used to determine the attitude angles of the three directions of the cross section of the flexible cable. are the arc coordinates of the flexible cable.

4. The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension isolator according to claim 2 is characterized in that: The shear strain model of the flexible cable is expressed as: in: , , They are the shear strain of the flexible cable in the inertial coordinate system. flat, Plane and The weight of the plane, , , They are the lower edge of the flexible cable in the inertial coordinate system. x , y , z Displacement in three directions, , , They are respectively used to determine the attitude angles of the three directions of the cross section of the flexible cable. are the arc coordinates of the flexible cable.

5. The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension isolator according to claim 2 is characterized in that: The velocity model of the flexible cable is expressed as: in: , , They are the flexible cable in the inertial coordinate system. x , y , z The speed in three directions, , , They are the lower edge of the flexible cable in the inertial coordinate system. x , y , z Displacement in three directions, , , They are respectively used to determine the attitude angles of the three directions of the cross section of the flexible cable. For time.

6. The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension isolator according to claim 2 is characterized in that: The internal force model of the flexible cable is expressed as: in: , , They are the flexible cable in the inertial coordinate system. x , y , z Internal forces in three directions, , , They are the flexible cable in the inertial coordinate system. x , y , z The normal strain in three directions, , , The flexible cable is in the non-drag state. x , y , z The normal strain in three directions, is the shear stiffness coefficient, is the tensile stiffness coefficient.

7. The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension isolator according to claim 1 is characterized in that: In step S2, the partial differential equation solving model includes an input layer, a hidden layer and an output layer which are connected in sequence and take arc coordinates and time as inputs, and take cross-sectional attitude angles, arc coordinate components, forward strain, shear strain, velocity and internal force as outputs.

8. The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension isolator according to claim 1 is characterized in that: Step S3 includes the following steps: S31. Using the residual and residual gradient of the partial differential equation, construct a residual function and residual gradient function of the partial differential equation solving model; S32, constructing a loss function of a partial differential equation solving model using initial conditions and boundary conditions of the partial differential equation, and constructing an initial condition loss function and a boundary condition loss function of the partial differential equation solving model; S33. Based on the residual function, residual gradient function, initial condition loss function and boundary condition loss function of the partial differential equation solution model, a loss function of the partial differential equation solution model is constructed, which is expressed as: in: is the loss function of the partial differential equation solving model, is the weight of the residual term, Find the total residual value for the partial differential equation solution model, is the weight of the residual gradient term, Find the total residual gradient value for the partial differential equation solution model, is the weight of the initial condition term, Initial condition loss for solving partial differential equation models, is the weight of the boundary condition term, Solve boundary condition losses for partial differential equation models.

9. The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension isolator according to claim 8 is characterized in that: In step S31, the residual function and residual gradient function of the partial differential equation solving model are expressed as: in: Find the total residual value for the partial differential equation solution model, Find the first sample residual value for the partial differential equation solution model, The second sample residual value of the partial differential equation solution model is: Solve the model for the partial differential equation Sample residual values, is the number of samples used for residual calculation, is the residual function, , the dynamic equations of the flexible cable include 18 partial differential equations, corresponding to 18 residual functions, among which is the solution predicted by the neural network, and are the partial derivatives of the solution with respect to space and time, respectively. Find the total residual gradient value for the partial differential equation solution model, is the gradient symbol, and the gradient of the residual is the rate of change of the residual with respect to the input variable.

10. The method for solving the micro-vibration transmission model of the flexible cable of the space magnetic suspension isolator according to claim 8, characterized in that: In step S32, the initial condition loss function and boundary condition loss function of the partial differential equation solving model are expressed as: in: Initial condition loss for solving partial differential equation models, The boundary condition loss for solving the partial differential equation model is, is the number of samples used for initial condition calculation, are the values ​​of the initial conditions predicted by solving the model through the partial differential equations, are known initial conditions, is the number of samples used for boundary condition calculation, is the value of the boundary condition predicted by solving the model through the partial differential equations, The boundary conditions are known.

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