Bridge distortion differential equation calculation method based on physical information neural network

Abstract

The application discloses a bridge distortion differential equation calculation method based on a physical information neural network and belongs to the technical field of bridge engineering, and comprises the following steps: determining structural size, material parameters and load mode based on the minimum potential energy principle, and constructing a bridge distortion differential equation in a calculation domain; defining a data set, boundary conditions and constraint conditions in the calculation domain according to the bridge distortion differential equation in the calculation domain; calculating a distortion differential equation residual loss and a boundary loss, and constructing a total loss function; and adjusting neural network parameters by a gradient descent multi-objective optimization algorithm based on the total loss function, generating a neural network model, and calculating distortion angles of different sections of a bridge structure. The bridge distortion differential equation calculation method based on the physical information neural network can significantly improve the precision of partial differential equation solving, can solve complex mechanical problems more quickly and more economically, and effectively improves the practicality and the application range.

Description

Technical Field

[0001] This invention relates to the field of bridge engineering technology, and in particular to a method for calculating the differential equation of bridge distortion based on a physical information neural network. Background Technology

[0002] With the continuous development of modern bridge engineering, the design and construction of viaducts and other bridges have received increasing attention. These bridges, including those with closed cross-sections (such as box girders) and open cross-sections (such as channel beams), may exhibit significant distortion and warping deformation under vertical eccentric loading of vehicle loads, whether in straight bridges or curved beam bridges. Therefore, sufficient diaphragms are necessary in the design process to control distortion effects. However, due to the complexity of distortion stress calculation and the lack of reliable and accurate theoretical basis, quantitatively predicting the distortion stress after diaphragm arrangement is very difficult, leading to potentially unsafe designs. Bridge designers tend to adopt conservative and uneconomical diaphragm arrangement schemes. Therefore, proposing an optimized diaphragm design method based on certain distortion control indicators is essential for controlling construction costs and improving the stress performance of long-span composite beam bridges.

[0003] Physical information neural networks (PINNs) combine the underlying physical information describing partial differential equations with neural networks. By fully utilizing physical information as prior knowledge, they can be trained with little or no labeled data as alternative models, thus achieving accurate solutions to the equations. PINN trains the partial differential equations to accurately approximate the exact solution by using an optimizer to minimize the sum of residuals from the initial conditions, boundary conditions, governing equations, and measurement data. Due to the superior ability of neural networks to describe complex relationships between inputs and outputs, PINN demonstrates significant advantages in solving complex constraint equations such as the distortion differential equations of bridge structures by directly embedding the constraints of the physical equations into the network's loss function, ensuring that the network output satisfies physical laws and boundary conditions.

[0004] Existing methods for addressing the distortion stress problem in bridge structures with diaphragms have the following drawbacks:

[0005] (1) The finite element method has high computational cost and low computational efficiency, and the calculation results cannot isolate distorted stress. When traditional methods are extended to multi-span, multi-main-beam and multi-load conditions, the complexity of the calculation model and the computational efficiency increase exponentially, making it difficult to achieve generalization and high efficiency.

[0006] (2) Theoretical methods can only be used for box girders with internal diaphragms and cannot consider the influence of transverse connections between multiple main girders. Moreover, existing theoretical solutions require solving many nonlinear equations simultaneously even for beams with few diaphragms, making them inconvenient to apply. The treatment of diaphragms and transverse connections often uses empirical formulas or simplified models, which cannot fully consider the impact of their quantity and stiffness on bridge performance, especially in multi-girder bridge structures, where such simplified methods cannot accurately describe the stress state of the bridge. Summary of the Invention

[0007] The purpose of this invention is to provide a method for calculating the differential equation of bridge distortion based on a physical information neural network, which significantly improves the accuracy of solving partial differential equations and enables more efficient and economical solutions to complex mechanical problems, thereby improving practicality and applicability.

[0008] To achieve the above objectives, this invention provides a method for calculating the differential equation of bridge distortion based on a physical information neural network, comprising the following steps:

[0009] Step S1: Based on the principle of minimum potential energy, determine the structural dimensions, material parameters, and loading methods, and construct the differential equation of bridge distortion within the domain.

[0010] Step S2: Define the dataset and boundary constraints in the computational domain based on the differential equation of bridge distortion within the computational domain.

[0011] Step S3: Based on the bridge distortion differential equation, dataset, and boundary constraints within the computational domain, calculate the residual loss and boundary loss of the distortion differential equation, and construct the total loss function;

[0012] Step S4: Based on the total loss function, adjust the neural network parameters using a multi-objective optimization algorithm of gradient descent to generate a neural network model and calculate the distortion angle of different sections of the bridge structure.

[0013] Preferably, in step S1, based on the principle of minimum potential energy, the distortion differential equation of the bridge structure is expressed in the following form, which is applicable to both straight beam bridges and curved beam bridges:

[0014]

[0015] Where E is the elastic modulus of the material; I DW γ is the distortion warping stiffness; K is the distortion frame stiffness; γ is the distortion angle; m is the external torque; M Z / 2R represents the torque generated by the curvature effect of the curved beam; M Z This represents the bending moment of the calculated section; R is the radius of curvature of the curved beam (for a straight beam, R is taken as ∞ here);

[0016] Preferably, based on the cross-sectional deformation compatibility condition, the expression for the distortion angle γ is:

[0017]

[0018] Where Δv1 is the vertical displacement of corner point D of the cross section; Δv2 is the vertical displacement of corner point B of the cross section; Δh1 is the horizontal displacement of corner point D of the cross section; and Δh2 is the horizontal displacement of corner point B of the cross section.

[0019] Preferably, for the stress condition of a curved beam under concentrated load and concentrated torque, the expression for the bending moment of the calculated section is:

[0020]

[0021]

[0022] in, The central angle of the curved beam; To calculate the central angle corresponding to the cross section; denoted as the central angle at the location where the load is applied; z is the length of the curved beam along the arc length direction; P0 is the concentrated load; T0 is the concentrated torque.

[0023] Preferably, the dataset in step S2 includes the coordinates of points inside the bridge and boundary points; the boundary constraint condition is that the distortion angle at bridge support points A and B is 0.

[0024] The process of constructing the dataset includes: assuming the longitudinal coordinate z∈[0,L] along the bridge, generating M points z within the interval [0,L] of the computational domain in the longitudinal direction of the bridge. i ;

[0025]

[0026] Where i = 1, 2, 3, ... M; L is the arc length of the curved beam;

[0027] The specific boundary constraints are as follows:

[0028] γ A =γ″ A =0;

[0029] γ B =γ″ B =0;

[0030] γ D =0;

[0031] Where D = 1, 2, 3, ... N, represents the number of diaphragms N, the distortion angle at the diaphragm is set to 0, and the boundary conditions of the model output are defined using hard constraint functions at the locations with diaphragms; γ A γB All are bridge supports; γ″ A γ″ is the curvature at bridge support point A; B γ is the curvature at bridge support point B; D This refers to the distortion angle at the location of the bridge diaphragm.

[0032] Preferably, step S3 is as follows:

[0033] A multi-scale deep neural network is constructed sequentially, consisting of an input layer, H linear layers and H-1 activation function layers, a multi-scale feature layer and an output layer. The number of neurons in the linear layers is s, and the hyperbolic sine function sinh(x) is selected for the activation function layers.

[0034] Based on a multi-scale deep neural network, a total loss function is defined; the total loss function includes: the loss function of the distorted differential equation and the loss function of the left and right boundary conditions of the simply supported beam.

[0035] Preferably, the total loss function Loss is:

[0036] Loss = W PDE Loss PDE +W BC Loss2+W BC Loss3;

[0037]

[0038] Among them, Loss PDE Loss1 represents the loss function of the distorted differential equation; Loss2 represents the loss function of the left boundary condition of the simply supported beam; Loss3 represents the loss function of the right boundary condition of the simply supported beam; W PDE and W BC These represent the weighting coefficients of the corresponding loss terms, with values ​​of W. PDE =10; W BC =1.

[0039] Preferably, the core idea of ​​adjusting the neural network parameters using the multi-objective optimization algorithm of gradient descent in step S4 is to simultaneously optimize multiple loss functions and update the network parameters through gradient backpropagation. Specifically:

[0040] First, multiple objective functions are defined, including physical equation loss (ensuring that the network prediction satisfies the fourth-order differential equation of beam bending) and boundary condition loss (forcing the network prediction to satisfy the bending moment boundary condition), and weights are assigned to each loss function to balance their importance.

[0041] Next, the weighted total loss function is used as the optimization objective. The gradient of the loss with respect to the network parameters is calculated using the gradient descent method, and the parameters are updated through backpropagation.

[0042] Through iterative optimization, the network gradually learns prediction results that simultaneously satisfy physical laws and engineering constraints, ultimately achieving high-precision, physically consistent beam bending deformation calculations.

[0043] Preferably, the specific process of generating the neural network model in step S4 is as follows:

[0044] Step S401: Determine the neural network structure and construct a fully connected neural network, including:

[0045] One input layer with 1 node, the input is the normalized axis coordinate z;

[0046] There are 3 hidden layers, each containing 64 neurons. The activation function of the hidden layers is a sine function to improve the network's ability to fit high-frequency oscillations.

[0047] One output layer with 1 node outputs the deflection v of the beam.

[0048] Step S402: Data preprocessing, linearly transforming the physical coordinate z to the interval [-1,1] to enhance the network's adaptability to numerical range.

[0049] Step S403: Apply hard constraints to the output layer of the neural network to ensure that the boundary conditions are strictly satisfied.

[0050] v(z)=|(z-0)(z-0.205)(z-1.313)(z-2.298)(z-3.283)(z-4.268)(z-5.253)(z-6. 361)(z-6.566)|*N(z) / (0.205*1.313*2.298*3.283*4.268*5.253*6.361*6.566);

[0051] Where N(z) is the original output of the neural network;

[0052] This constraint ensures that the deflection v is zero at nine points z = 0, 0.205, ..., 6.566 (these points include the beam's support points), thus precisely satisfying the multi-support boundary conditions.

[0053] Step S404: Define the physical equation loss function. According to beam bending theory, the fourth-order differential equation is:

[0054] P1(z)=EI*d 4 v / dz 4 +Kv;

[0055] Where EI is the bending stiffness; P1(z) is the distributed load, specifically defined as:

[0056] When z≤3.283: P1(z)=(P / 2)(sin(22.5°) / sin(45°))sin(45°z / 6.566);

[0057] When z>3.283: P1(z)=(P / 2)*(sin(22.5°) / sin(45°))sin(45°-45°z / 6.566);

[0058] Where P = 50 kN, is a concentrated load;

[0059] Loss function calculation steps:

[0060] (1) Calculate the fourth derivative (v_xxxx) of the network output v with respect to the input z:

[0061] v = net(z) # Network output;

[0062] v_z = grad(v,z) # First derivative;

[0063] v_zz = grad(v_z, z) # Second derivative;

[0064] v_zzz = grad(v_zz, z) # Third derivative;

[0065] v_zzzz = grad(v_zzz, z) # Fourth derivative;

[0066] (2) Calculate the residuals of the physical equations:

[0067] f=v_zzzz+(K / EI)*v-P1(z) / EI;

[0068] (3) Physical loss is taken as SmoothL1 Loss (Robust L1 loss):

[0069] loss_pde=SmoothL1Loss(f,0);

[0070] Step S405: Define the boundary condition loss function, and apply a boundary condition with zero bending moment (i.e., d) at the hinge support points of the beam (z = 0 and z = 6.566). 2 v / dz 2 =0):

[0071] (1) Calculate at the boundary points:

[0072] v = net(z_boundary);

[0073] v_z = grad(v, z_boundary);

[0074] v_zz=grad(v_z,z_boundary);

[0075] (2) Boundary loss:

[0076] loss_bc=SmoothL1Loss(v_zz,0).

[0077] Step S406: Set up training. The optimizer in the training device uses the AdamW algorithm with 10,000 iterations. The model parameters are saved once every 100 iterations. The learning rate is 0.001 by default. The device automatically selects GPU or CPU.

[0078] Step S407: Model Validation. After training, the saved model parameters are used to predict the results at 100 axial coordinate points to verify convergence.

[0079] This invention also provides a bridge distortion differential equation calculation system based on a physical information neural network, comprising:

[0080] The equation construction module is used to determine the structural dimensions, material parameters, and loading methods based on the principle of minimum potential energy, and to construct the differential equations of bridge distortion in the computational domain.

[0081] The definition module is used to define the dataset and boundary constraints in the computational domain based on the differential equation of bridge distortion in the computational domain.

[0082] The loss construction module is used to calculate the residual loss and boundary loss of the bridge distortion differential equation based on the bridge distortion differential equation, dataset and boundary constraints in the computational domain, and construct the total loss function.

[0083] The calculation module is used to adjust the neural network parameters based on the total loss function using a multi-objective optimization algorithm of gradient descent, generate a neural network model, and calculate the distortion angle of different sections of the bridge structure.

[0084] Therefore, the present invention employs the aforementioned method for calculating the differential equation of bridge distortion based on a physical information neural network, and the beneficial effects are as follows:

[0085] (1) This invention introduces specific hard constraints into the neural network model, which ensures that the model output meets preset mechanical constraints at specific boundary points, thereby significantly improving the accuracy of solving partial differential equations. This is crucial for applications in engineering design that require accurate simulation of beam deformation and stress distribution.

[0086] (2) By utilizing the efficient backpropagation algorithm of deep neural networks, this technique can quickly optimize model parameters, greatly reducing the computation time and resources required to solve partial differential equations. Especially in large-scale datasets or real-time simulation scenarios, this performance improvement enables complex mechanical problems to be solved faster and more economically.

[0087] (3) This invention can accurately solve the distortion equations of specific single or multi-box beam problems, and can also be extended to similar beam structures without redesigning the network structure. This improves the practicality and applicability of the model and reduces the development cycle for solving new problems.

[0088] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0089] Figure 1 This is a flowchart of an embodiment of a method for calculating the differential equation of bridge distortion based on a physical information neural network according to the present invention;

[0090] Figure 2 This is a diagram illustrating the cross-sectional distortion of a box girder, based on an embodiment of a bridge distortion differential equation calculation method using a physical information neural network according to the present invention.

[0091] Figure 3 This is a diagram of a simply supported curved beam bridge simulated by an embodiment of a bridge distortion differential equation calculation method based on a physical information neural network according to the present invention.

[0092] Figure 4 This is a network structure diagram of the distortion differential equation in an embodiment of the bridge distortion differential equation calculation method based on physical information neural network of the present invention;

[0093] Figure 5 This is the calculation result of the distortion angle of a single main beam section according to an embodiment of the bridge distortion differential equation calculation method based on physical information neural network of the present invention. Detailed Implementation

[0094] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0095] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0096] Example 1

[0097] This embodiment takes a curved single-box single-cell beam bridge under concentrated load at the mid-span section center point as an example. No external torque m is applied. According to the relevant parameters of the beam dimensions in the case study, EI = 413.957 kN·m 4 K = 294.52 kN, P = 50 kN L = 6.566m.

[0098] like Figure 1As shown, this invention provides a method for calculating the differential equation of bridge distortion based on a physical information neural network, including the following steps:

[0099] Step S1: Based on structural mechanics theory, establish the differential equations of bridge distortion within the computational domain; specifically, based on the principle of minimum potential energy, determine the structural dimensions, material parameters, and loading methods. The expression form of the differential equations of bridge distortion is as follows, applicable to both straight beam bridges and curved beam bridges:

[0100]

[0101] Where E is the elastic modulus of the material; I DW γ is the distortion warping stiffness; K is the distortion frame stiffness; γ is the distortion angle; m is the external torque; M Z / 2R represents the torque generated by the curvature effect of the curved beam; M Z This represents the bending moment of the calculated section; R is the radius of curvature of the curved beam, and for a straight beam, R is taken as ∞ here;

[0102] Furthermore, based on the cross-sectional deformation compatibility condition, the distortion angle γ is further expressed as:

[0103]

[0104] Where Δv1 is the vertical displacement of corner point D of section; Δv2 is the vertical displacement of corner point B of section; Δh1 is the horizontal displacement of corner point D of section; Δh2 is the horizontal displacement of corner point B of section; the cross-sectional distortion of the box is illustrated as follows. Figure 2 As shown.

[0105] like Figure 3 As shown, based on the curved beam structure in structural mechanics, and based on the force equilibrium and deformation compatibility equations, the differential equation for the curved beam under concentrated load P0 and concentrated torque T0 is as follows:

[0106] when hour,

[0107]

[0108] when hour,

[0109]

[0110] in, The central angle of the curved beam; To calculate the central angle corresponding to the cross section; θ is the central angle corresponding to the location where the load is applied; z is the length of the curved beam along the arc length direction.

[0111] Step S2: Based on the differential equation of bridge distortion within the computational domain, define the dataset, boundary conditions, and constraints within the computational domain; specifically, the dataset consists of collected points z inside the bridge. PDE and boundary point coordinates z BC Let the longitudinal coordinate of the beam be z∈[0,6.566]. M (M=100 here) points z are generated within the interval [0,6.566] of the computational domain along the beam's length direction. i The details are as follows:

[0112]

[0113] Where i = 1, 2, 3, ... M.

[0114] Boundary constraints include bridge support γ A (z=0), γ B (z=L), and the distortion angle γ at the location of the bridge diaphragm. D =0:

[0115] γ A =γ″ A =0;

[0116] γ B =γ″ B =0;

[0117] γ D =0;

[0118] Where D = 1, 2, 3, ... N represents the number of diaphragms N, the distortion angle at the diaphragm is set to 0, γ = 0, and the boundary conditions of the model output are defined using hard constraint functions at the locations with diaphragms.

[0119] The specific constraints are as follows:

[0120] ①z=0,γ=γ”=0 (simply supported boundary conditions);

[0121] ②z=0.205,1.313,2.298,3.283,4.268,5.253,6.361,γ=0 (where the diaphragm is set);

[0122] ③z=6.566,γ=γ”=0 (simply supported boundary conditions);

[0123] Step S3: Based on the bridge distortion differential equation in the computational domain, the dataset and boundary conditions of Step S2, calculate the residual loss and boundary loss of the distortion differential equation, and construct the total loss function. In this process, a multi-scale deep neural network consisting of an input layer, H linear layers and H-1 activation function layers, a multi-scale feature layer and an output layer is constructed sequentially. The number of neurons in the linear layer is s, and the activation function layer is selected as the hyperbolic sine function sinh(x).

[0124]

[0125] Traditional numerical models solve for structural stress by providing the initial state, boundary state, and physical parameters at any point (x,y). However, it is difficult to calculate the specific value of distortion stress using the finite element method (FEM) in numerical simulations, as the FEM yields the total structural stress. Therefore, it is necessary to propose an accurate analytical method to solve the distortion differential equation. PINN, based on deep neural networks, embeds physical information into the network framework to establish a physical information neural network to approximate u(x,y). The model residuals consist of two parts: the residuals of the predicted structure and the residuals of the physical information constraints. The loss function of PINN can be expressed as:

[0126] Loss = W PDE Loss PDE +W BC Loss2+W BC Loss3;

[0127] Among them, Loss PDE represents the loss function of the distorted differential equation; Loss2 and Loss3 represent the loss functions of the left and right boundary conditions of the simply supported beam; W PDE and W BC These represent the weighting coefficients of the corresponding loss terms, with values ​​of W. PDE =10; W BC =1.

[0128] Since the distorted differential equation cannot be pre-input as the true solution into the neural network as training data through the finite element method, the following functional expression is constructed. The loss function of the equation represents the relationship between the output value of this functional expression and 0. The closer the value is to 0, the more accurate the output result is.

[0129]

[0130] Step S4: Based on the total loss function, adjust the neural network parameters using a multi-objective optimization algorithm of gradient descent to generate a neural network model and calculate the distortion angle of different sections of the bridge structure.

[0131] The core idea behind adjusting neural network parameters using a multi-objective optimization algorithm with gradient descent is to simultaneously optimize multiple loss functions and update the network parameters through gradient backpropagation. Specifically:

[0132] First, multiple objective functions are defined, including physical equation loss (ensuring that the network prediction satisfies the fourth-order differential equation of beam bending) and boundary condition loss (forcing the network prediction to satisfy the bending moment boundary condition), and weights are assigned to each loss function to balance their importance.

[0133] Next, the weighted total loss function is used as the optimization objective. The gradient of the loss with respect to the network parameters is calculated using the gradient descent method, and the parameters are updated through backpropagation.

[0134] Through iterative optimization, the network gradually learns prediction results that simultaneously satisfy physical laws and engineering constraints, ultimately achieving high-precision, physically consistent beam bending deformation calculations.

[0135] The specific process of generating a neural network model is as follows:

[0136] Step S401: Determine the neural network structure and construct a fully connected neural network, including:

[0137] One input layer with 1 node, the input is the normalized axis coordinate z;

[0138] There are 3 hidden layers, each containing 64 neurons. The activation function of the hidden layers is a sine function to improve the network's ability to fit high-frequency oscillations.

[0139] One output layer with 1 node outputs the deflection v of the beam.

[0140] Step S402: Data preprocessing, linearly transforming the physical coordinate z to the interval [-1,1] to enhance the network's adaptability to numerical range.

[0141] Step S403: Apply hard constraints to the output layer of the neural network to ensure that the boundary conditions are strictly satisfied.

[0142] v(z)=|(z-0)(z-0.205)(z-1.313)(z-2.298)(z-3.283)(z-4.268)(z-5.253)(z-6. 361)(z-6.566)|*N(z) / (0.205*1.313*2.298*3.283*4.268*5.253*6.361*6.566);

[0143] Where N(z) is the original output of the neural network;

[0144] This constraint ensures that the deflection v is zero at nine points z = 0, 0.205, ..., 6.566 (these points include the beam's support points), thus precisely satisfying the multi-support boundary conditions.

[0145] Step S404: Define the physical equation loss function. According to beam bending theory, the fourth-order differential equation is:

[0146] P1(z)=EI*d 4 v / dz 4 +Kv;

[0147] Where EI is the bending stiffness; P1(z) is the distributed load, specifically defined as:

[0148] When z≤3.283: P1(z)=(P / 2)(sin(22.5°) / sin(45°))sin(45°z / 6.566);

[0149] When z>3.283: P1(z)=(P / 2)*(sin(22.5°) / sin(45°))sin(45°-45°z / 6.566);

[0150] Where P = 50kN, is a concentrated load;

[0151] Loss function calculation steps:

[0152] (1) Calculate the fourth derivative (v_xxxx) of the network output v with respect to the input z:

[0153] v = net(z) # Network output;

[0154] v_z = grad(v,z) # First derivative;

[0155] v_zz = grad(v_z, z) # Second derivative;

[0156] v_zzz = grad(v_zz, z) # Third derivative;

[0157] v_zzzz = grad(v_zzz, z) # Fourth derivative;

[0158] (2) Calculate the residuals of the physical equations:

[0159] f=v_zzzz+(K / EI)*v-P1(z) / EI;

[0160] (3) Physical loss is taken as SmoothL1 Loss (Robust L1 loss):

[0161] loss_pde=SmoothL1Loss(f,0);

[0162] Step S405: Define the boundary condition loss function, and apply a boundary condition with zero bending moment (i.e., d) at the hinge support points of the beam (z = 0 and z = 6.566). 2 v / dz 2 =0):

[0163] (1) Calculate at the boundary points:

[0164] v = net(z_boundary);

[0165] v_z = grad(v, z_boundary);

[0166] v_zz=grad(v_z,z_boundary);

[0167] (2) Boundary loss:

[0168] loss_bc=SmoothL1Loss(v_zz,0).

[0169] Step S406: Set up training. The optimizer in the training device uses the AdamW algorithm with 10,000 iterations. The model parameters are saved once every 100 iterations. The learning rate is 0.001 by default. The device automatically selects GPU or CPU.

[0170] Step S407: Model Validation. After training, the saved model parameters are used to predict the results at 100 axial coordinate points to verify convergence.

[0171] The network structure for solving the distorted differential equation is as follows: Figure 4 As shown, where:

[0172] The leftmost z-coordinate is the input layer, representing the beam's coordinates.

[0173] There are three hidden layers in the middle, with 64 circles in each column.

[0174] The first hidden layer is a fully connected layer (nn.Linear(1,64)), which receives the output of the input layer.

[0175] The second hidden layer is a fully connected layer (nn.Linear(64,64)) that receives the output of the first hidden layer.

[0176] The third hidden layer is a fully connected layer (nn.Linear(64,64)) that receives the output of the second hidden layer.

[0177] Output layer: γ maps the 64-dimensional output of the third hidden layer to a scalar (the original neural network output value).

[0178] Physical hard constraint module: Embedded constraint conditions.

[0179] For the condition that the distortion angle is 0 when there is a diaphragm, hard contact is used in the network to achieve this. After 10,000 training iterations, the calculation results are relatively stable, and the loss function value does not change significantly. The output distortion angle is as follows: Figure 5 As shown.

[0180] Example 2

[0181] A system for calculating the differential equations of bridge distortion based on a physical information neural network includes:

[0182] The equation construction module is used to determine the structural dimensions, material parameters, and loading methods based on the principle of minimum potential energy, and to construct the differential equations of bridge distortion in the computational domain.

[0183] The definition module is used to define the dataset and boundary constraints in the computational domain based on the differential equation of bridge distortion in the computational domain.

[0184] The loss construction module is used to calculate the residual loss and boundary loss of the bridge distortion differential equation based on the bridge distortion differential equation, dataset and boundary constraints in the computational domain, and construct the total loss function.

[0185] The calculation module is used to adjust the neural network parameters based on the total loss function using a multi-objective optimization algorithm of gradient descent, generate a neural network model, and calculate the distortion angle of different sections of the bridge structure.

[0186] If the aforementioned functions are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0187] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a processor-included system, or other system that can fetch and execute instructions from, an instruction execution system, apparatus, or device). For the purposes of this specification, "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transmit programs for use by, or in conjunction with, an instruction execution system, apparatus, or device.

[0188] More specific examples of computer-readable media (a non-exhaustive list) include: electrical connections (electronic devices) having one or more wires, portable computer disk drives (magnetic devices), random access memory (RAM), read-only memory (ROM), erasable and editable read-only memory (EPROM or flash memory), fiber optic devices, and portable optical disc read-only memory (CDROM). Furthermore, computer-readable media can even be paper or other suitable media on which the program can be printed, because the program can be obtained electronically, for example, by optically scanning the paper or other medium, followed by editing, interpreting, or otherwise processing as necessary, and then stored in computer memory.

[0189] Therefore, the present invention adopts the above-mentioned method for calculating the differential equation of bridge distortion based on physical information neural network, which significantly improves the accuracy of solving partial differential equations, and can solve complex mechanical problems faster and more economically, effectively improving practicality and applicability.

[0190] It is worth noting that all the contents not described in detail in this invention are existing technologies and are well known to those skilled in the art.

[0191] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for calculating the differential equation of bridge distortion based on a physical information neural network, characterized in that, Includes the following steps: Step S1: Based on the principle of minimum potential energy, determine the structural dimensions, material parameters, and loading methods, and construct the differential equation of bridge distortion in the computational domain; The expression for the differential equation of bridge distortion in step S1 is: ； in, E The elastic modulus of the material; I DW The distortion warping stiffness is z; z is the length of the curved beam along the arc length direction. K For the stiffness of the distorted frame; γ This is the distortion angle; m External torque; M Z / 2 R The torque generated by the curvature effect of the curved beam; M Z This represents the bending moment of the calculated section. R Let be the radius of curvature of the curved beam; Step S2: Define the dataset and boundary constraints in the computational domain based on the differential equation of bridge distortion in the computational domain. Step S3: Based on the bridge distortion differential equation, dataset, and boundary constraints within the computational domain, calculate the residual loss and boundary loss of the bridge distortion differential equation, and construct the total loss function; Step S4: Based on the total loss function, adjust the neural network parameters using the multi-objective optimization algorithm of gradient descent to generate a neural network model and calculate the distortion angle of different sections of the bridge structure; In step S4, adjusting the neural network parameters using the gradient descent multi-objective optimization algorithm specifically involves: First, we define multiple objective functions, including physical equation loss and boundary condition loss, and assign weights to each loss function to balance their importance. Next, the weighted total loss function is used as the optimization objective. The gradient of the loss with respect to the network parameters is calculated using the gradient descent method, and the parameters are updated through backpropagation. The specific process of generating the neural network model in step S4 is as follows: Step S401: Determine the neural network structure and construct a fully connected neural network; Step S402: Data preprocessing, converting physical coordinates z Linearly transform to the interval [-1, 1]; Step S403: Apply hard constraints to the output layer of the neural network. Step S404: Define the physical equation loss function. According to beam bending theory, the fourth-order differential equation is: P 1( z ) =EI*d 4 v / dz 4 +Kv ； in, EI For bending stiffness; P 1 ( z ) represents the distributed load; v For deflection; Step S405: Define the boundary condition loss function and apply a boundary condition with zero bending moment at the hinge support of the beam. Step S406: Set up training. The optimizer in the training device uses the AdamW algorithm with 10,000 iterations. The model parameters are saved once every 100 iterations. The learning rate is 0.001 by default. The device automatically selects GPU or CPU. Step S407: Model Validation. After training, the saved model parameters are used to predict the results at 100 axial coordinate points to verify convergence.

2. The method for calculating the differential equation of bridge distortion based on a physical information neural network according to claim 1, characterized in that, The expression for the distortion angle is: ； Where, Δ v 1 represents the vertical displacement of the corner point D of the cross section; Δ v 2 represents the vertical displacement of corner point B of the cross section; Δ h 1 represents the horizontal displacement of the corner point D of the cross section; Δ h 2 represents the horizontal displacement of corner point B of the cross section.

3. The method for calculating the differential equation of bridge distortion based on a physical information neural network according to claim 2, characterized in that, The expression for the bending moment of the calculated section is: ，0≤ ， in, The central angle of the curved beam; To calculate the central angle corresponding to the cross section; The central angle corresponding to the location where the load is applied; P 0 represents a concentrated load; T 0 represents concentrated torque.

4. The method for calculating the differential equation of bridge distortion based on a physical information neural network according to claim 3, characterized in that, In step S2, the dataset includes the coordinates of points inside the bridge and boundary points; the boundary constraint condition is that the distortion angle at bridge support points A and B is 0. The process of constructing the dataset includes: setting the longitudinal coordinates along the bridge. z ∈[0, L ], in the interval [0, ] of the calculation domain in the longitudinal direction of the bridge L [Generated within] M Points z i ; ； in, i =1,2,3,… M ; L Let be the arc length of the curved beam; The specific boundary constraints are as follows: ； ； ； in, D =1,2,3,…N, representing the number of diaphragms N, where hard constraint functions are used to define the boundary conditions of the model output at the locations with diaphragms; γ A , γ B All of them are bridge supports; Let be the curvature at point A, the support point of the bridge. Let be the curvature at bridge support point B; γ D This refers to the distortion angle at the location of the bridge diaphragm.

5. The method for calculating the differential equation of bridge distortion based on a physical information neural network according to claim 4, characterized in that, Step S3 is as follows: A multi-scale deep neural network is constructed sequentially, consisting of an input layer, H linear layers, H-1 activation function layers, a multi-scale feature layer, and an output layer. The number of neurons in the linear layers is s, and the activation function layer is a hyperbolic sine function. A total loss function is constructed based on a multi-scale deep neural network; the total loss function includes: the loss function of the distorted differential equation and the loss function of the left and right boundary conditions of the simply supported beam.

6. The method for calculating the differential equation of bridge distortion based on a physical information neural network according to claim 5, characterized in that, Total loss function for: ； ； in, The loss function represents the distorted differential equation; The loss function representing the left boundary conditions of a simply supported beam; The loss function representing the right boundary conditions of a simply supported beam; W PDE and W BC These represent the weight coefficients of the corresponding loss terms, with values ​​ranging from 1 to 2. W PDE =10; W BC =1.

7. A bridge distortion differential equation calculation system based on a physical information neural network, employing the bridge distortion differential equation calculation method according to any one of claims 1-6, characterized in that, include: The equation construction module is used to determine the structural dimensions, material parameters, and loading methods based on the principle of minimum potential energy, and to construct the differential equations of bridge distortion in the computational domain. The definition module is used to define the dataset and boundary constraints in the computational domain based on the differential equation of bridge distortion in the computational domain. The loss construction module is used to calculate the residual loss and boundary loss of the bridge distortion differential equation based on the bridge distortion differential equation, dataset and boundary constraints in the computational domain, and construct the total loss function. The calculation module is used to adjust the neural network parameters based on the total loss function using a multi-objective optimization algorithm of gradient descent, generate a neural network model, and calculate the distortion angle of different sections of the bridge structure.

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