Solving Method for Micro-Vibration Transfer Model of Flexible Cable of Spatial Magnetic Levitation Vibration Isolator

Abstract

The present invention discloses a method for solving the micro-vibration transfer model of a flexible cable of a space magnetic levitation isolator, which relates to the technical field of space vibration isolation. The method includes the following steps: establishing a micro-vibration transfer model of the flexible cable of the space magnetic levitation isolator based on the telescopic deformation of the center line and the shear deformation of the cross section; constructing a partial differential equation solving model based on a physics-informed neural network; using the residual, residual gradient, initial conditions, and boundary conditions of the partial differential equation to construct a loss function of the partial differential equation solving model to train the partial differential equation solving model and save the trained model offline; and then solving the micro-vibration transfer model of the flexible cable of the space magnetic levitation isolator to obtain a solution result. The present invention can improve the accuracy of the micro-vibration transfer model of the flexible cable in the construction and solution processes, provide support for the design of vibration isolation compensation control, and further solve the dynamic disturbance problem introduced by the flexible cable in the space magnetic levitation isolation system.

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 techniques, although the spring-damping model can capture some dynamic characteristics of the cable, it cannot give a sufficient description of the disturbance characteristics in 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 telescopic deformation of the centerline and the shear deformation of the cross-section assumed by it 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 limitation of the effectiveness of this model and being unable 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 solutions of partial differential equations are usually difficult to obtain, so numerical methods need to be used to find approximate solutions. Traditional numerical methods such as the finite element method require a large amount of data to be prepared in advance (such as geometric models, material properties, boundary conditions, etc.), and the preparation process is cumbersome and error-prone. Especially when dealing with complex or irregular geometric shapes, the computational amount increases significantly. The finite difference method replaces the differential with the difference quotient, with poor adaptability and stability. For complex partial differential equations, it is difficult to converge and find approximate solutions. Applying PINN (Physics-Informed Neural Network) does not require mesh generation and can better solve nonlinear and high-dimensional partial differential equation problems. However, when dealing with large and complex equation systems, PINN requires a large amount of data for training, which will significantly increase the time for network training and learning, and this 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 solution accuracy, it is very difficult to control the time to meet the requirements by simply relying on optimizing control algorithms to shorten the solution time. If the offline calculation and online call method can be adopted, not only can the problem of solution time be avoided, but also the design of the controller can be carried out, so as to achieve real-time control. Therefore, there is an urgent need in this field for a model that can accurately reflect the dynamic characteristics of the cable, and an effective method for solving the dynamics model mainly composed of partial differential equations. Summary of the Invention

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

[0005] In order to achieve the above invention purpose, the technical solution adopted by the present invention is as follows:

[0006] The method for solving the micro-vibration transfer model of the flexible cable of the spatial magnetic levitation isolator includes the following steps:

[0007] S1. Based on the telescopic deformation of the centerline and the shear deformation of the cross-section, establish a micro-vibration transfer model of the flexible cable of the spatial magnetic levitation isolator;

[0008] S2. Based on the physics-informed neural network, construct a partial differential equation solution model;

[0009] 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;

[0010] 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.

[0011] 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.

[0012] Furthermore, the forward strain model of the flexible cable is expressed as:

[0013]

[0014] 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.

[0015] Furthermore, the shear strain model of the flexible cable is expressed as:

[0016]

[0017] in: , , are the shear strain components of the flexible cable in the inertial coordinate system in the plane, plane and plane, , , are the displacements of the flexible cable in the inertial coordinate system along x , y , z in three directions, , , are the attitude angles for determining the three directions of the flexible cable cross-section, is the arc coordinate of the flexible cable.

[0018] Furthermore, the velocity model of the flexible cable is expressed as:

[0019]

[0020] where: , , are the velocities of the flexible cable in the inertial coordinate system in x , y , z in three directions, , , are the displacements of the flexible cable in the inertial coordinate system along x , y , z in three directions, , , are the attitude angles for determining the three directions of the flexible cable cross-section, is the time.

[0021] Furthermore, the internal force model of the flexible cable is expressed as:

[0022]

[0023] where: , , are the internal forces of the flexible cable in the inertial coordinate system in x , y , z in three directions, , , are the positive strains of the flexible cable in the inertial coordinate system in x , y , z in three directions, , , are the positive strains of the flexible cable in the non-dragging state in x , y , z three directions respectively, is the shear stiffness coefficient, is the tensile stiffness coefficient.

[0024] Furthermore, in step S2, the partial differential equation solving model includes an input layer that takes arc coordinates and time as inputs, a hidden layer, and an output layer that outputs cross-sectional attitude angles, arc coordinate components, positive strains, shear strains, velocities, and internal forces, which are connected in sequence.

[0025] Furthermore, step S3 includes the following steps:

[0026] S31. Use the residuals and residual gradients of the partial differential equation to construct the residual function and residual gradient function of the partial differential equation solving model;

[0027] S32. Use the initial conditions and boundary conditions of the partial differential equation to construct the loss function of the partial differential equation solving model, and construct the initial condition loss function and boundary condition loss function of the partial differential equation solving model;

[0028] S33. Based on the residual function, residual gradient function, initial condition loss function, and boundary condition loss function of the partial differential equation solving model, construct the loss function of the partial differential equation solving model, expressed as:

[0029]

[0030] where: is the loss function of the partial differential equation solving model, is the weight of the residual term, is the total residual value of the partial differential equation solving model, is the weight of the residual gradient term, is the total residual gradient value of the partial differential equation solving model, is the weight of the initial condition term, is the initial condition loss of the partial differential equation solving model, is the weight of the boundary condition term, is the boundary condition loss of the partial differential equation solving model.

[0031] Furthermore, in step S31, the residual function and residual gradient function of the partial differential equation solving model are expressed as:

[0032]

[0033]

[0034]

[0035] Wherein: is the total residual value of the partial differential equation solving model, is the first sample residual value of the partial differential equation solving model, is the second sample residual value of the partial differential equation solving model, is the th sample residual value of the partial differential equation solving model, 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 respectively, wherein is the solution predicted by the neural network, and are the partial derivatives of the solution with respect to space and time respectively, is the total residual gradient value of the partial differential equation solving 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.

[0036] Furthermore, in step S32, the initial condition loss function and the boundary condition loss function of the partial differential equation solving model are expressed as:

[0037]

[0038]

[0039] Wherein: is the initial condition loss of the partial differential equation solving model, is the boundary condition loss of the partial differential equation solving model, is the number of samples used for initial condition calculation, is the initial condition value predicted by the partial differential equation solving model, is the known initial condition, is the number of samples used for boundary condition calculation, is the boundary condition value predicted by the partial differential equation solving model, is the known boundary condition.

[0040] The present invention has the following beneficial effects:

[0041] (1) The present invention constructs a micro-vibration transfer model of the flexible cable of the space magnetic levitation isolator based on the telescopic deformation of the center line and the shear deformation of the cross-section, enabling the constructed micro-vibration transfer model of the flexible cable of the space magnetic levitation isolator to more accurately reflect the true dynamic characteristics of the cable, thereby providing more reliable prediction results in complex environments;

[0042] (2) The present invention constructs a partial differential equation solving model based on a physics-informed neural network, and constructs a loss function of the partial differential equation solving model by using the residual, residual gradient, initial conditions, and boundary conditions of the partial differential equation. Training the partial differential equation solving model based on the loss function of the partial differential equation solving model can improve the accuracy of the partial differential equation solving model. Then, by saving the trained partial differential equation solving model offline, the trained partial differential equation solving model can be repeatedly called for prediction, providing a model reference and technical guarantee for the design of the subsequent vibration isolation compensation controller, and ensuring the effectiveness and reliability of the controller in cable disturbance suppression. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a schematic flow chart of the solution method for the micro-vibration transfer model of the flexible cable of the space magnetic levitation isolator;

[0044] Figure 2 It is a diagram of the partial differential equation solving model in the present invention;

[0045] Figure 3 It is a block diagram of the solution process of the flexible cable in the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0046] The following describes the specific embodiments of the present invention to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions made using the concept of the present invention are within the scope of protection.

[0047] As Figure 1 shown, the solution method for the micro-vibration transfer model of the flexible cable of the space magnetic levitation isolator includes steps S1-S4, specifically as follows:

[0048] S1. Based on the telescopic deformation of the center line and the shear deformation of the cross-section, establish a micro-vibration transfer model of the flexible cable of the space magnetic levitation isolator.

[0049] 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.

[0050] 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 , and then 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.

[0051] 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:

[0052] .

[0053] 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 , , Thus, the cable cross-section attitude angle is obtained by the present invention. , , and displacement projections of the positive strain and the shear silhouette , , kinematic relationships.

[0054] The present invention differentiates both sides of the equation with respect to time t to obtain the cable cross-section attitude angle , , and the deformation velocity of each point on the cable centerline and displacement projections kinematic relationships therebetween.

[0055] Then, when the present invention considers the infinitesimal arc segment between points and in terms of arc coordinate and , point is affected by internal forces and at time , and point is affected by forces and at time . Under the action of distributed forces and , according to the momentum theorem and the moment of momentum theorem of Newtonian mechanics and the kinematic relationships established above, a closed system of equations with 18 unknown variables can be obtained, including the cross-section attitude angle , , , the components of the arc coordinate of each point , , , internal forces , , , velocity , , , positive strain , , and shear strain , , .

[0056] Before the center line of the cable undergoes telescopic deformation and shear deformation, the center line is represented by arc coordinates (s) and time (t) in three-dimensional space, and its radius vector can be expressed as:

[0057] 。

[0058] The telescopic deformation of the center line can be expressed by displacement as:

[0059] 。

[0060] The radius vector of the cable after telescoping can be expressed as:

[0061] 。

[0062] In the case of considering the telescopic deformation of the cable center line and the cross-section shear deformation in the present invention, the strain of the cable not only comes from the rotation and shear deformation of the cross-section, but also from the tension or compression of the center line. The strain will include the normal strain caused by the linear displacement , , and the shear strain caused by the cross-section rotation 、 、 。These strain components will jointly describe the deformation state of the cable.

[0063] The positive strain model of the flexible cable, expressed as:

[0064]

[0065] Where: 、 、 are the components of the positive strain of the flexible cable in the inertial coordinate system along x 、 y 、 z three directions respectively, 、 、 are the displacements of the flexible cable along x 、 y 、 z three directions in the inertial coordinate system respectively, 、 、 are the attitude angles determining the three directions of the flexible cable cross-section respectively, is the arc coordinate of the flexible cable.

[0066] The shear strain model of the flexible cable, expressed as:

[0067]

[0068] Wherein: , , are the components of the shear strain of the flexible cable in the inertial coordinate system in the plane, plane and plane respectively, , , are the displacements of the flexible cable in three directions along x , y , z in the inertial coordinate system respectively, , , are the attitude angles for determining the three directions of the cross-section of the flexible cable respectively, is the arc coordinate of the flexible cable.

[0069] The velocity model of the flexible cable is expressed as:

[0070]

[0071] Wherein: , , are the velocities of the flexible cable in three directions in the inertial coordinate system in x , y , z respectively, , , are the displacements of the flexible cable in three directions along x , y , z in the inertial coordinate system respectively, , , are the attitude angles for determining the three directions of the cross-section of the flexible cable respectively, is the time.

[0072] The present invention takes the derivative of both sides of the equal sign with respect to time , is the deformation velocity of the P point, the initial velocity when the cable is in the equilibrium state, and there is , and there is

[0073] .

[0074] Furthermore, the velocity model of the above flexible cable is constructed.

[0075] The internal force model of the flexible cable is expressed as:

[0076]

[0077] Among them: 、 、 are respectively the internal forces of the flexible cable in the inertial coordinate system in the x 、 y 、 z three directions, , , are respectively the positive strains of the flexible cable in the inertial coordinate system in the x 、 y 、 z three directions, , , are respectively the positive strains of the flexible cable in the non-dragging state in the x 、 y 、 z three directions, is the shear stiffness coefficient, is the tensile stiffness coefficient.

[0078] The initial values of the bending-torsion degree and strain of the cable in the non-dragging state of the present invention are respectively , , and , , . According to the linear constitutive relationship, the scalar form of the internal moment of the cable projected onto the principal axis coordinate system is:

[0079]

[0080] Among them: and are respectively the bending stiffnesses of the cross-section around the axis and the axis, is the torsional stiffness of the cross-section around the axis, and are respectively the shear and tensile stiffness coefficients, and are respectively the elastic modulus and shear modulus of the cable, is the cable cross-section shape coefficient, and are respectively the moments of inertia of the cross-section relative to the axis and the axis, is the polar moment of inertia of the cross-section relative to the axis.

[0081] The present invention considers the infinitesimal arc segment between the arc coordinates and of and points. When considering the infinitesimal arc segment between the arc coordinates point at time is affected by internal forces and . The point at time is affected by forces and . Under the action of distributed forces and , according to the momentum theorem of Newtonian mechanics and the momentum moment theorem about the center of mass, the equilibrium equation of the cable can be derived:

[0082]

[0083] where, 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. The components along the axis and axis are the same principal moment of inertia . The principal moment of inertia along the axis is . and are the moment of inertia and polar moment of inertia of the cross-section. Differentiating the above equation with respect to the arc length in the coordinate system , we get:

[0084]

[0085] Furthermore, we get:

[0086]

[0087] .

[0088] Ultimately, according to the above model, the present invention can obtain , , , , , , , , , , , , , , , , , to form a closed system of equations with 18 unknown variables. If the constraint conditions at both ends and and the initial conditions of each variable, as well as the initial shape of the cable before deformation, are given, then the above model can be solved by numerical integration.

[0089] S2. Based on the physics-informed neural network, a partial differential equation solving model is constructed.

[0090] 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 that are connected in sequence. The input layer takes the arc coordinate and time as inputs, the hidden layer takes the cross-sectional attitude angle, arc coordinate components, normal strain, shear strain, velocity, and internal force as outputs.

[0091] As Figure 2 shown, the neural network architecture suitable for this partial differential equation problem is designed in the present invention. The input layer is designed as 2 (arc coordinate s and time t), the hidden layer is designed as H according to the simulation and experimental results, the neurons in the hidden layer are designed as K, and the output layer is designed as 18. To improve the running speed, the activation function is selected as ReLU.

[0092] S3. The loss function of the partial differential equation solving model is constructed using the control equation and boundary conditions of the partial differential equation. The partial differential equation solving model is trained based on the loss function of the partial differential equation solving model, and the trained partial differential equation solving model is saved offline.

[0093] In an alternative embodiment of the present invention, the present invention constructs a loss function by using the governing equation and boundary conditions of partial differential equations. The loss function includes various relevant residuals, initial conditions, and boundary conditions to evaluate the difference between the neural network output and the solution of the equation. To address the possible problem of weakened gradients during the solution process of PINN, the present invention embeds the gradient information of the partial differential equation residuals into the loss function. In this way, the neural network is not only constrained by the PDE residuals but also by their gradient residuals during the training process, thereby improving the accuracy and stability of the solution. And to balance the influence of boundary conditions, initial conditions, and PDE residuals on the solution result, 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 result. 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 PDEs without manually deriving the gradient formula and writing complex code. This greatly reduces the complexity and workload of the solution process and improves the solution efficiency. During the automatic differentiation process, the present invention adopts the backpropagation algorithm to calculate gradients and update the weights of the neural network. In this way, the present invention can solve PDEs more efficiently and obtain more accurate results.

[0094] 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 backpropagation is performed to obtain gradients. The network parameters are updated using the L-BFGS optimizer, and the process is iterated to train the neural network. By minimizing the loss function, the network parameters are optimized so that the neural network learns the accurate solution of the partial differential equation. To improve the solution speed of PINN, the present invention utilizes 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. During the parallel computing process, the present invention adopts strategies of data parallelism and model parallelism to fully utilize the computing resources of the GPUs. 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 applications. The offline data is downloaded and repeatedly called to perform real-time prediction on the cable model, thereby optimizing the control strategy of micro-vibrations.

[0095] Step S3 includes the following steps:

[0096] S31. Using the residuals and residual gradients of the partial differential equation, construct the residual function and residual gradient function of the partial differential equation solution model, expressed as:

[0097]

[0098]

[0099]

[0100] Wherein: is the total residual value of the partial differential equation solving model, is the first sample residual value of the partial differential equation solving model, is the second sample residual value of the partial differential equation solving model, is the th sample residual value of the partial differential equation solving model, 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 respectively, wherein is the solution predicted by the neural network, and are the partial derivatives of the solution with respect to space and time respectively, is the total residual gradient value of the partial differential equation solving 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.

[0101] S32. Construct the loss function of the partial differential equation solving model using the initial conditions and boundary conditions of the partial differential equation, and construct the initial condition loss function and boundary condition loss function of the partial differential equation solving model, expressed as:

[0102]

[0103]

[0104] Wherein: is the initial condition loss of the partial differential equation solving model, is the boundary condition loss of the partial differential equation solving model, is the number of samples used for initial condition calculation, is the initial condition value predicted by the partial differential equation solving model, is the known initial condition, is the number of samples used for boundary condition calculation, is the boundary condition value predicted by the partial differential equation solving model, is the known boundary condition.

[0105] S33. Based on the residual function, residual gradient function, initial condition loss function, and boundary condition loss function of the partial differential equation solving model, construct the loss function of the partial differential equation solving model, which is expressed as:

[0106]

[0107] Where: is the loss function of the partial differential equation solving model, is the weight of the residual term, is the total residual value of the partial differential equation solving model, is the weight of the residual gradient term, is the total residual gradient value of the partial differential equation solving model, is the weight of the initial condition term, is the initial condition loss of the partial differential equation solving model, is the weight of the boundary condition term, is the boundary condition loss of the partial differential equation solving model.

[0108] In the process of constructing the loss function, in order to simplify the manual derivation and programming workload in the PDE solving 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 code. This greatly reduces the complexity and workload of the solving process and improves the solving efficiency. In the process of automatic differentiation, the present invention adopts the 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.

[0109] Next, use the L-BFGS optimizer for network training. During the training process, the L-BFGS optimizer is optimized through the following steps:

[0110] (1) Calculate the loss:

[0111] According to the parameters of the current network, use forward propagation. Calculate the activation values of the neurons in the hidden layer through weighted summation from the input layer and the hidden layer, and finally output , and calculate the loss function .

[0112] (2) Calculate the gradient:

[0113] Use automatic differentiation technology to calculate the gradient of the loss function with respect to each network parameter:

[0114] .

[0115] (3) L-BFGS update step:

[0116] The core of L-BFGS lies in the process of updating parameters. The L-BFGS optimization algorithm updates parameters according to the following rules:

[0117]

[0118] where: is the search direction calculated in the L-BFGS algorithm, is the step size, which is determined by line search.

[0119] (4) Approximate Hessian matrix:

[0120] By maintaining the information of the previous several iterations (step size and gradient change), L-BFGS uses this information to construct an approximation of the Hessian matrix without calculating the Hessian matrix itself. This method greatly reduces memory usage and is suitable for large-scale problems.

[0121] (5) Iteration:

[0122] Repeatedly calculate the loss, gradient, and update the parameters until the convergence criterion is met (the convergence criterion is set as the change in the loss function being lower than a certain threshold, or reaching the maximum number of iterations N max )

[0123] 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. During the parallel computing process, the present invention adopts the strategies of data parallelism and model parallelism to fully utilize the computing resources of the GPU. In this way, the present invention can obtain more accurate solution results in a shorter time.

[0124] Finally, the trained PINN model will be saved offline and used to solve the cable. A set of independent test points can be used as the input of the cable, download the offline data, repeatedly call and perform real-time prediction on the cable model, and then obtain the output response of the cable online to 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 a real-time controller. The flexible cable solution process is as Figure 3 shown.

[0125] S4. Based on the trained partial differential equation solution model saved offline, solve the flexible cable micro-vibration transfer model of the spatial magnetic levitation isolator to obtain the solution result.

[0126] Simulation experiment:

[0127] Currently, PINN can only be used for simple conditions such as periodic boundary conditions and fixed boundary conditions, and it is not yet applicable to complex conditions such as random boundary conditions. First, the complexity and diversity of random boundary conditions make it difficult for neural networks to effectively learn, and the requirement for data sparsity may lead to low training efficiency. In addition, the changes in random boundaries may affect the convergence and numerical stability of the model. Especially in nonlinear problems, it 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 the model performance. Therefore, although PINN provides a new idea for solving partial differential equations, its application under random boundary conditions still requires in-depth research and improvement.

[0128] To test the effectiveness of the PINN solution method in the present invention, a numerical example of a two-dimensional wave equation is taken as an example to verify this method.

[0129] Solve the two-dimensional wave equation:

[0130]

[0131]

[0132] The initial conditions are:

[0133]

[0134]

[0135] The analytical solution of the partial differential equation system is:

[0136]

[0137]

[0138] The present invention uses PINN for solution. The simulation parameters are: the input layer is 2, the output layer is 2, the hidden layer is 50, the maximum number of iterations is 1500 times, and the iteration loss threshold is loss < 1e-4. The number of initial values is 50, 0 < u <1, 0 < v <1, 0 < t <0.1, and the results are compared with the analytical solution results as shown in the following table:

[0139] Table 1 Comparison of the solution results of the numerical example

[0140]

[0141] As can be seen from the simulation results, the root mean square errors between the numerical solutions and the analytical solutions of u and v obtained by using the finite difference method are 0.06 and 0.30 respectively. The root mean square errors between the numerical solutions and the analytical solutions of u and v obtained by using the method of the present invention are 0.01 and 0.04 respectively. It can be thus concluded that the method of the present invention has higher solution accuracy compared with the traditional finite difference method, thereby verifying the effectiveness of this method.

[0142] The present invention is described with reference to the flowcharts and / or block diagrams of methods, apparatuses (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowcharts and / or block diagrams, and the combination of processes and / or blocks in the flowcharts and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, such that the instructions executed by the processor of the computer or other programmable data processing devices produce a means for implementing the functions specified in Figure 1 one process or multiple processes and / or blocks Figure 1 one block or multiple blocks.

[0143] These computer program instructions can also be stored in a computer-readable memory capable of guiding a computer or other programmable data processing devices to work in a specific manner, such that the instructions stored in the computer-readable memory produce a manufactured article including an instruction means, and the instruction means implements the functions specified in Figure 1 one process or multiple processes and / or blocks Figure 1 one block or multiple blocks.

[0144] These computer program instructions can also be loaded onto a computer or other programmable data processing devices, such that a series of operation steps are executed on the computer or other programmable devices to generate a computer-implemented process, and thus the instructions executed on the computer or other programmable devices provide steps for implementing the functions specified in Figure 1 one process or multiple processes and / or blocks Figure 1 one block or multiple blocks.

[0145] Specific embodiments are applied in the present invention to elaborate on the principles and implementation manners 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 of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation manners and application scopes. In summary, the content of this specification should not be construed as a limitation to the present invention.

[0146] Those of ordinary skill in the art will realize that the embodiments described herein are provided to assist the reader in understanding the principles of the present invention, and it should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific deformations and combinations that do not depart from the essence of the present invention based on the technical revelations disclosed in the present invention, and these deformations and combinations are still within the scope of protection 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 and contraction deformation and the cross-sectional shear deformation, a micro-vibration transmission model of the flexible cable of the space magnetic levitation vibration isolator is established. The specific process is as follows: 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 vibration isolator; 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; 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, Loss of boundary conditions for solving models for partial differential equations; 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: 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.

3. 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: 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.

4. 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: 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.

5. 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: 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.

6. 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 connected in sequence with arc coordinates and time as input, a hidden layer, and an output layer with cross-sectional attitude angles, arc coordinate components, forward strain, shear strain, velocity and internal force as output.

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 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, For Random sampling points generated at For The time corresponding to the random sampling point generated at 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.

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: 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, are the values ​​of the boundary conditions predicted by solving the model through partial differential equations, The boundary conditions are known.

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