A complex dynamic system modeling method, application method and related device

Abstract

The application discloses a complex dynamic system modeling method, an application method and related devices, and relates to the field of automation.The modeling method comprises the following steps: constructing an actual physical complex dynamic system; determining a solver of the actual physical complex dynamic system; performing time discretization on the actual physical complex dynamic system; constructing a deep neural network based on the solver and the discretized actual physical complex dynamic system; fusing the residual error of the deep neural network and an ordinary differential equation to construct a deep neural network model driven by physical equations and data fusion; obtaining an actual data set, and constructing a data residual loss function and an ordinary differential equation residual loss function, and calculating a total loss function through weighted summation; and determining the optimal shared parameters between the residual error of the deep neural network and the ordinary differential equation by minimizing the total loss function, so as to obtain an optimal deep neural network model driven by physical equations and data fusion.The application can realize deep fusion and complementation of mechanism models and data models.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of automation, in particular to a complex dynamic system modeling method, an application method and related devices. BACKGROUND

[0002] Precise modeling of complex dynamic systems is the key to simulating and analyzing the evolution of complex physical systems, and is the prerequisite for realizing precise prediction, optimal control and efficient optimization of complex physical systems in the field of automation. At present, the modeling methods of complex dynamic systems mainly include mechanism modeling methods driven by physical knowledge and deep learning modeling methods driven by data. Mechanism models essentially reveal the laws of actual physical systems, and have the advantages of high reliability, good generalization performance and strong interpretability. However, mechanism models are usually composed of complex, high-order nonlinear equations, which are difficult to solve and have a large amount of calculation, making online calculation and application difficult, and it is also difficult to identify the model parameters. Unlike mechanism models, deep learning models are mainly driven by data, such as DNN, CNN, RNN, LSTM and Transformer, which directly establish the mapping relationship between input data and output data without analyzing the physical mechanism of the process object, and have good real-time performance in online calculation. However, such data models have the disadvantages of poor generalization ability, easy overfitting, insufficient interpretability, and dependence on the completeness and accuracy of the training set data.

[0003] Therefore, how to overcome the respective defects of mechanism models and data models of complex dynamic systems and achieve the effect of taking the advantages of both, has become a technical problem to be solved in the field. SUMMARY

[0004] The purpose of the present application is to provide a complex dynamic system modeling method, an application method and related devices, which can realize deep fusion and complementarity of mechanism models and data models, and provide a complex dynamic system model that is physically reasonable, mathematically accurate and computationally efficient for prediction, control and optimization of complex physical systems in the field of automation.

[0005] To achieve the above purpose, the present application provides the following solutions:

[0006] In a first aspect, the present application provides a complex dynamic system modeling method, which comprises:

[0007] constructing an actual physical complex dynamic system; the actual physical complex dynamic system is a nonlinear steady-state space model.

[0008] determining a solver of the actual physical complex dynamic system; the solver of the actual physical complex dynamic system comprises a control variable and an initial state value.

[0009] Discretize the actual physical complex dynamic system in time to obtain a discretized actual physical complex dynamic system.

[0010] Based on the solver of the actual physical complex dynamic system and the discretized actual physical complex dynamic system, a deep neural network is constructed; the deep neural network approximates the solver of the actual physical complex dynamic system in each time interval.

[0011] Fusing the deep neural network and the residual of ordinary differential equation, a physical equation and data fusion driven deep neural network model is constructed.

[0012] An actual data set is obtained, and a data residual loss function and an ordinary differential equation residual loss function are constructed; the actual data set is data measured by an actual production process.

[0013] The data residual loss function and the ordinary differential equation residual loss function are summed by weighting to obtain a total loss function.

[0014] By minimizing the total loss function, the optimal shared parameters between the residual of the deep neural network and the ordinary differential equation are determined, and an optimal physical equation and data fusion driven deep neural network model is obtained.

[0015] In a second aspect, the present application provides a complex dynamic system application method, which comprises:

[0016] According to the optimal physical equation and data fusion driven deep neural network model, a prediction used deep neural network is determined; the optimal physical equation and data fusion driven deep neural network model is a model trained based on the above-mentioned complex dynamic system modeling method.

[0017] Obtain to-be-predicted data; the to-be-predicted data comprises a time variable, a control variable and a first time interval system initial state.

[0018] The to-be-predicted data is input into the prediction used deep neural network to obtain a prediction result.

[0019] In a third aspect, the present application provides a computer device, which comprises a memory, a processor and a computer program stored in the memory and executable on the processor, and the processor executes the computer program to implement the complex dynamic system modeling method or the complex dynamic system application method in any one of the above.

[0020] In a fourth aspect, the present application provides a computer readable storage medium, which stores a computer program executable by a processor to implement the complex dynamic system modeling method or the complex dynamic system application method in any one of the above.

[0021] In a fifth aspect, the present application provides a computer program product comprising a computer program which, when executed by a processor, implements the complex dynamic system modeling method or the complex dynamic system application method of any one of the above.

[0022] According to the specific embodiments provided by the present application, the following technical effects are disclosed:

[0023] The present application provides a complex dynamic system modeling method, an application method and related devices. First, an actual physical complex dynamic system is constructed, and a solver of the actual physical complex dynamic system is determined. The actual physical complex dynamic system is time-discretized to obtain a discretized actual physical complex dynamic system. A deep neural network is constructed based on the solver of the actual physical complex dynamic system and the discretized actual physical complex dynamic system. The deep neural network and the residual error of an ordinary differential equation are fused to construct a deep neural network model driven by physical equations and data fusion. Second, an actual data set is obtained, and a data residual error loss function and an ordinary differential equation residual error loss function are constructed. The total loss function is calculated by weighted sum of the data residual error loss function and the ordinary differential equation residual error loss function. Finally, the optimal shared parameters between the deep neural network and the residual error of the ordinary differential equation are determined by minimizing the total loss function, and the optimal deep neural network model driven by physical equations and data fusion is obtained. The deep neural network framework driven by physical equations and data fusion (PDFD-DNN) proposed in the present application uses the basic idea of the neural network model of physical information (Physics-informed Neural Networks, PINN) to design the loss function of the deep neural network. The residual error of the ordinary differential equation is constructed by using the automatic differentiation technology widely used in the deep neural network, and is integrated into the loss function design process of the deep neural network. This can be regarded as embedding the physical law represented by the ordinary differential equation into the deep neural network, so as to obtain the deep neural network with physical equation constraint (the differential form constraint condition formed by the ordinary differential equation), obtain the deep neural network model driven by physical equations and data fusion, realize the deep fusion and complementarity of the mechanism model and the data model, and obtain a complex dynamic system model which is reasonable in physics, accurate in mathematics and efficient in calculation. BRIEF DESCRIPTION OF DRAWINGS

[0024] In order to more clearly illustrate the technical solutions of the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed in the embodiments. Obviously, the drawings described below are only some embodiments of the present application, and those skilled in the art can obtain other drawings according to these drawings without creative labor.

[0025] Figure 1 A flowchart of a complex dynamic system modeling method provided by an embodiment of the present application.

[0026] Figure 2 A physical equation and data fusion driven deep neural network model structure diagram provided by an embodiment of the present application.

[0027] Figure 3 A flowchart of a complex dynamic system application method provided by an embodiment of the present application.

[0028] Figure 4 A schematic diagram of the evolution of a self-loop prediction model over time provided by an embodiment of the present application.

[0029] Figure 5 A schematic diagram of a double-link manipulator structure provided by an embodiment of the present application.

[0030] Figure 6 A schematic diagram of the PDFD-DNN prediction effect of the generalized coordinate included angle a provided by an embodiment of the present application.

[0031] Figure 7 A schematic diagram of the PDFD-DNN prediction effect of the generalized coordinate included angle b provided by an embodiment of the present application.

[0032] Figure 8 A schematic diagram of the PDFD-DNN prediction effect of the generalized coordinate included angle a provided by an embodiment of the present application.

[0033] Figure 9 A schematic diagram of the PDFD-DNN prediction effect of the generalized coordinate included angle b provided by an embodiment of the present application.

[0034] Figure 10 A schematic diagram of the PDFD-DNN prediction effect of the generalized coordinate included angle a provided by an embodiment of the present application.

[0035] Figure 11 A schematic diagram of the PDFD-DNN prediction effect of the generalized coordinate included angle b provided by an embodiment of the present application.

[0036] Figure 12 A schematic diagram of a computer device provided by an embodiment of the present application. DETAILED DESCRIPTION

[0037] With reference to the accompanying drawings, the technical solutions in the embodiments of the present application will be described clearly and completely. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative efforts belong to the scope of the present application.

[0038] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the present application will be further described in detail below with reference to the drawings and specific embodiments.

[0039] Currently, in the field of automatic complex dynamic system modeling, there are mainly two kinds of physical knowledge and data hybrid driving modeling methods. One is to mainly use mechanism model, and to identify and correct the mechanism model based on data. The other is to parallel the two methods, and to obtain the final result based on the combination evaluation of the results of the two methods. However, the above two methods lack deeper fusion and optimization. In 2017, Raissi proposed a neural network model based on physical information, which realized the construction of a deep neural network model driven by physical equation and data fusion. In recent years, PINN has gradually become a research hotspot in the cross discipline of machine learning and computational mathematics, and has made great progress in theory and application. However, the above PINN model and its variants are only applicable to simulation and analysis design in fields such as fluid mechanics, material mechanics and biomedical field, and cannot be applied to the fields of complex physical system prediction, control, optimization and other automatic complex system modeling. The main reason is that the PINN model and its variants assume that the boundary conditions of the physical field are fixed and unchanged. Therefore, in order to be applicable to the field of automatic control (the real-time change of the boundary conditions of the physical field is a common feature), it is necessary to specially study the modeling method for automatic complex dynamic system driven by physical equation and data fusion, to construct a deep neural network model driven by physical equation and data fusion, to realize the deep fusion and complementarity of mechanism model and data model, and to provide a complex dynamic system model which is physically reasonable, mathematically accurate and computationally efficient for the prediction, control and optimization of complex physical systems in the field of automation.

[0040] In one exemplary embodiment, as shown in Figure 1 a complex dynamic system modeling method is provided, which comprises the following steps:

[0041] S1: constructing an actual physical complex dynamic system; the actual physical complex dynamic system is a nonlinear steady-state space model.

[0042] S2: determining the solution operator of the actual physical complex dynamic system; the solution operator of the actual physical complex dynamic system comprises a control variable and an initial state value.

[0043] S3: Time-discretizing the actual physical complex dynamic system to obtain a discretized actual physical complex dynamic system.

[0044] S4: Constructing a deep neural network based on the solver of the actual physical complex dynamic system and the discretized actual physical complex dynamic system; the deep neural network approximates the solver of the actual physical complex dynamic system in each time interval.

[0045] S5: Fusing the deep neural network and the residual of ordinary differential equation to construct a physical equation and data fusion driven deep neural network model.

[0046] S6: Obtaining an actual data set and constructing a data residual loss function and an ordinary differential equation residual loss function; the actual data set is data measured by an actual production process.

[0047] S7: Calculating a total loss function by weighted summing the data residual loss function and the ordinary differential equation residual loss function.

[0048] S8: Determining the optimal shared parameters between the deep neural network and the residual of ordinary differential equation by minimizing the total loss function to obtain an optimal physical equation and data fusion driven deep neural network model.

[0049] Implementing the above steps S1 to S8 has the following technical effects:

[0050] (1) The residual of ordinary differential equation is integrated into the loss function design process of the deep neural network, and the trained deep neural network can not only approximate the observation data, but also automatically satisfy the symmetry, invariance, conservation and other physical laws followed by the ordinary differential equation, thereby improving the physical interpretability of the deep neural network model.

[0051] (2) The residual term of ordinary differential equation plays a regularization role, which can enhance the generalization ability of the deep neural network, avoid the occurrence of overfitting phenomenon, reduce the dependence on training data, and solve the bottleneck that the deep neural network is difficult to be applied to small sample scenarios in industrial field.

[0052] (3) The PDFD-DNN framework divides the entire simulation process into multiple equidistant control time intervals, each time interval is τ seconds, and the solution x(t) in each time interval is calculated by repeated iteration of a single PDFD-DNN network, which significantly alleviates the problem that the prediction performance of the traditional PINN rapidly degrades when exceeding the training time range.

[0053] (4) Unlike the traditional PINN, the initial state is added to the input space of the deep neural network in the PDFD-DNN framework, which can handle complex dynamic systems with varying initial states and can be applied in the field of automation.

[0054] (5) The deep neural network with ordinary differential equation residual loss (i.e., the physical equation and data fusion driven deep neural network model) performs better than the deep neural network without ordinary differential equation residual loss (i.e., the conventional pure data driven deep neural network, DNN) in complex dynamic system modeling and prediction.

[0055] In an exemplary embodiment, the steps of the physical equation and data fusion driven complex dynamic system modeling scheme are as follows:

[0056] A1: The actual physical complex dynamic system is represented by the following nonlinear constant state space model (composed of ordinary differential equations):

[0057]

[0058] where, is the time variable, x: is the system state, is the initial state of the system, u: is the system control variable, t0is the initial time, t N is the termination time, and N() represents a nonlinear function. It is assumed that the nonlinear function N() is continuous and locally Lipschitz continuous with respect to the state x, such that for the initial value problem of the actual physical complex dynamic system (1), there is a unique (weak) solution for every .

[0059] A2: Define the solver of the actual physical complex dynamic system, i.e., map the control variable and initial state value to the current solution of the system, expressed as:

[0060]

[0061] A3: Considering the practical application of the controller in the digital control of the physical system, the actual physical complex dynamic system (1) is time-discretized, and the following time grid is introduced:

[0062] t0<t1<t2<...<t N (3);

[0063] An equidistant time grid is selected, i.e., t k =t0+kτ, τ>0 is a constant sampling period corresponding to the sampling period of the discrete-time control system, and the setting of τ usually depends on the specific dynamic characteristics of the actual physical system.

[0064] A4: Introduce the transfer input u k : = u(t0+kτ), k = 0,1,2,...,N-1, we have the following expression:

[0065]

[0066] where x k = x(t k ). If the control input satisfies the zero-order hold assumption, i.e. then formula (4) is similar to a discrete-time control system.

[0067] A5: Construct a deep neural network f N (t, u, x0; w) approximates the system's solution operator in each time interval The input of the deep neural network includes: time variable t, control variable u, and the initial state x0 of the system in each time interval, assuming that the control input remains constant in the time interval t ∈ [0, τ], similar to the zero-order holder, the expression is as follows:

[0068] x(t) = f N (t, u, x0; w), t ∈ [0, τ] (5);

[0069] where f N represents the deep neural network mapping relationship parameterized by w, w is a set of trainable weight parameters and bias parameters, and the nonlinear activation function of the deep neural network is selected as the hyperbolic tangent activation function tanh(x) or the sine activation function sin(x).

[0070] A6: Fuse the deep neural network f N (t, u, x0; w) with the residual of the ordinary differential equation to form a deep neural network model driven by physical equations and data fusion, the model structure is as shown in Figure 2 , where the residual of the ordinary differential equation is defined as:

[0071]

[0072] where F(t, u, x0; w) represents the residual of the ordinary differential equation, represents the time variable, u: represents the system control variable, x0 represents the initial state of the system in each time interval, and w represents a set of trainable weight parameters and bias parameters. N (t, u, x0; w) represents a deep neural network, f N represents the deep neural network mapping relationship parameterized by w, and N() represents a nonlinear function.

[0073] A7: Obtain the differential of function combination by applying chain rule using automatic differentiation technique

[0074] A8: Actual dataset is composed of actual production process measurement data and construct data residual loss function L data (w), as shown in equation 7.

[0075]

[0076] where L data (w) represents the data residual loss function, data pair represents the i-th training data sample, X represents the actual real state of the system, represents all inputs of the deep neural network (including time, control input, initial state), represents the time variable, u: represents the system control variable, X0 represents the initial state of the system at each time interval, represents the deep neural network of the i-th training data sample, f N represents the mapping relationship of the deep neural network parameterized by w, w represents the set of trainable weight parameters and bias parameters, N data represents the number of samples in the actual dataset.

[0077] A9: Randomly sample the residual of ordinary differential equation in the 3-dimensional space of (time, control input, initial state) from their respective specific interval range to form a configuration point set and construct the ordinary differential equation residual loss function L phys (w), as shown in equation 8.

[0078]

[0079] where L phys (w) represents the ordinary differential equation residual loss function, represents the residual of the j-th training data sample ordinary differential equation, represents the time variable, u: represents the system control variable, X0 represents the initial state of the system at each time interval, w represents the set of trainable weight parameters and bias parameters, N phys represents the number of samples in the configuration point set.

[0080] It should be particularly pointed out that the interval range of time t is [0, τ].

[0081] A10: Calculate the total loss function L(w) by weighted sum of data residual loss and partial differential equation residual loss, the total loss function is defined as:

[0082] L(w) = λ data L data (w) + λ phys L phys (w) (9);

[0083] wherein λ data represents a first weight coefficient used to balance the interaction between the data residual loss and the partial differential equation residual loss, λ phys represents a second weight coefficient used to balance the interaction between the data residual loss and the partial differential equation residual loss, so that the sizes of the two losses are on the same scale. These weight coefficients can be given empirically before the neural network training, or can be adaptively adjusted during the training. Reasonable weight coefficient settings play an important role in improving the trainability and convergence speed of the PDFD-DNN model.

[0084] A11: After the PDFD-DNN network structure, the actual data set, the configuration point set and the loss function are determined, the deep neural network f N (t, u, x0; w) and the residual F(t, u, x0; w) of the ordinary differential equation are learned by minimizing the total loss function L(w), first K1 iterations are performed using the ADAM optimizer, and then K2 iterations are performed using the L-BFGS optimizer, and then the optimal physical equation and data fusion driven deep neural network model parameter w * .

[0085] In another exemplary embodiment of the present application, as shown in Figure 3 , the present application also provides a complex dynamic system application method, which comprises:

[0086] B1: determining a deep neural network for prediction according to the optimal physical equation and data fusion driven deep neural network model; the optimal physical equation and data fusion driven deep neural network model is a model trained based on the complex dynamic system modeling method described above.

[0087] B2: obtaining to-be-predicted data; the to-be-predicted data comprises a time variable, a control variable and a first time interval system initial state.

[0088] B3: inputting the to-be-predicted data into the deep neural network for prediction to obtain a prediction result.

[0089] That is, after obtaining the optimal physical equation and data fusion driven deep neural network model parameter w * in step A11, a deep neural network for prediction f N (t, u, x0; w)* ), given the input of a deep neural network, including time variable, control variable and the first time interval system initial state, the deep neural network f N (t,u,x0;w * ) can directly calculate the solution in the time interval t∈[0,τ]. The calculation result of the deep neural network at time t=τ f N (τ,u,x0;w * ) is used as the initial state of the time interval t∈[τ,2τ], and the solving calculation process of formula (5) is iteratively repeated, forming a self-loop prediction model, so as to obtain the solution of all time of the system at time t>τ, as shown in Figure 4 .

[0090] The effectiveness of the physical equation and the data fusion driven deep neural network framework proposed in the application is verified by a double-link manipulator dynamics simulation system.

[0091] The structural diagram of the double-link manipulator and the inertial reference frame {e1, e2, e3} used for modeling are shown in Figure 5 . The links are connected by rotating joints representing rotating units. The system considered has 2 degrees of freedom, and the kinematics of the double-link manipulator can be described by generalized coordinates q=[α,β] T , α and β represent the angles of the two generalized coordinates, respectively. The system parameters include mass m1 and m2, moment of inertia J1 and J2, link length l0, l1 and l2, etc.

[0092] The system modeling of the rigid double-link manipulator system considered is carried out, and the minimum form of the nonlinear motion equation is obtained:

[0093]

[0094] Where the input vector is composed of the motor currents of the first and second rotating joints u=[I1, I2] T . The dynamic system in formula (10) is represented by a positive definite symmetric generalized mass matrix M(q), and the vector is composed of centrifugal force, Coriolis force and gyroscopic force. The input matrix D is represented, and the expression is as follows:

[0095]

[0096] Where T fr,1 , T fr,2 are the friction torques of the first and second rotating joints, respectively, R1 and R2 are the gear transmission ratios of the first and second rotating joints, respectively, and k1 and k2 are the motor constants of the first and second rotating joints, respectively.

[0097] The The first-order nonlinear state-space model expression of formula (10) is obtained:

[0098]

[0099] The deep neural network used in this simulation experiment is composed of 4 fully connected layers, uses a hyperbolic tangent activation function on the hidden layer neurons, and includes 32 neurons. For the data residual loss function L data (w) quantization, N data initial state data pairs are used to form actual data samples, and these data pairs are only a subset of all possible initial states. Therefore, for the quantization of the ordinary differential equation residual loss function L phys (w), N phys configuration points are randomly generated in the allowed 7-dimensional input space domain, and the range is:

[0100]

[0101] All hyperparameters are the results of grid search, including the number of deep neural network layers, the number of hidden layer neurons, the number of actual data set samples N data , and the number of configuration point set samples N phys . For the training of the PDFD-DNN network, first use ADAM to optimize the loss function for K1=10000 iterations, and then use L-BFGS to optimize the loss function for K2=20000 iterations to enhance the stability of the training process.

[0102] To verify the effectiveness of the PDFD-DNN method proposed in the present application and its approximate prediction ability, run the PDFD-DNN in the self-loop mode (see Figure 4 ), and set the sampling time interval to τ=0.2S. There are a total of 20 self-loop iterations in this experiment.

[0103] Figure 6 and Figure 7 show the changes of the prediction effect of PDFD-DNN with the number of actual data set samples (N phys remains unchanged at 20000, and N data is 100, 500, and 1000, respectively). As can be seen from the figure, as the number of actual data set samples increases, the prediction effect of PDFD-DNN becomes better (i.e., the predicted values α-pre, β-pre of the generalized coordinates angle α, β are closer to the actual values α-rel, β-rel).

[0104] Figure 8 and Figure 9 show the changes of the prediction effect of PDFD-DNN with the number of configuration point set samples (Ndata N is kept as 1000 phys 5000, 10000, 20000 respectively, it can be seen from the figure that with the increase of the number of configuration point set samples, the prediction effect of PDFD-DNN becomes better (that is, the predicted values of the generalized coordinate angles α, β, α-pre, β-pre, are closer to the actual values α-rel, β-rel).

[0105] Figure 10 and Figure 11 The comparison of the prediction effects of PDFD-DNN and DNN is shown in the figure, it can be seen from the figure that the prediction effect of the physical equation and data fusion driven deep neural network (PDFD-DNN) is obviously better than that of the pure data driven deep neural network (DNN), even if the prediction effect of the PDFD-DNN with the actual data set sample number of 1000 is much better than that of the DNN with the actual data set sample number of 10000. This is because the PDFD-DNN embeds the physical law represented by the ordinary differential equation into the deep neural network, the residual term of the ordinary differential equation plays a regularization role, enhances the generalization ability of the deep neural network, and at the same time reduces the dependence on the training data, which can be applied to the small sample scene in the industrial field.

[0106] In an exemplary embodiment, a computer device, which can be a server or a terminal, is provided, and an internal structure diagram thereof can be as shown in Figure 12 The computer device includes a processor, a memory, an input / output interface (I / O) and a communication interface. Among them, the processor, the memory and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. Among them, the processor of the computer device is used to provide computing and control ability. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The database of the computer device is used to store an actual data set. The input / output interface of the computer device is used to exchange information between the processor and external devices. The communication interface of the computer device is used to communicate with external terminals through network connection. The computer program is executed by the processor to implement a complex dynamic system modeling method or a complex dynamic system application method.

[0107] Those skilled in the art can understand, Figure 12The structure shown in the figure is only a block diagram of part of the structure related to the scheme of the present application, and does not constitute a limitation on the computer device to which the scheme of the present application is applied. The specific computer device can include more or fewer components than those shown in the figure, or combine certain components, or have a different arrangement of components.

[0108] In an exemplary embodiment, a computer device is also provided, including a memory and a processor, the memory storing a computer program, and the processor implementing the above method embodiments when executing the computer program.

[0109] In an exemplary embodiment, a computer readable storage medium is provided, storing a computer program, and the computer program implements the above method embodiments when executed by a processor.

[0110] In an exemplary embodiment, a computer program product is provided, including a computer program, and the computer program implements the above method embodiments when executed by a processor.

[0111] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data for analysis, stored data, displayed data, etc.) involved in the present application are all information and data authorized by the user or authorized by all parties, and the collection, use and processing of related data need to comply with relevant regulations.

[0112] Those skilled in the art can understand that all or part of the processes in the above-mentioned embodiment methods can be completed by instructing the relevant hardware through a computer program. The computer program can be stored in a non-volatile computer readable storage medium, and when the computer program is executed, the processes of the above-mentioned embodiments of the methods can be included. Any reference to memory, database or other medium used in the embodiments provided by the present application can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical storage, high-density embedded non-volatile memory, resistive memory (ReRAM), magnetoresistive random access memory (MRAM), ferroelectric memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. As an illustration but not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM), etc.

[0113] The database involved in the embodiments provided by the present application can include at least one of a relational database and a non-relational database. The non-relational database can include a distributed database based on a blockchain, etc., without being limited thereto. The processor involved in the embodiments provided by the present application can be a general-purpose processor, a central processing unit, a graphics processing unit, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, etc., without being limited thereto.

[0114] The technical features of the above embodiments can be combined arbitrarily. In order to make the description concise, all possible combinations of the technical features in the above embodiments are not described, but as long as the combinations of the technical features do not exist contradictory, they should be considered as the scope of the present application.

[0115] The principles and implementation modes of the present application are described by using specific examples in this paper, and the above-mentioned embodiments are only used to help understand the method of the present application and its core idea; at the same time, for those skilled in the art, according to the idea of the present application, the specific implementation mode and application range will be changed. In conclusion, the content of the present application should not be understood as a limitation.

Claims

1. A method of modeling a complex dynamic system, characterized by, The complex dynamic system modeling method is applied to a double-link manipulator dynamics simulation system, and kinematics of the double-link manipulator is described by using generalized coordinates q=[α,β] T , where α and β respectively represent two generalized coordinate angles; system parameters include masses m1 and m2, rotational inertias J1 and J2, and link lengths l0, l1 and l2; A system model is established for the rigid double-link manipulator system under consideration, and a minimal form of nonlinear motion equation is obtained: where the input vector is composed of the motor currents of the first and second revolute joints u = [I1, I2] T ; the dynamics of the system in the minimal form of the nonlinear motion equation is given by the positive definite symmetric generalized mass matrix M(q), the vector of centrifugal, Coriolis and gyroscopic forces force vector The input matrix D is characterized by the expression where T fr,1 , T fr,2 are the friction torques of the first and second revolute joints, respectively, R1, R2 are the gear ratios of the first and second revolute joints, respectively, and k1, k2 are the motor constants of the first and second revolute joints, respectively. Introduction Obtain the first order nonlinear state space model expression of the minimal form of the nonlinear motion equation: The complex dynamic system modeling method comprises: An actual physical complex dynamic system is constructed; the actual physical complex dynamic system is a nonlinear constant state space model; A solver of the actual physical complex dynamic system is determined; the solver of the actual physical complex dynamic system comprises a control variable and an initial state value; The actual physical complex dynamic system is time-discretized to obtain a discretized actual physical complex dynamic system; Based on the solver of the actual physical complex dynamic system and the discretized actual physical complex dynamic system, a deep neural network is constructed; the deep neural network approximates the solver of the actual physical complex dynamic system in each time interval; The deep neural network and the residual error of the ordinary differential equation are fused to construct a deep neural network model driven by physical equation and data fusion; An actual data set is obtained, and a data residual error loss function and an ordinary differential equation residual error loss function are constructed; the actual data set is data measured by an actual production process; A total loss function is calculated by weighted summing the data residual error loss function and the ordinary differential equation residual error loss function; By minimizing the total loss function, optimal shared parameters between the residual error of the deep neural network and the ordinary differential equation are determined, and an optimal deep neural network model driven by physical equation and data fusion is obtained; An expression of the data residual error loss function is: where L data (w) denotes a data residual loss function, data pair denotes the i-th training data sample, X denotes the actual real state of the system, denotes a time variable, denotes the system control variable, x0 denotes the initial state of the system at each time interval, denotes the deep neural network of the i-th training data sample, f N denotes the deep neural network mapping relationship parameterized by w, w denotes a set of trainable weight parameters and bias parameters, N data denotes the number of actual data set samples; An expression of the ordinary differential equation residual error loss function is: wherein L phys (w) represents the ordinary differential equation residual loss function, represents the residual of the jth training data sample ordinary differential equation, represents the time variable, represents the system control variable, x0 represents the initial state of the system at each time interval, w represents a set of trainable weight parameters and bias parameters, N phys represents the number of configuration point set samples.

2. The complex dynamic system modeling method of claim 1, wherein, An expression of the actual physical complex dynamic system is: wherein, denotes a time variable, denotes a system state, denotes a system initial state, denotes a system control variable, t0denotes an initial time, t N denotes a final time, N() denotes a nonlinear function.

3. The complex dynamic system modeling method of claim 1, wherein, An expression of the residual error of the ordinary differential equation is: where F(t, u, x0; w) represents the residual of the ordinary differential equation, denotes the time variable, denotes the system control variable, x0 denotes the initial state of the system at each time interval, w denotes a set of trainable weight parameters and bias parameters, f N (t, u, x0; w) represents the deep neural network, f N denotes the deep neural network mapping relationship parameterized by w, N() denotes a nonlinear function.

4. The complex dynamic system modeling method of claim 1, wherein, An expression of the total loss function is: L(w) = λ data L data (w) + λ phys L phys (w); wherein L(w) represents a total loss function, L data (w) represents a data residual loss function, L phys (w) represents an ordinary differential equation residual loss function, λ data represents a first weight coefficient used to balance the interaction between the data residual loss and the partial differential equation residual loss, λ phys represents a second weight coefficient used to balance the interaction between the data residual loss and the partial differential equation residual loss.

5. A method of complex dynamic system application, characterized by, The complex dynamic system application method comprises: According to the optimal deep neural network model driven by physical equation and data fusion, a deep neural network for prediction is determined; the optimal deep neural network model driven by physical equation and data fusion is a model trained based on the complex dynamic system modeling method in any one of claims 1-4; Obtaining to-be-predicted data; the to-be-predicted data comprises a time variable, a control variable, and a first time interval system initial state; The to-be-predicted data is input into the deep neural network for prediction to obtain a prediction result.

6. A computer device comprising: A memory, a processor, and a computer program stored on the memory and executable on the processor, characterized in that the processor executes the computer program to implement the complex dynamic system modeling method in any one of claims 1-4 or the complex dynamic system application method in claim 5.

7. A computer-readable storage medium having stored thereon a computer program, characterized in that, The computer program is executed by the processor to implement the complex dynamic system modeling method in any one of claims 1-4 or the complex dynamic system application method in claim 5.

8. A computer program product comprising a computer program, characterized in that, The computer program is executed by the processor to implement the complex dynamic system modeling method in any one of claims 1-4 or the complex dynamic system application method in claim 5.

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