CN116151152A - Hydrologic numerical simulation calculation method based on gridless calculation - Google Patents
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Abstract
The invention relates to the field of hydrologic numerical simulation calculation. A hydrological numerical simulation calculation method based on gridless calculation comprises the following steps: (1) Establishing a conceptual model of an underground water flow simulation project and a dam break flood wave evolution simulation project, determining boundary conditions and initial conditions, and collecting sampling data of a simulated time-space domain; (2) Determining mathematical physical control equations of underground water flow simulation engineering and dam break flood wave evolution simulation engineering, and giving substitution functions meeting initial conditions or boundary conditions; (3) constructing a double-layer residual cooperative neural network model; (4) Constructing a distance function meeting the initial conditions and boundary conditions of underground water flow simulation engineering and dam break flood wave evolution simulation engineering, embedding the distance function into a double-layer residual collaborative neural network model for integration, and (5) carrying out numerical simulation prediction on water level simulation of underground water flow and dam break flood wave evolution. The invention can improve the accuracy of hydrodynamic numerical simulation.
Description
Technical Field
The invention relates to the field of hydrologic numerical simulation calculation, in particular to a hydrologic numerical simulation calculation method based on gridless calculation.
Background
With the rapid development of modern industrial society, human activities have had unprecedented impact on water resources. At the same time, the influence of climate change on the water resource amount and distribution thereof is increasingly remarkable, especially on the hydrodynamic field and water circulation conditions, and a series of water resource problems such as water resource shortage, water ecological deterioration, urban flood and the like can be induced. Therefore, how to accurately describe the basic state of the water power field and the water flow motion law under the influence of complex conditions is the key to solve the problems of water resources, and is also a focus and difficulty of current students.
Partial differential equations are mathematical representations containing unknown physical quantities and their partial derivatives, which are one of the important mathematical tools describing objective physical world laws, also called mathematical physical control equations. However, for most partial differential equations, the general solution is often difficult to express explicitly, i.e. the result cannot be represented by a first order function. In particular, for the solution of higher order partial differential equations, it is more difficult to determine the function using a definite solution condition. Therefore, solving the partial differential equation in a numerical calculation manner becomes the first choice for exploring the physical law in various disciplines. Currently, numerical computation is mainly based on finite difference method and finite element method. The solution ideas of the two methods are basically the same, and the partial differential equation is solved in a discrete mode, but the discrete mode and the applicability are different. For example, the finite difference method is to approximate differentiation using a difference, and the finite element method is approximated using an interpolation function; the finite difference method is mainly applicable to the problem of regular areas, and the finite element method can calculate the problem of nonlinear complex areas. In popular terms, the finite difference method is to solve discrete iterations of equations, which aims at the original problem, but the obtained result is an approximate solution. The finite element method is to limit the problem to one subdomain, and aims at the approximate problem, but the accurate solution on the subdomain is obtained. Although these conventional solutions are widely used in the current engineering problems of solids, fluids and structural analysis, certain limitations remain, and these disadvantages are mostly derived from the grid. Firstly, in numerical simulation, not all grids can be constructed to be proper interpolation functions, when the grids are concave, grid singular is easy to appear, and solving is limited; second, the construction of the interpolation function needs to take into account the continuity between the grids, which becomes exceptionally difficult once the model requires C1 continuity (i.e., the first derivative of the function is continuous); furthermore, the equation solving method based on the grid can only obtain the numerical value on the divided grid or the node, and the topological structure of the divided grid can be changed by re-dividing the grid, so that the calculation cost is increased; furthermore, discrete processing of partial differential equations is also a difficult task, especially for high-dimensional non-homogeneous partial differential equations.
Hydrodynamic force simulation is an effective technical approach for researching the basic state of a natural hydrodynamic field and the motion rule of water flow. Currently, research in this regard is largely multi-modal, employing grid-based discrete methods, in addition to a small number of data-driven black-box simulations. The black box model learns historical data by constructing a specific network model, summarizes historical data rules and gives feedback. The model can obtain values in any time-space domain, and the characteristics of partial differential equations are not considered, so that the model cannot be used for carrying out fine solution on the global values, and a large amount of historical data is needed. In contrast, the grid-discrete-based method completely satisfies the partial differential equation and can acquire values in the time-space domain, but is limited by the grid itself, and the acquired value solutions are not continuous. As global climate is changed and human activities are exacerbated, hydrologic laws become abnormal, so how to construct a count value model that satisfies both physical laws and enables grid-free computation is a problem to be solved in the hydrodynamic simulation field.
Disclosure of Invention
The invention overcomes the defects of the technical problems, provides a hydrographic numerical simulation calculation method based on gridless calculation, is suitable for hydrographic numerical simulation under large-scale complex conditions, can solve high-order non-homogeneous partial differential equations describing different hydrographic working conditions, and improves the accuracy of hydrographic numerical simulation.
In order to solve the technical problems, the invention adopts the following technical scheme:
a hydrological numerical simulation calculation method based on gridless calculation comprises the following steps:
(1) Establishing a conceptual model of an underground water flow simulation project and a dam-break flood wave evolution simulation project, determining boundary conditions and initial conditions of the underground water flow simulation project and the dam-break flood wave evolution simulation project, and uniformly sampling an analog time-space domain to obtain sampling data of the analog time-space domain;
(2) Determining mathematical physical control equations of underground water flow simulation engineering and dam break flood wave evolution simulation engineering, and giving substitution functions of underground water exploitation and dam break flood wave simulation meeting initial conditions and boundary conditions according to simulation working conditions;
(3) On the basis of the fully-connected neural network, a residual calculation channel is constructed, and a double-layer residual cooperative neural network model is constructed by adding the residual calculation channel so as to enhance the fitting capacity and convergence of the fully-connected neural network;
(4) Constructing a distance function meeting the initial conditions and boundary conditions of underground water flow simulation engineering and dam break flood wave evolution simulation engineering on the basis of the second step, embedding the distance function into the double-layer residual collaborative neural network model in the third step for integration, obtaining an integrated double-layer residual collaborative neural network model, and forming a new network output;
(5) And inputting the sampled data into the integrated double-layer residual cooperative neural network model, constructing a loss function term with an equation trend of 0 by utilizing an automatic differentiation technology according to an equation form of a mathematical physical control equation, and carrying out water level simulation of underground water flow and numerical simulation prediction of dam break flood wave evolution.
As a further improvement of the present invention, in the step (2), substitution functions of groundwater exploitation and dam break flood wave simulation are as follows:
wherein: the formula (1) represents an underground water exploitation substitution function, and the formula (2) represents a two-dimensional circular dam break and rectangular dam break substitution function; q represents the single well production capacity; sigma represents a smoothing parameter; x is x 0 、y 0 Representing the position of an underground water exploitation point or a dam break position; A. b, dam break initial static water level regulating coefficient; x and y represent the coordinates of the simulated time-space domain.
As a further improvement of the present invention, in the step (3), the calculation process of the dual-layer residual cooperative neural network may be expressed as:
X res L-1 ＝σ res L-1 (W res L-1 X res L-2 +b res L-1 +X 1 ) (3)
X L-1 ＝σ L-1 (W L-1 (X L-2 +X res L-2 )+b L-1 ) (4)
Y＝W L X L-1 +b L (5)
wherein: x is X res The input of the residual network layer; sigma (sigma) res Representing an activation function of the residual network layer; w (W) res And b res Respectively representing the weight and bias of a residual network layer; x represents the input of a fully connected neural network; l represents the hidden layer number of the network of the fully connected nerve; sigma represents the activation function of the fully connected neural network; w and b represent the weight and bias of the fully connected neural network, respectively; y represents the output of the neural network.
As a further improvement of the present invention, in the step (4), a distance function satisfying the initial conditions and boundary conditions of the groundwater flow simulation project and the dam-break flood wave evolution simulation project is constructed, and is embedded into a double-layer residual cooperative neural network model to form a new network output, which is expressed by the following specific formulas:
d(X)＝(x-x bc )·(y-y bc )·t (6)
Y[h]＝f(X)+d(X)Net(X)/N s (7)
wherein: d (X) is a distance function; net (X) is the output of the neural network in step three; y [ h ]]New network output after embedding initial conditions and boundary conditions; n (N) s For scaling adjustment coefficients, for balancing magnitudes between data; x and y represent the coordinates of the simulated time-space domain; x is x bc 、y bc Representing the boundary position.
Compared with the prior art, the invention has the following beneficial effects:
1. the hydrodynamic force numerical simulation method based on gridless calculation is suitable for hydrodynamic force numerical simulation under large-scale complex conditions, can solve high-order non-homogeneous partial differential equations describing different hydrologic conditions, and improves accuracy of hydrodynamic force numerical simulation. The invention is based on automatic differentiation technology and neural network, constructs the partial differential equation to be a term tending to 0 as the loss function of the neural network, and solves the partial differential equation by utilizing the characteristic that the neural network can fit any function. Firstly, uniformly sampling a simulated time-space domain to be solved, and determining model input data; secondly, reconstructing a partial differential equation by utilizing an automatic differential technology, and establishing a substitution function of groundwater exploitation and dam-break flood waves so as to adapt to model calculation; and then integrating the initial condition, the boundary condition and the original equation (namely the double-layer residual cooperative neural network) by utilizing a distance function, constructing a new approximation equation (namely the integrated double-layer residual cooperative neural network), and establishing an unsupervised learning network model based on the conservation characteristic of the partial differential equation so as to solve the partial differential equation. And finally, inputting the sampling data into a double-layer residual cooperative neural network model, and performing water level simulation of underground water flow and numerical simulation prediction of dam break flood wave evolution.
2. The method is based on a gridless calculation mode, can effectively overcome the defects caused by meshing of the traditional numerical simulation method, has higher resolution theoretically, and can lead the result to be infinitely close to an analytic solution.
3. The computing core of the invention is a double-layer residual cooperative neural network, has more convergence than a fully-connected neural network (common neural network), and can effectively solve the problem of gradient explosion or gradient disappearance of a unidirectional deep network.
4. The method has good applicability, can meet the complex hydrodynamic force simulation in various large solving fields, and has simple operation, clear regulations and high calculation precision.
Drawings
FIG. 1 is a flow chart of a hydrodynamic force numerical simulation method based on gridless calculation of the present invention;
FIG. 2 shows that the one-dimensional inter-canal time-varying rainfall replenishment intensity is 0.1×e -0.05t A water level simulation diagram under a working condition, wherein a is an RCPINN simulation result; b is the simulation result of the finite element method; c is comparison of simulation results at specific moments; d is the RCPINN model loss dip curve.
FIG. 3 shows that the one-dimensional inter-canal time-varying rainfall replenishment intensity is 0.001×e 0.05t A water level simulation diagram under a working condition, wherein a is a simulation result of an RCPINN model; b is the simulation result of the finite element method; c is the comparison of the simulation results at key moments; d is the RCPINN model loss dip curve.
FIG. 4 is a two-dimensional simulation diagram of the groundwater twin-well production water level between channels, wherein a is the simulation result of continuous production of the RCPINN model for 20 days; b is simulation results of finite element method mining for 20 days; c is the comparison of simulation results of two models when mining for 5 days; d is the comparison of simulation results of two models when mining for 10 days; e is the comparison of simulation results of two models when mining for 15 days; f is the model loss dip curve.
Fig. 5 is a one-dimensional instantaneous dam-break flood wave evolution simulation diagram, wherein a is a simulation result of an RCPINN model on a water head h; b is the simulation result of the finite difference method on the water head h; c is the simulation result of the RCPINN model on the speed v; d is the simulation result of the finite difference method on the velocity v; e is the comparison of simulation results of the representative moments of the two models; f is the RCPINN model loss dip curve.
FIG. 6 is a two-dimensional instantaneous circular dam-break flood wave evolution simulation diagram, wherein a is a simulation result of an RCPINN model on a water head h; b is the simulation result of the finite difference method on the water head h; c is a simulation result of the RCPINN model on the transverse speed u; d is the simulation result of the finite difference method on the transverse speed u; e is the simulation result of the RCPINN model on the radial velocity v; f is the simulation result of the finite difference method on the radial velocity v; g is the comparison of simulation results of the water head h at the moment represented by the two models; h is the RCPINN model loss dip curve.
Detailed Description
The invention will be further described with reference to the drawings and examples. It should be noted that the specific embodiments of the present invention are only for describing the technical solution more clearly, and should not be taken as limiting the scope of the present invention.
Referring to fig. 1-6, the invention discloses a hydrodynamic force numerical simulation method based on gridless calculation, which adopts a neural network to solve partial differential equations, and comprises five main steps of determining a ground water flow simulation engineering conceptual model, constructing initial conditions and boundary conditions, determining a mathematical physical control equation, constructing an initial condition and boundary condition substitution function, constructing a double-layer residual cooperative neural network, embedding constraint of the mathematical physical control equation and the control condition and simulating and predicting. Taking two-dimensional groundwater flow simulation as an example, the specific implementation of the invention is carried out according to the following steps:
as shown in fig. 1, a hydrographic numerical simulation calculation method based on gridless calculation includes the following steps:
(1) Establishing a conceptual model of an underground water flow simulation project and a dam-break flood wave evolution simulation project, determining boundary conditions and initial conditions of the underground water flow simulation project and the dam-break flood wave evolution simulation project, and uniformly sampling an analog time-space domain to obtain sampling data of the analog time-space domain; the boundary condition is the condition of the boundary of the seepage area, and is used for representing the condition that the water head h should meet on the boundary of the seepage area, namely the relationship between the water flow in the seepage area and the surrounding environment. The initial condition is the distribution of the water head h in the seepage area at the initial moment of a certain selected point. The initial conditions and boundary conditions for groundwater flow simulation can be expressed as:
h(x,y,0)＝h 0 (x,y)(8)
h(x Di ,y Di ,t)＝h Di (x,y,t)(9)
wherein: h is a 0 The water head distribution at the initial moment is shown; h is a Di Representing dirichlet boundary conditions; g Ne Representing a Newman boundary condition;
a unit normal vector representing the direction in which the Newman boundary conditions act; x is x Di 、y Di 、x Ne 、y Ne Respectively representing coordinate points acted by the Dirichlet boundary condition and the Newman boundary condition;
(2) Determining mathematical physical control equations of underground water flow simulation engineering and dam break flood wave evolution simulation engineering through conceptual equations, and giving substitution functions of underground water exploitation and dam break flood wave simulation meeting initial conditions and boundary conditions according to simulation working conditions, wherein the mathematical physical control equations can be expressed as:
wherein:
representing a gradient operator; k represents a hydraulic conductivity; m represents the thickness of the aquifer; h (x, y, t) represents a head value; w (x, y, t) represents a sourceCollecting items; s is S s Indicating the water storage rate of the aqueous medium; x and y represent directions; t represents time;
the substitution functions of groundwater exploitation and dam-break flood wave are established, and the substitution functions of groundwater exploitation and dam-break flood wave simulation are respectively as follows:
wherein: the formula (1) represents an underground water exploitation substitution function; because dam break flood wave simulation is a large class and is divided into two-dimensional circular dam break and rectangular dam break, the substitution functions of the dam break flood wave simulation are described in the formula (2), namely the formula (2) represents the substitution functions of the two-dimensional circular dam break and the rectangular dam break; q represents the single well production capacity; sigma represents a smoothing parameter; x is x 0 、y 0 Representing the position of an underground water exploitation point or a dam break position; A. b, dam break initial static water level regulating coefficient; x and y represent the coordinates of the simulated time-space domain.
The mathematical physical control equation of the underground water flow simulation engineering and the dam break flood wave evolution simulation engineering is a formula accepted by the academic community, but the boundary conditions and the initial conditions are determined according to the simulation working conditions, and the purpose of the step (2) is to provide substitution functions of the initial conditions and the boundary conditions according to the simulation working conditions.
(3) On the basis of the fully-connected neural network, a residual calculation channel is constructed, a double-layer residual cooperative neural network model is constructed by adding the residual calculation channel, and each layer of operation information of the fully-connected neural network is transmitted to the next layer in a jumping manner by the residual calculation channel, so that information transmission loss during operation of the fully-connected neural network is reduced, and fitting capacity and convergence of the fully-connected neural network are improved; the double-layer residual collaborative neural network calculation process can be expressed as:
X res L-1 ＝σ res L-1 (W res L-1 X res L-2 +b res L-1 +X 1 ) (3)
X L-1 ＝σ L-1 (W L-1 (X L-2 +X res L-2 )+b L-1 ) (4)
Y＝W L X L-1 +b L (5)
wherein: x is X res The input of the residual network layer; sigma (sigma) res Representing an activation function of the residual network layer; w (W) res And b res Respectively representing the weight and bias of a residual network layer; x represents the input of a fully connected neural network; l represents the hidden layer number of the network of the fully connected nerve; sigma represents the activation function of the fully connected neural network; w and b represent the weight and bias of the fully connected neural network, respectively; y represents the output of the double-layer residual cooperative neural network;
(4) Constructing a distance function meeting the initial conditions and boundary conditions of underground water flow simulation engineering and dam break flood wave evolution simulation engineering on the basis of the second step, embedding the distance function into the double-layer residual collaborative neural network model in the third step for integration, obtaining an integrated double-layer residual collaborative neural network model, and forming a new network output; is expressed by the following specific formula:
d(X)＝(x-x bc )·(y-y bc )·t (6)
Y[h]＝f(X)+d(X)Net(X)/N s (7)
wherein: d (X) is a distance function; net (X) is the output of the neural network in step three; y [ h ]]New network output after embedding initial conditions and boundary conditions; n (N) s For scaling adjustment coefficients, for balancing magnitudes between data; x and y represent the coordinates of the simulated time-space domain; x is x bc 、y bc Representing the boundary position.
(5) And inputting the sampled data into the integrated double-layer residual cooperative neural network model, constructing a loss function term with an equation trend of 0 by utilizing an automatic differentiation technology according to an equation form of a mathematical physical control equation, and carrying out water level simulation of underground water flow and numerical simulation prediction of dam break flood wave evolution. And the actual measured underground water flow process is utilized to simulate and obtain the underground water exploitation water level and dam break flood wave range, the simulation result and the actual measured data are compared, and if the errors of the simulation result and the actual measured data meet the set requirements, the input parameters of the model are reasonably selected, and the model precision is good. The numerical simulation prediction includes: and inputting the sampling data of the time-space domain into a model, solving to obtain the groundwater exploitation water level and dam break flood wave range and the dynamic change process thereof in the calculation region, and realizing the visualization of the simulation result in the Python environment.
In order to verify the applicability of the method, the invention respectively changes the rainfall replenishment intensity between one-dimensional channels to 0.1×e -0.05t Water level simulation under working conditions, one-dimensional inter-canal time-varying rainfall replenishment intensity of 0.001×e 0.05 the water level simulation under the t working condition, the two-dimensional canal underground water double-well exploitation water level simulation, the one-dimensional instantaneous dam-break flood wave evolution simulation and the two-dimensional instantaneous circular dam-break flood wave evolution simulation are analyzed as typical cases, and compared with the traditional numerical simulation method (finite element method and finite difference method), and the results are respectively shown in fig. 2, 3, 4, 5 and 6.
Case 3 simulates the one-dimensional instantaneous dam break flood wave evolution. As shown in fig. 5, it is assumed that a dam is provided in the middle of a 1500m long river, the upstream static water level of the dam is 20m, the downstream water level is 15m, and the dam suddenly breaks at a certain moment. The comparative model uses the WENO format finite difference scheme.
Case 4 simulates two-dimensional instantaneous dam break flood wave evolution. As shown in fig. 6, it is assumed that a circular reservoir is provided in the middle of a rectangular area of 50×50m, the radius of the reservoir is 11m, the static water level in the dam body is 10m, the external water level is 1m, and the dam body suddenly breaks at a certain moment. The comparative model uses the WENO format finite difference scheme.
In conclusion, the method can have higher resolution, the result can be infinitely close to an analytic solution, and complex hydrodynamic force simulation under various large solving domains can be satisfied. The invention has high precision and high efficiency, and is a new choice for simulating the groundwater flowing area in a large range and high precision for a long time.
To verify the rationality of the simulation result and to quantify the simulation accuracy, the root mean square error RMSE and the determination coefficient R are used respectively 2 As simulation error and fitting degree evaluation criteria, the smaller the RMSE is, the smaller the model error is, R is 2 The closer to 1, the higher the fitting degree of the model is, and the calculation formulas of the model and the model are expressed as follows:
wherein: n represents the number of sampling coordinate points; y [ h ]]Representing an RCPINN analog output value; y is Y obs Representing a comparison numerical model output; y is Y obs，mean Representing the mean of the outputs of the comparison model.
The method is based on a gridless calculation mode, does not need to preprocess equation terms, and is simple to operate. The method has high precision and applicability in the process of groundwater flow simulation and dam break flood wave evolution simulation, the overall fitting precision reaches more than 0.99, and the method can provide a new thought and guiding direction for hydrodynamic force simulation and has good application prospect.
The foregoing description is directed to the preferred embodiments of the present invention, but the embodiments are not intended to limit the scope of the invention, and all equivalent changes or modifications made under the technical spirit of the present invention should be construed to fall within the scope of the present invention.
Claims (4)
1. The hydrological numerical simulation calculation method based on gridless calculation is characterized by comprising the following steps of:
(1) Establishing a conceptual model of an underground water flow simulation project and a dam-break flood wave evolution simulation project, determining boundary conditions and initial conditions of the underground water flow simulation project and the dam-break flood wave evolution simulation project, and uniformly sampling an analog time-space domain to obtain sampling data of the analog time-space domain;
(2) Determining mathematical physical control equations of underground water flow simulation engineering and dam break flood wave evolution simulation engineering, and giving substitution functions of underground water exploitation and dam break flood wave simulation meeting initial conditions and boundary conditions according to simulation working conditions;
(3) On the basis of the fully-connected neural network, a residual calculation channel is constructed, and a double-layer residual cooperative neural network model is constructed by adding the residual calculation channel so as to enhance the fitting capacity and convergence of the fully-connected neural network;
(4) Constructing a distance function meeting the initial conditions and boundary conditions of underground water flow simulation engineering and dam break flood wave evolution simulation engineering on the basis of the second step, embedding the distance function into the double-layer residual collaborative neural network model in the third step for integration, obtaining an integrated double-layer residual collaborative neural network model, and forming a new network output;
(5) And inputting the sampled data into the integrated double-layer residual cooperative neural network model, constructing a loss function term with an equation trend of 0 by utilizing an automatic differentiation technology according to an equation form of a mathematical physical control equation, and carrying out water level simulation of underground water flow and numerical simulation prediction of dam break flood wave evolution.
2. The method of calculation of a hydrographic numerical simulation based on gridless calculation according to claim 1, wherein in the step (2), substitution functions of groundwater exploitation and dam break flood wave simulation are as follows:
wherein: the formula (1) represents an underground water exploitation substitution function, and the formula (2) represents a two-dimensional circular dam break and rectangular dam break substitution function; q represents the single well production capacity; sigma represents a smoothing parameter; x is x 0 、y 0 Representing the position of an underground water exploitation point or a dam break position; A. b, dam break initial static water level regulating coefficient; x and y represent the coordinates of the simulated time-space domain.
3. The method of grid-less calculation-based hydrologic numerical simulation calculation according to claim 1, wherein in the step (3), the double-layer residual collaborative neural network calculation process can be expressed as:
X res L-1 ＝σ res L-1 (W res L-1 X res L-2 +b res L-1 +X 1 ) (3)
X L-1 ＝σ L-1 (W L-1 (X L-2 +X res L-2 )+b L-1 ) (4)
Y＝W L X L-1 +b L (5)
wherein: x is X res Input for residual network layer；σ res Representing an activation function of the residual network layer; w (W) res And b res Respectively representing the weight and bias of a residual network layer; x represents the input of a fully connected neural network; l represents the hidden layer number of the network of the fully connected nerve; sigma represents the activation function of the fully connected neural network; w and b represent the weight and bias of the fully connected neural network, respectively; y represents the output of the neural network.
4. The method of calculation of the hydrographic numerical simulation based on gridless calculation according to claim 1, wherein in the step (4), a distance function satisfying initial conditions and boundary conditions of the groundwater flow simulation project and the dam-break flood wave evolution simulation project is constructed, and is embedded into a double-layer residual collaborative neural network model to form a new network output, and the new network output is represented by the following specific formula:
d(X)＝(x-x bc )·(y-y bc )·t (6)
Y[h]＝f(X)+d(X)Net(X)/N s (7)
wherein: d (X) is a distance function; net (X) is the output of the neural network in step three; y [ h ]]New network output after embedding initial conditions and boundary conditions; n (N) s For scaling adjustment coefficients, for balancing magnitudes between data; x and y represent the coordinates of the simulated time-space domain; x is x bc 、y bc Representing the boundary position.
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