Method and device for solving mBL equation based on parallel physical information neural network

Abstract

The present invention belongs to the field of artificial intelligence technology, and discloses a method and device for solving mBL equations based on parallel physical information neural networks, including time-space division of the overall velocity potential field, reasonable selection of multi-time slice data, and inputting the reasonably selected data of the high-speed field and the low-speed field into the parallel physical information neural networks of different servers for sub-domain parallel calculation, so as to obtain the solved high-speed field and the solved low-speed field; and then spatially merge the solved sub-domains of these two parts to obtain the solved velocity potential field, which is used to solve the mBL equation. The present invention optimizes the solution based on time-space domain division, and at the same time reduces the influence of the difficulty in transmitting the physical information of the initial point and the boundary point, and improves the accuracy of the solution.

Description

Technical Field

[0001] The present invention belongs to the field of artificial intelligence technology, and in particular relates to a method and a device for solving mBL equations based on a parallel physical information neural network. Background Art

[0002] The modified Benney-Luke equation (mBL equation) is a relatively complex new type of partial differential equation in the field of physical oceanography, with multiple high partial derivatives, complex space-time boundaries, and large spatial and temporal ranges. However, solving the mBL equation with traditional numerical methods is not only time-consuming but also heavily dependent on manual experience: on the one hand, the iterative solution of the mBL equation requires expensive computational overhead; on the other hand, in order to avoid computational failure, frequent human-computer interaction is usually required to identify and optimize the mesh quality during the meshing process to meet the requirements of prediction accuracy.

[0003] Physical information neural network is a kind of neural network used to solve supervised learning tasks. It adds physical equations as constraints in the neural network so that the trained model can satisfy the laws of physics and the final prediction results can fit the real value and satisfy the physical laws described by partial differential equations. However, directly using physical information neural network to solve mBL equations cannot obtain high-precision solutions, mainly because physical information neural network has the following problems:

[0004] (1) It is difficult to solve complex partial differential equations with high-dimensional non-convex properties. High-dimensional partial differential equations usually have complex structures and high-order derivatives, and are likely to be high-dimensional non-convex functions. High-dimensional non-convex optimization problems are NP-hard problems and are difficult to solve. Therefore, due to the inaccuracies involved in solving high-dimensional non-convex optimization problems, physical information neural networks can easily fall into local optimal solutions when solving high-dimensional partial differential equations, making it difficult to obtain exact solutions to high-dimensional partial differential equations. (2) It is difficult to transmit the physical information of boundary points and initial points. Physical information neural networks start training from initial points and boundary points, but the initial conditions or boundary conditions of partial differential equations are sometimes difficult to determine, and as the number of training times increases, the physical information corresponding to the initial points and boundary points is difficult to transmit to a longer time range or deeper space. (3) It is difficult to perform low-cost fast training based on automatic differentiation. The backbone network of the physical information neural network is the deep neural network. The deep neural network introduces the physical information of partial differential equations in iterative training through automatic derivation. The derivation includes spatial partial derivatives and temporal partial derivatives. The complex derivation process will greatly increase the cost and time of training. Summary of the invention

[0005] In view of the shortcomings of the prior art, the present invention provides a method and device for solving mBL equations based on a parallel physical information neural network. The present invention uses selected data of different time slices as input, focuses on the physical information of the time slice to be solved and associates the physical information of the previous time slice, accelerates the solution and merging of high-speed fields and low-speed fields through multiple servers, and uses the merged overall velocity potential field as output. Specifically, the following three aspects are improved: (1) Optimized solution of high-dimensional non-convex equations: The overall velocity potential field time-space division module performs time dimension reduction and space division on the overall velocity potential field, divides multiple groups of velocity potential fields under different time-space conditions, and reduces the complex high-dimensional non-convexity of the overall velocity potential field. (2) In-depth transmission of effective physical information: When the initial point and boundary point only exist in the initial field where the velocity potential value is all zero, the multi-time slice data reasonable selection module reasonably selects data from different time slices to start training, avoiding the problem that the physical information corresponding to the initial point and boundary point is difficult to transmit. (3) Low cost and fast training of the network: The high-speed and low-speed field parallel calculation modules accelerate the solution of high-speed and low-speed fields based on the spatiotemporal division of the overall velocity potential field and the reasonable selection of multi-time slice data, thereby reducing the training cost and time of the network.

[0006] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0007] The method for solving the mBL equation based on a parallel physical information neural network includes the following steps:

[0008] Step 1: Spatiotemporal division of the overall velocity potential field;

[0009] First, the overall velocity potential field with three-dimensional space-time data is reduced in time dimension to obtain the velocity potential field of different time slices; then the velocity potential field of different time slices is spatially divided to obtain the high-speed field of different time slices and the low-speed field of different time slices;

[0010] Step 2: Reasonable selection of multi-time slice data;

[0011] First, the high-speed field or low-speed field data of different time slices are divided into the data of the time slice to be solved and the data of the previous time slice; then the two parts of data are reasonably selected to obtain the reasonably selected high-speed field data and low-speed field data; wherein the reasonable selection method is: selecting the data of the previous time slice associated with the data of the time slice to be solved, and making the number of the selected data of the previous time slice less than the data of the time slice to be solved;

[0012] Step 3, high-speed field and low-speed field are calculated in parallel;

[0013] First, the reasonably selected data of the high-speed field and the reasonably selected data of the low-speed field are respectively input into the parallel physical information neural network of different servers for sub-domain parallel calculation to obtain the solved high-speed field and the solved low-speed field; then the solved sub-domains of these two parts are spatially merged to obtain the solved velocity potential field, which is used to solve the mBL equation.

[0014] Furthermore, the specific steps of step 1 are as follows:

[0015] Step 1.1, normalize the mBL equation;

[0016] The mBL equation is shown below:

[0017] (3);

[0018] in, is the velocity potential, is the wave speed, , , , b is the coefficient, Velocity potential The first-order partial derivative of represents the second-order derivative of the velocity potential ξ with respect to time t, represents the Laplace operator, represents the gradient operator;

[0019] The equation coefficients, initial conditions and boundary conditions of the mBL equation are set so that the velocity potential field of the mBL equation has a regular shape; the mBL equation after setting is shown as follows:

[0020] (12);

[0021] Setting the equation coefficient b in equation (3) to 0, we get equation (12); for the mBL equation represented by equation (12), the initial point is the space point at t = 0, the boundary point is ξ = 0 and = 0 spatial point;

[0022] Step 1.2: Time dimension reduction;

[0023] For the mBL equation, the overall velocity potential field is divided according to time slices to obtain velocity potential fields of different time slices, and the velocity potential fields of different time slices have different shapes;

[0024] Step 1.3: Space division;

[0025] For the mBL equation, except for the initial field at t=0 which is a gradient-free uniform field, the velocity potential fields of other time slices have a regular shape: high in the center and low at the edge; therefore, the velocity potential fields of different time slices are spatially divided according to this regular shape to obtain high-speed fields and low-speed fields of different time slices; specifically, the spatial division is based on the velocity potential threshold to divide the velocity potential field of a certain time slice into a high-speed potential field and a low-speed potential field, that is, a high-speed field and a low-speed field.

[0026] Furthermore, the velocity potential threshold has the following characteristics: it is smaller than the value of all points in the high-speed field and larger than the value of all points in the low-speed field.

[0027] Furthermore, after the time-space division of the overall velocity potential field in step 1, the initial point and boundary point of the mBL equation are all located in the initial field, and the velocity potential values ​​of all spatial points in the initial field are zero, and the overall velocity potential field of the mBL equation is divided into high-speed fields and low-speed fields of different time slices;

[0028] Step 2 reasonably selects data of different time slices and reasonably allocates physical information of different time slices; for high-speed fields of different time slices, the data input into the parallel physical information neural network for training includes two parts, as shown in the following formula:

[0029] (13);

[0030] in, Represents all the data input into the network for training in the current time slice. Represents the data selected from the current time slice, Represents the data selected from the previous time slice; the selected data includes the spatial point and the corresponding true velocity potential value; The amount of data is greater than The amount of data, Contains the data of the initial field, and the initial points and boundary points are located within the initial field.

[0031] Furthermore, after steps 1 and 2, the overall velocity potential field of the mBL equation is divided into high-speed fields and low-speed fields of different time slices. When solving the high-speed field or low-speed field of a certain time slice in step 3, the data input to the network includes not only the data of the current time slice, but also the data of the previous time slice. Therefore, different servers are used to calculate the high-speed field or low-speed field of a certain time slice in parallel, and then they are merged into a complete velocity potential field according to the spatial coordinates, as shown in the following formula:

[0032] (14);

[0033] (15);

[0034] (16);

[0035] Among them, x and y represent spatial coordinates, and t represents time. represents the velocity potential value of the current velocity potential field being solved, represents the velocity potential value of the current high-speed field being solved, Indicates the velocity potential value of the current low-velocity field being solved; Represents the deep neural network currently used in the high-speed field, Indicates the current high-speed field and the previous high-speed field; Represents the deep neural network used in the current low-speed field, Indicates the current low-speed field and the previous low-speed field, including the initial field, and the initial point and boundary point are located in the initial field; Depend on and Take the union to get; and Having the same network structure, the two constitute the parallel physical information neural network.

[0036] Furthermore, the parallel physical information neural network inputs the spatial coordinates x, y and time t, and outputs the velocity potential value ξ and the solution f; specifically:

[0037] The network first determines whether x and y belong to the high-speed field. If yes, it is trained with the deep neural network 1 in server 1. If not, it is trained with the deep neural network 2 in server 2. The structures of the two deep neural networks are exactly the same, except for the input and output. The spatial coordinates and time are outputted as velocity potential values ​​after passing through the deep neural network. The velocity potential values ​​are automatically differentiated into multiple first-order partial derivatives and second-order partial derivatives. The multiple partial derivatives are substituted into equation (12) to obtain the solution f of the equation.

[0038] (17);

[0039] (18);

[0040] (19);

[0041] (20);

[0042] Among them, ξ is the velocity potential value solved by the neural network, is the velocity potential value solved by pseudospectral method, is the first-order partial derivative of the velocity potential with respect to time solved by the neural network, is the first-order partial derivative of the velocity potential with respect to time obtained by the pseudo-spectral method, N is the number of input spatial points, including the initial point and the boundary point; Zhong ξ is ξ high ,exist Zhong ξ is ξ low ; It is also responsible for processing the data of the initial field, and the initial point and the boundary point are both located in the initial field, and the loss calculation of the initial point and the boundary point also follows equations (17) to (20);

[0043] For the mBL equation, MSE is the overall loss function, which is composed of the loss function MSE_ξ of the velocity potential value and the loss function MSE_ξ of the first-order time partial derivative. t and the loss function MSE_f of physical information added together; when MSE approaches zero, it is considered that the predicted value of each point in the training data set is close to the true value. In this way, solving the mBL equation is transformed into optimizing the loss function, which is optimized using the back propagation mechanism of the neural network and the two optimizers, L-BFGS and Adam.

[0044] The present invention also provides a device for solving mBL equations based on parallel physical information neural network, comprising: an overall velocity potential field time-space division module, a multi-time slice data reasonable selection module, and a high-speed and low-speed field parallel calculation module, wherein the overall velocity potential field time-space division module is used to perform time dimension reduction and space division on the overall velocity potential field of the input three-dimensional time-space data to obtain high-speed fields of different time slices and low-speed fields of different time slices;

[0045] The multi-time slice data reasonable selection module is used to reasonably select data from high-speed fields of different time slices and low-speed fields of different time slices to obtain reasonably selected data from high-speed fields or low-speed fields;

[0046] The high-speed and low-speed field parallel calculation module is used to input the reasonably selected data of the high-speed field and the low-speed field into the parallel physical information neural network of different servers for sub-domain parallel calculation to obtain the solved high-speed field and the solved low-speed field; then the solved sub-domains of these two parts are spatially merged to obtain the solved velocity potential field for solving the mBL equation.

[0047] Compared with the prior art, the present invention has the advantages of:

[0048] First, the optimal solution of high-dimensional non-convex equations is achieved. First, the mBL equation is normalized, and the equation coefficients, initial conditions, and boundary conditions are set so that the velocity potential field can present a regular shape. Secondly, the velocity potential field is reduced in time dimension, and the velocity potential field of different time slices has a regular convex shape. Finally, the velocity potential field of different time slices is spatially divided to obtain high-speed fields and low-speed fields that can be solved separately. After the above processing, the mBL equation has a low-dimensional convex property that is easy to solve, and can be optimized based on the division of time and space domains.

[0049] Second, the in-depth transmission of effective physical information is achieved. After the mBL equation is normalized, the boundary points are all located in the initial field like the initial points, and the velocity potential values ​​of all points in the initial field are zero. By selecting most of the data in the velocity potential field of the current time slice, selecting a small amount of data in the velocity potential field of the previous time slice (including the initial field), and then associating the physical information of different time slices to start training, the accuracy of the neural network solution on different time slices can be enhanced. After the above processing, the neural network starts training not only with the physical information of the initial points and boundary points with limitations, but also with the physical information of other time slices, transmitting more effective physical information to a longer time range and deeper space.

[0050] Third, low-cost and fast training of the network is achieved. The extended physical information neural network proves that it is feasible to parallelize partial differential equations in space and time. After time dimension reduction and space partitioning, the high-speed field and low-speed field of different time slices obtained by mBL can be calculated in parallel by multiple servers, and then merged into the entire velocity potential field according to the spatial coordinates. After the above processing, compared with serial calculation, parallel calculation based on time-space domain decomposition significantly improves the training speed and solution accuracy, and solves the problem of too many network parameters caused by too much input data, reducing the cost of training. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings required for use in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other accompanying drawings can be obtained based on these accompanying drawings without paying creative work.

[0052] Figure 1 is a flow chart of the method of the present invention;

[0053] Figure 2 It is a structural diagram of the device of the present invention;

[0054] Figure 3 It is a working schematic diagram of the space-time division module of the overall velocity potential field of the present invention;

[0055] Figure 4 This is a working diagram of the multi-time slice data reasonable selection module of the present invention;

[0056] Figure 5 It is a schematic diagram of the operation of the high-speed and low-speed field parallel processing modules of the present invention;

[0057] Figure 6 A network structure diagram of a parallel physical information neural network of the present invention;

[0058] Figure 7 It is a work flow chart of the present invention;

[0059] Figure 8 The velocity potential field of some time slices of an embodiment of the present invention is given as an example, wherein Figure (a) is the velocity potential field at t=0s, Figure (b) is the velocity potential field at t=1500s, Figure (c) is the velocity potential field at t=3000s, Figure (d) is the velocity potential field at t=4500s, Figure (e) is the velocity potential field at t=6000s, and Figure (f) is the velocity potential field at t=7500s. DETAILED DESCRIPTION

[0060] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0061] Before elaborating on the overall architecture of the parallel physical information neural network of the present invention, it is necessary to clearly introduce the BL equation, the KP equation and the mBL equation.

[0062] (1) BL equation and KP equation

[0063] The Benney-Luke equation (BL equation) is used to describe the three-dimensional weak nonlinear small amplitude long water wave problem, as shown in the following equation:

[0064] (1);

[0065] in, is the velocity potential, express The first-order partial derivative of represents the second-order derivative of ξ with respect to time t, represents the first-order derivative of ξ with respect to the spatial variable, represents the second-order derivative of ξ with respect to the spatial variable, represents the fourth-order derivative of ξ with respect to the spatial variable, represents the product of the second-order derivative of ξ with respect to time t and the second-order derivative with respect to the spatial variable, represents the product of the first-order derivative of ξ with respect to time t and the first-order derivative with respect to the spatial variable, and is a positive coefficient, so , is the Bond number, which measures the effect of surface tension and is a formally valid approximation for describing the two modes of water wave propagation in the presence of surface tension.

[0066] The Kadomtsev-Petviashvili equation (KP equation) is used to describe the evolution of nonlinear small-amplitude long waves that slowly depend on the transverse coordinates, as shown in the following equation:

[0067] (2).

[0068] (2) mBL equation

[0069] In the context of three-dimensional ocean internal waves, considering the influence of terrain, Yuan Chunxin et al. proposed a modified Benney-Luke equation (mBL equation) to describe the internal wave-wave interaction on the inclined seabed, as shown in the following equation:

[0070] (3);

[0071] in, is the velocity potential, is the wave speed, , , , b is the coefficient, Velocity potential The first-order partial derivative of represents the second-order derivative of the velocity potential ξ with respect to time t, represents the Laplace operator, Represents the gradient operator.

[0072] , , , The specific expression is shown as follows:

[0073] (4);

[0074] (5);

[0075] (6);

[0076] (7);

[0077] In the above four equations, h + and h - are the water depth of the upper fluid layer and the water depth of the lower fluid layer respectively, R is the ratio of the density of the upper fluid layer to the density of the lower fluid layer, and g is the gravitational acceleration.

[0078] The mBL equation does not need to involve fourth-order partial derivatives like the BL equation, and its numerical calculation results are in good agreement with those of the BL equation. The mBL equation also has the characteristics of isotropy and bidirectional propagation, which are not available in the widely used KP equation.

[0079] Without considering the terrain, the mBL equation has an analytical solution for the interior solitary wave, which can be explicitly expressed as:

[0080] (8);

[0081] (9);

[0082] In the above two equations, is a vector in any direction on the horizontal plane, is the corresponding wave number vector, whose magnitude is , the amplitude A(k) and nonlinear wave velocity v(k) are:

[0083] (10);

[0084] (11).

[0085] Example 1

[0086] Combination Figure 1-Figure 4 ,The method based on parallel physical information neural network for solving mBL equations implemented in this paper improves the three aspects of the optimization solution of high-dimensional non-convex equations, the in-depth transmission of effective physical information, and the low cost and fast training of the network through the three modules of the overall velocity potential field time-space division, the rational selection of multi-time slice data, and the parallel calculation of high-speed and low-speed fields.

[0087] In the overall velocity potential field partition module ( Figure 3 ), firstly, the overall velocity potential field with three-dimensional space-time data is reduced in time dimension to obtain the velocity potential fields of different time slices; then, the velocity potential fields of different time slices are divided into spaces to obtain the high-speed fields (the raised ladder in the middle marked with high) and the low-speed fields (the smooth bottom around marked with low) of different time slices.

[0088] Reasonable selection of modules in multiple time slice data ( Figure 4 ), first divide the high-speed field (low-speed field) data of different time slices into the data of the time slice to be solved and the data of the previous time slice; then reasonably select the data of these two parts of data, focus on selecting the data of the time slice to be solved, and associate and select the data of the previous time slice to obtain the reasonably selected data of the high-speed field (low-speed field).

[0089] Parallel processing modules in high-speed and low-speed fields ( Figure 5 ), firstly, the reasonably selected data of the high-speed field and the reasonably selected data of the low-speed field are input into the parallel physical information neural network of different servers respectively for sub-domain parallel calculation to obtain the solved high-speed field and the solved low-speed field; then, the solved sub-domains of these two parts are spatially merged to obtain the solved velocity potential field.

[0090] Each step is described in detail below.

[0091] Step 1: Spatiotemporal division of the overall velocity potential field.

[0092] First, the overall velocity potential field with three-dimensional space-time data is reduced in time dimension to obtain the velocity potential field of different time slices; then the velocity potential field of different time slices is divided into spaces to obtain the high-speed field of different time slices ( Figure 3 The middle raised ladder in the middle) and the low-speed field at different time slices ( Figure 3 smooth bottom all around).

[0093] The specific steps are as follows:

[0094] Step 1.1, normalize the mBL equation;

[0095] The mBL equation has different velocity potential fields under different equation coefficients, initial conditions and boundary conditions. The more equation coefficients there are, the more complex the initial conditions and boundary conditions are, and the more irregular the shape of the velocity potential field of the mBL equation is. For the velocity potential field of the mBL equation, space division is performed according to a regular shape, which can reduce complex high-dimensional non-convexity. Therefore, in order to obtain a regular velocity potential field shape, the present invention sets the equation coefficients, initial conditions and boundary conditions of the mBL equation so that the velocity potential field shape of the mBL equation is regular.

[0096] The mBL equation after setting is as follows:

[0097] (12);

[0098] Setting the equation coefficient b in equation (3) to 0, we get equation (12); for the mBL equation represented by equation (12), the initial point is the space point at t = 0, the boundary point is ξ = 0 and =0 point in space.

[0099] Step 1.2, time dimension reduction;

[0100] For the mBL equation, the velocity potential field in different time slices may have different shapes. Therefore, before the velocity potential field is spatially divided, time dimension reduction is required to divide the overall velocity potential field according to the time slices to obtain the velocity potential fields in different time slices. The velocity potential fields in different time slices have different shapes.

[0101] Take the mBL equation with x∈[-1,1], y∈[-2,2], t∈[0,7500] as an example to perform time dimension reduction. Among them, x contains 512 points, y contains 108 points, and t contains 51 time slices. After time dimension reduction, the overall velocity potential field is divided into 51 velocity potential fields, corresponding to 51 time slices. The velocity potential field of some time slices is as follows Figure 8 shown.

[0102] exist Figure 8 In the figure, t=0s is the velocity potential field of the 1st time slice, t=1500s is the velocity potential field of the 11th time slice, t=3000s is the velocity potential field of the 21st time slice, t=4500s is the velocity potential field of the 31st time slice, t=6000s is the velocity potential field of the 41st time slice, and t=7500s is the velocity potential field of the 51st time slice.

[0103] Step 1.3: Space division;

[0104] Depend on Figure 8 It can be seen that for the mBL equation, except for the initial field at t=0 which is a gradient-free uniform field, the velocity potential fields of other time slices have a regular shape: high in the center and low at the edge. Therefore, according to this regular shape, the velocity potential fields of different time slices are spatially divided to obtain high-speed fields and low-speed fields of different time slices; in this embodiment, 51 groups of high-speed fields and low-speed fields can be obtained for 51 time slices.

[0105] Specifically, spatial division is based on the velocity potential threshold to divide the velocity potential field of a certain time slice into a high velocity potential field and a low velocity potential field, that is, a high velocity field and a low velocity field.

[0106] Take the velocity potential field of the mBL equation at t=7500s (the 51st time slice) as an example for spatial division. At this time, the velocity potential value of some spatial points has a large range of variation and is all greater than the velocity potential threshold. These spatial points are all distributed in the middle of the velocity potential field; the velocity potential values ​​of other spatial points have almost no change and are all less than the velocity potential threshold. These points are all distributed around the velocity potential field. Therefore, the velocity potential field at this time can be divided into a high velocity potential field (referred to as "high-speed field") and a low velocity potential field (referred to as "low-speed field") based on the velocity potential threshold.

[0107] The velocity potential threshold has the following characteristics: it is smaller than the value of all points in the high-speed field and larger than the value of all points in the low-speed field. Because there is no spatial point whose velocity potential value is equal to the velocity potential threshold, there are no boundary points and boundary domains when dividing the velocity potential field based on the velocity potential threshold, and there is no sub-neural network for the boundary domain.

[0108] Step 2: Reasonable selection of multi-time slice data;

[0109] For the overall velocity potential field of the mBL equation, it is difficult to obtain a high-precision solution by directly randomly selecting a part of the data from all time slices and inputting it into the physical information neural network to output the predicted velocity potential values ​​of all time slices. This is because if the overall velocity potential field is not divided, its high-dimensional non-convex nature cannot be reduced, and the interaction of physical information in different time slices increases the complexity and difficulty of training. Therefore, in addition to dividing the overall velocity potential field, the physical information of different time slices must be reasonably allocated. By reasonably selecting data from different time slices, the physical information of different time slices can be reasonably allocated.

[0110] The velocity potential value of the mBL equation at a certain time slice depends only on the physical information of the current time slice and the previous time slice, and mainly on the physical information of the current time slice. Therefore, for the velocity potential field of a certain time slice, the physical information of the current time slice should be emphasized during training and the physical information of the previous time slice should be associated. In other words, the data input to the network for training should include the data of the current time slice and the previous time slice, and mainly include the data of the current time slice.

[0111] Therefore, in step 2, the high-speed field or low-speed field data of different time slices are first divided into the data of the time slice to be solved and the data of the previous time slice; then the two parts of data are reasonably selected to obtain the reasonably selected high-speed field data and low-speed field data; wherein, the reasonable selection method is: select the data of the previous time slice associated with the data of the time slice to be solved, and make the number of selected data of the previous time slice less than the data of the time slice to be solved. That is, focus on selecting a large number of data of the time slice to be solved, and select a small number of data of the previous time slice in association, so as to obtain the reasonably selected high-speed field data and low-speed field data.

[0112] After the space-time division of the overall velocity potential field in step 1, the initial point and boundary point of the mBL equation are all located in the initial field, and the velocity potential values ​​of all spatial points in the initial field are zero, and the overall velocity potential field of the mBL equation is divided into high-speed fields and low-speed fields of different time slices. In this embodiment, the overall velocity potential field of the mBL equation is divided into 51 groups (corresponding to 51 time slices) of high-speed fields and low-speed fields.

[0113] Step 2 reasonably selects data of different time slices and reasonably allocates physical information of different time slices; for high-speed fields of different time slices, the data input into the parallel physical information neural network for training includes two parts, as shown in the following formula:

[0114] (13);

[0115] in, Represents all the data input into the network for training in the current time slice. Represents the data selected from the current time slice, Represents the data selected from the previous time slice; the selected data includes the spatial point and the corresponding true velocity potential value; The amount of data is greater than The amount of data, Contains the data of the initial field, and the initial points and boundary points are located within the initial field.

[0116] Step 3, high-speed field and low-speed field are calculated in parallel;

[0117] First, the reasonably selected data of the high-speed field and the reasonably selected data of the low-speed field are respectively input into the parallel physical information neural network of different servers for sub-domain parallel calculation to obtain the solved high-speed field and the solved low-speed field; then the solved sub-domains of these two parts are spatially merged to obtain the solved velocity potential field, which is used to solve the mBL equation.

[0118] After step 1 and step 2, the overall velocity potential field of the mBL equation is divided into high-speed fields and low-speed fields of different time slices (divided into 51 groups of high-speed fields and low-speed fields in this embodiment). When solving the high-speed field or low-speed field of a certain time slice in step 3, the data input to the network includes not only the data of the current time slice, but also the data of the previous time slice. Compared with using a single server to calculate the overall velocity potential field serially, using multiple servers to calculate the high-speed field and low-speed field of different time slices in parallel can speed up the calculation. Therefore, different servers are used to calculate the high-speed field or low-speed field of a certain time slice in parallel, and then they are merged into a complete velocity potential field according to the spatial coordinates, as shown in the following formula:

[0119] (14);

[0120] (15);

[0121] (16);

[0122] Among them, x and y represent spatial coordinates, and t represents time coordinates. represents the velocity potential value of the current velocity potential field being solved, represents the velocity potential value of the current high-speed field being solved, Indicates the velocity potential value of the current low-velocity field being solved; Represents the neural network currently used in the high-speed field, Indicates the current high-speed field and the previous high-speed field; Indicates the neural network used in the current low-speed field, Indicates the current low-speed field and the previous low-speed field, including the initial field, and the initial point and boundary point are located in the initial field; Depend on and Take the union to get; and Having the same network structure, the two constitute the parallel physical information neural network, such as Figure 6 shown.

[0123] The parallel physical information neural network inputs the spatial coordinates x, y and time t, and outputs the velocity potential value ξ and the solution f; specifically:

[0124] Figure 6 In the network, first, it determines whether x and y belong to the high-speed field. If yes, it is trained with the deep neural network 1 in server 1. If not, it is trained with the deep neural network 2 in server 2. The structures of the two deep neural networks are exactly the same, except for the input and output. The spatial coordinates and time are outputted as velocity potential values ​​after passing through the deep neural network. The velocity potential values ​​are converted into multiple first-order partial derivatives and second-order partial derivatives after automatic differentiation. The multiple partial derivatives are substituted into equation (2) to obtain the solution f of the equation.

[0125] (17);

[0126] (18);

[0127] (19);

[0128] (20);

[0129] Among them, ξ is the velocity potential value solved by the neural network, is the velocity potential value solved by pseudo-spectral method, is the first-order partial derivative of the velocity potential with respect to time solved by the neural network, is the first-order partial derivative of the velocity potential with respect to time obtained by the pseudo-spectral method, N is the number of input spatial points, including the initial point and the boundary point; Zhong ξ is ξ high ,exist Zhong ξ is ξ low ; It is also responsible for processing the data of the initial field, while the initial point and the boundary point are both located within the initial field, and the loss calculation of the initial point and the boundary point also follows equations (17) to (20).

[0130] For the mBL equation, MSE is the overall loss function, which is composed of the loss function MSE_ξ of the velocity potential value and the loss function MSE_ξ of the first-order time partial derivative. tand the loss function MSE_f of physical information. If the parallel physical information neural network based on time-space domain decomposition to solve the mBL equation can well solve the mBL equation, then the velocity potential value of MSE_ξ for each point will approach zero, and MSE_ξ t The first-order time partial derivative of each point approaches zero, and the physical equation value of MSE_f approaches zero for each point. In other words, when MSE approaches zero, it can be considered that the predicted value of each point in the training data set approaches the true value. In this way, solving the mBL equation is transformed into optimizing the loss function, which is optimized using the back propagation mechanism of the neural network and the two optimizers, L-BFGS and Adam.

[0131] Figure 6 The working process of the parallel physical information neural network is shown in Figure 7 As shown, in Figure 7 In the model, the workflow is divided into eight steps:

[0132] In the first step, some spatial points at different time slices, as well as the true velocity potential and the first-order time partial derivatives of the true velocity potential corresponding to these spatial points are taken as input.

[0133] The second step is to determine whether these spatial points belong to the high-speed field. If so, the parallel physical information neural network of the high-speed field is used; if not, the parallel physical information neural network of the low-speed field is used.

[0134] The third step is to set the constant parameters of the mBL equation and set the initial number of iterations k=0.

[0135] The fourth step is to output the training speed potential through the deep neural network, and calculate MSE_ξ through the real speed potential and the training speed potential.

[0136] The fifth step is to output multiple partial derivatives of the training velocity potential through automatic derivation, and calculate MSE_ξ through the first-order time partial derivative of the true velocity potential and the first-order time partial derivative of the training velocity potential t .

[0137] Step 6: Calculate MSE_f by training multiple partial derivatives of velocity potential, and calculate MSE=MSE_ξ+MSE_ξ t +MSE_f.

[0138] Step 7: Determine whether the number of iterations has been reached. If so, input all the spatial points of the time slice to be solved. If not, the number of iterations k=k+1.

[0139] In the eighth step, the high-speed field and the low-speed field are merged according to the position parameters of the spatial point, and the trained parallel physical information neural network and the merged velocity potential field are output.

[0140] Example 2

[0141] As another embodiment of the present invention, a device for solving mBL equations based on a parallel physical information neural network is provided. Figure 2 As shown, it includes: a module for time-space division of the overall velocity potential field, a module for reasonable selection of multi-time slice data, and a module for parallel calculation of high-speed and low-speed fields.

[0142] like Figure 3 As shown in FIG, the overall velocity potential field time-space division module is used to perform time dimension reduction and space division on the overall velocity potential field of the input three-dimensional time-space data to obtain high-speed fields at different time slices and low-speed fields at different time slices. Figure 3 In the figure, the "velocity potential field at different time slices" includes the velocity potential field at t=0s to the velocity potential field at t=7500s, the time interval is 90s, the x-axis range is -1 to 1, the y-axis range is -2 to 2, and the ξ-axis range is -1 to 2. The "high-speed field at different time slices" includes the high-speed field at t=0s to the high-speed field at t=7500s, and the time interval and the range of the three axes are the same as above. The "low-speed field at different time slices" includes the low-speed field at t=0s to the low-speed field at t=7500s, and the time interval and the range of the three axes are the same as above.

[0143] like Figure 4 As shown, the multi-time slice data reasonable selection module is used to reasonably select data for high-speed fields of different time slices and low-speed fields of different time slices, and obtain reasonably selected data for high-speed fields or low-speed fields;

[0144] The high-speed and low-speed field parallel calculation modules. Figure 4 In the data, "high-speed field (low-speed field) of different time slices" includes the high-speed field (low-speed field) at t=0s to the high-speed field (low-speed field) at t=7500s, the time interval is 90s, the x-axis range is -1 to 1, the y-axis range is -2 to 2, and the ξ-axis range is -1 to 2. "Data of previous time slices" includes the high-speed field (low-speed field) data at t=0s to the high-speed field (low-speed field) data at t=7350s, and the time interval and the range of the three axes are the same as above. "Data of time slices to be solved" includes the high-speed field (low-speed field) data at t=7500s, and the range of the three axes are the same as above. "Reasonably selected data of high-speed field (low-speed field)" includes a large amount of high-speed field (low-speed field) data at t=7500s, and a small amount of high-speed field (low-speed field) data from t=0s to t=7350s, and the time interval and the range of the three axes are the same as above.

[0145] like Figure 5As shown, the high-speed field and low-speed field parallel calculation modules are used to input the reasonably selected data of the high-speed field and the low-speed field into the parallel physical information neural network of different servers for sub-domain parallel calculation to obtain the solved high-speed field and the solved low-speed field; then the solved sub-domains of these two parts are spatially merged to obtain the solved velocity potential field, which is used to solve the mBL equation. Figure 5 In the figure, "reasonably selected data of high-speed field" includes a large amount of high-speed field data at t=7500s, and a small amount of high-speed field data from t=0s to t=7350s. The time interval is 90s, the x-axis range is -1 to 1, the y-axis range is -2 to 2, and the ξ-axis range is -1 to 2. "Reasonably selected data of low-speed field" includes a large amount of low-speed field data at t=7500s, and a small amount of low-speed field data from t=0s to t=7350s. The time interval and the range of the three axes are the same as those mentioned above. "The solved high-speed field" is the solved high-speed field at t=7500s, and the range of the three axes are the same as those mentioned above. "The solved low-speed field" is the solved low-speed field at t=7500s, and the range of the three axes are the same as those mentioned above. "The solved high-speed field" solves the velocity potential field at t=7500s, and the range of the three axes is the same as those mentioned above.

[0146] In practical applications, the present invention may be used to study problems in the fields of ocean waves, river floods, and electromagnetic wave propagation.

[0147] Example 3

[0148] This embodiment provides an application of a method for solving mBL equations based on a parallel physical information neural network for simulating a flow field. Specifically, by using the method described in the previous embodiment 1, the velocity potential field is solved and a fluid flow field simulation visualization image is output, wherein the visualization method can be implemented using visualization software of the prior art, which is not a design point of the present invention and will not be described in detail here.

[0149] This embodiment introduces a specific use method / application of the present invention, which models the propagation of water waves and accurately describes the characteristics of water wave motion in the context of three-dimensional ocean internal waves. The method of the present invention can better capture the nonlinear behavior and interaction effects of waves.

[0150] Example 4

[0151] This embodiment provides an application of a method for solving mBL equations based on a parallel physical information neural network, which is used to predict flow velocity, and then analyze and predict fluid behavior. Specifically, by using the method described above, a velocity potential field is solved, and the velocity of the fluid at any point in the flow field can be determined based on the velocity potential field. Therefore, it can be used to predict the flow velocity of the fluid under specific conditions, and then predict the fluid behavior. In summary, the present invention (1) designs a time-space division module for the overall velocity potential field. The module first divides the overall velocity potential field into velocity potential fields of different time slices based on the time axis, and then divides the velocity potential fields of different time slices into high-speed fields and low-speed fields based on regular shapes. The module performs time dimensionality reduction and space division on the overall velocity potential field, and divides multiple groups of velocity potential fields under different time and space conditions, thereby reducing the complex high-dimensional non-convexity of the overall velocity potential field.

[0152] (2) A module for rationally selecting data from multiple time slices was designed. This module selects data from different time slices to start training by rationally allocating physical information: on the basis of associating the data from the previous time slice, it focuses on selecting the data from the time slice to be solved, and combines the two selected parts of data to start training. This module rationally selects data from the velocity potential field of different time slices to start training, reducing the impact of the difficulty in transmitting physical information at the initial point and boundary point. This module rationally selects data from the velocity potential field of different time slices to start training, reducing the impact of the difficulty in transmitting physical information at the initial point and boundary point.

[0153] (3) A high-speed and low-speed field parallel computing module was designed. This module uses multiple servers to accelerate the solution of high-speed and low-speed fields, and then merges the high-speed and low-speed fields into the overall velocity potential field according to the spatial coordinates. This module accelerates the solution and merging of high-speed and low-speed fields based on velocity potential field division and data optimization, reducing the network training cost and time.

[0154] (4) A parallel physical information neural network based on time-space domain decomposition was designed. The network inputs space coordinates and time coordinates, and outputs velocity potential and function values. Multiple partial derivatives can be obtained by automatically differentiating the velocity potential, and the function value can be obtained by calculating multiple partial derivatives. The loss function consists of three parts: the loss of velocity potential, the loss of the first-order time partial derivative, and the loss of the function value.

[0155] Of course, the above description is not a limitation of the present invention, and the present invention is not limited to the above examples. Any changes, modifications, additions or substitutions made by ordinary technicians in this technical field within the essential scope of the present invention should fall within the protection scope of the present invention.

Claims

1. A method for solving mBL equations based on a parallel physical information neural network, characterized in that: The following steps are involved: Step 1: Spatiotemporal division of the overall velocity potential field; First, the overall velocity potential field with three-dimensional space-time data is reduced in time dimension to obtain the velocity potential field of different time slices; then the velocity potential field of different time slices is spatially divided to obtain the high-speed field of different time slices and the low-speed field of different time slices; The specific steps of step 1 are as follows: Step 1.1, normalize the mBL equation; The mBL equation is shown below: Among them, ξ is the velocity potential, c is the wave speed, α, β, γ, b are coefficients, ξ t represents the first-order partial derivative of the velocity potential ξ, ξ tt represents the second-order derivative of the velocity potential ξ with respect to time t, Δ represents the Laplace operator, represents the gradient operator; The equation coefficients, initial conditions and boundary conditions of the mBL equation are set so that the velocity potential field of the mBL equation has a regular shape; the mBL equation after setting is shown as follows: The equation coefficient b in equation (3) is set to 0, and equation (12) is obtained; for the mBL equation represented by equation (12), the initial point is the space point at t = 0, the boundary point is ξ = 0 and ξ t = 0, that is, there is no velocity change on the boundary; Step 1.2, time dimension reduction; For the mBL equation, the overall velocity potential field is divided according to time slices to obtain velocity potential fields of different time slices, and the velocity potential fields of different time slices have different shapes; Step 1.3: Space division; For the mBL equation, except for the initial field at t=0 which is a gradient-free uniform field, the velocity potential fields of other time slices have a regular shape: high in the center and low at the edge; therefore, the velocity potential fields of different time slices are spatially divided according to this regular shape to obtain high-speed fields and low-speed fields of different time slices; specifically, the spatial division is based on the velocity potential threshold to divide the velocity potential field of a certain time slice into a high-speed potential field and a low-speed potential field, that is, a high-speed field and a low-speed field; Step 2: Reasonable selection of multi-time slice data; First, the high-speed field or low-speed field data of different time slices are divided into the data of the time slice to be solved and the data of the previous time slice; then the two parts of data are reasonably selected to obtain the reasonably selected high-speed field data and low-speed field data; wherein the reasonable selection method is: selecting the data of the previous time slice associated with the data of the time slice to be solved, and making the number of the selected data of the previous time slice less than the data of the time slice to be solved; Step 3, high-speed field and low-speed field are calculated in parallel; First, the reasonably selected data of the high-speed field and the reasonably selected data of the low-speed field are respectively input into the parallel physical information neural network of different servers for sub-domain parallel calculation to obtain the solved high-speed field and the solved low-speed field; then the solved sub-domains of these two parts are spatially merged to obtain the solved velocity potential field, which is used to solve the mBL equation.

2. The method for solving mBL equations based on parallel physical information neural network according to claim 1, characterized in that: The velocity potential threshold has the following characteristics: it is smaller than the value of all points in the high-speed field and larger than the value of all points in the low-speed field.

3. The method for solving mBL equations based on parallel physical information neural network according to claim 1, characterized in that: After the time-space division of the overall velocity potential field in step 1, the initial point and boundary point of the mBL equation are all located in the initial field, and the velocity potential values ​​of all spatial points in the initial field are zero, and the overall velocity potential field of the mBL equation is divided into high-speed fields and low-speed fields of different time slices; Step 2 reasonably selects data of different time slices and reasonably allocates physical information of different time slices; for high-speed fields of different time slices, the data input into the parallel physical information neural network for training includes two parts, as shown in the following formula: date=date l +date l- (13); Among them, data represents all the data input to the network for training in the current time slice, data l Indicates the data selected from the current time slice, data l- Represents the data selected from the previous time slice; the selected data includes the spatial point and the corresponding true velocity potential value; data l The amount of data is greater than data l- The amount of data, data l- Contains the data of the initial field, and the initial points and boundary points are located within the initial field.

4. The method for solving mBL equations based on parallel physical information neural network according to claim 1, characterized in that: After steps 1 and 2, the overall velocity potential field of the mBL equation is divided into high-speed fields and low-speed fields of different time slices. When solving the high-speed field or low-speed field of a certain time slice in step 3, the data input to the network contains not only the data of the current time slice, but also the data of the previous time slice. Therefore, different servers are used to calculate the high-speed field or low-speed field of a certain time slice in parallel, and then they are merged into a complete velocity potential field according to the spatial coordinates, as shown in the following formula: in=in high ∪u low (14); u high =[θ high (x,y,t)|(x,y)∈HFs] (15); u low =[θ low (x,y,t)|(x,y)∈LFs] (16); Among them, x and y represent spatial coordinates, t represents time, u represents the velocity potential value of the current velocity potential field to be solved, and u high Indicates the velocity potential value of the current high-speed field being solved, u low Represents the velocity potential value of the current low-speed field being solved; θ high represents the deep neural network used in the current high-speed field, HFs represents the current high-speed field and the previous high-speed field; θ low represents the deep neural network used in the current low-speed field, LFs represents the current low-speed field and the previous low-speed field, including the initial field, and the initial point and boundary point are located in the initial field; u is composed of u high and u Low Take the union and get; θ high and θ low Having the same network structure, the two constitute the parallel physical information neural network.

5. The method for solving mBL equations based on parallel physical information neural network according to claim 4, characterized in that: The parallel physical information neural network inputs the spatial coordinates x, y and time t, and outputs the velocity potential value ξ and the solution f; the parallel physical information neural network includes two deep neural networks, specifically: The network first determines whether x and y belong to the high-speed field. If yes, it is trained with the deep neural network 1 in server 1. If not, it is trained with the deep neural network 2 in server 2. The structures of the two deep neural networks are exactly the same, except for the input and output. The spatial coordinates and time are outputted as velocity potential values ​​after passing through the deep neural network. The velocity potential values ​​are automatically differentiated into multiple first-order partial derivatives and second-order partial derivatives. The multiple partial derivatives are substituted into equation (12) to obtain the solution f of the equation. MSE=MSE_ξ+MSE_ξ t +MSE_f (17); Among them, ξ is the velocity potential value solved by the neural network, ξ p is the velocity potential value solved by pseudospectral method, ξ t is the first-order partial derivative of the velocity potential with respect to time solved by the neural network, is the first-order partial derivative of the velocity potential with respect to time obtained by the pseudo-spectral method, N is the number of input spatial points, including the initial point and the boundary point; in θ high Zhong ξ is ξ high , in θ low Zhong ξ is ξ low θ low It is also responsible for processing the data of the initial field, and the initial point and the boundary point are both located in the initial field, and the loss calculation of the initial point and the boundary point also follows equations (17) to (20); For the mBL equation, MSE is the overall loss function, which is composed of the loss function MSE_ξ of the velocity potential value and the loss function MSE_ξ of the first-order time partial derivative. t and the loss function MSE_f of physical information added together; when MSE approaches zero, it is considered that the predicted value of each point in the training data set is close to the true value. In this way, solving the mBL equation is transformed into optimizing the loss function, which is optimized using the back propagation mechanism of the neural network and the two optimizers, L-BFGS and Adam.

6. A device for solving mBL equations based on parallel physical information neural network, characterized in that: include: The overall velocity potential field time-space division module, the multi-time slice data reasonable selection module, and the high-speed and low-speed field parallel calculation module are used to perform time dimension reduction and space division on the overall velocity potential field of the input three-dimensional time-space data to obtain high-speed fields of different time slices and low-speed fields of different time slices; The specific steps of time-space division of the overall velocity potential field are as follows: Step 1.1, normalize the mBL equation; The mBL equation is shown below: Among them, ξ is the velocity potential, c is the wave speed, α, β, γ, b are coefficients, ξ t represents the first-order partial derivative of the velocity potential ξ, ξ tt represents the second-order derivative of the velocity potential ξ with respect to time t, Δ represents the Laplace operator, represents the gradient operator; The equation coefficients, initial conditions and boundary conditions of the mBL equation are set so that the velocity potential field of the mBL equation has a regular shape; the mBL equation after setting is shown as follows: The equation coefficient b in equation (3) is set to 0, and equation (12) is obtained; for the mBL equation represented by equation (12), the initial point is the space point at t = 0, the boundary point is ξ = 0 and ξ t = 0, that is, there is no velocity change on the boundary; Step 1.2, time dimension reduction; For the mBL equation, the overall velocity potential field is divided according to time slices to obtain velocity potential fields of different time slices, and the velocity potential fields of different time slices have different shapes; Step 1.3: Space division; For the mBL equation, except for the initial field at t=0 which is a gradient-free uniform field, the velocity potential fields of other time slices have a regular shape: high in the center and low at the edge; therefore, the velocity potential fields of different time slices are spatially divided according to this regular shape to obtain high-speed fields and low-speed fields of different time slices; specifically, the spatial division is based on the velocity potential threshold to divide the velocity potential field of a certain time slice into a high-speed potential field and a low-speed potential field, that is, a high-speed field and a low-speed field; The multi-time slice data reasonable selection module is used to reasonably select data from high-speed fields of different time slices and low-speed fields of different time slices to obtain reasonably selected data from high-speed fields or low-speed fields; The high-speed and low-speed field parallel calculation module is used to input the reasonably selected data of the high-speed field and the low-speed field into the parallel physical information neural network of different servers for sub-domain parallel calculation to obtain the solved high-speed field and the solved low-speed field; then the solved sub-domains of these two parts are spatially merged to obtain the solved velocity potential field for solving the mBL equation.

Images