Physical information neural network-based flow rate prediction model training system, method, device and medium
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- TSINGHUA UNIVERSITY
- Filing Date
- 2026-04-02
- Publication Date
- 2026-06-19
Smart Images
Figure CN122242261A_ABST
Abstract
Description
Technical Field
[0001] This application relates to the fields of scientific computing and machine learning technology, and in particular to a training system, method, device and medium for a flow velocity prediction model based on a physical information neural network. Background Technology
[0002] Physics-Informed Neural Networks (PINNs) are widely used in scientific computing fields such as fluid mechanics, materials analysis, and electromagnetic field simulation by embedding the physical constraints described by partial differential equations into the training process of neural networks in the form of residuals. Compared with traditional numerical methods, PINNs do not require explicit construction of discrete meshes and have the potential advantage of handling high-dimensional problems and inverse problems.
[0003] In the PINNs framework, the activation function, as the fundamental operator of the neural network, directly influences the network's ability to express continuous functions and its gradient propagation characteristics. Different types of partial differential equations typically correspond to solutions with different structural features: wave-like equations usually exhibit periodic oscillations, diffusion-like equations show exponential decay, and reaction-diffusion systems may form local interfaces or spike structures. Due to the significant differences in the mathematical structure of these physical characteristics, it is difficult to simultaneously meet the requirements for characterizing multiple physical properties using activation functions with fixed forms.
[0004] Currently, the selection of activation functions often relies on human experience or adopts globally shared fixed combinations. On the one hand, activation function weights are usually randomly initialized or set based on experience, lacking structural analysis of the physical categories of partial differential equations, making it difficult to form an optimization direction that matches the physical characteristics of the equations in the early stages of training. On the other hand, activation function weights are shared throughout the entire spatiotemporal domain, making it impossible to differentiate them according to different physical behaviors in local regions. When the solution object includes locally rapidly changing regions such as boundary layers and shock waves, the globally uniform operator form limits the network's ability to finely characterize complex physical fields, leading to problems such as slow convergence and unstable optimization. Summary of the Invention
[0005] The purpose of this application is to provide a training system, method, device, and medium for a flow velocity prediction model based on a physical information neural network. This aims to solve the problems that physical information neural networks face in the process of modeling physical fields such as flow velocity prediction. These problems include the lack of structural awareness of the physical categories of partial differential equations in the activation function, which leads to uncertainty in the optimization direction in the early stage of training, and the difficulty in differentiated modeling of local complex physical regions due to the global sharing of the combined weights of the activation function throughout the entire spatiotemporal domain.
[0006] To solve the above-mentioned technical problems, this application is implemented as follows: A first aspect of this application discloses a flow velocity prediction model training system based on a physical information neural network, comprising: The physical category identification module identifies the physical category of the partial differential equations related to flow velocity, wherein the partial differential equations related to flow velocity characterize the physical prior knowledge of flow velocity changes; The combination coefficient initialization module generates the initial combination coefficients of each activation function in the activation function set based on the recognition results from the physical category recognition module. The activation function configuration module obtains the target activation function based on the initial combination coefficients from the combination coefficient initialization module, and constructs the physical information neural network to be trained according to the target activation function. The training module aims to learn the physical prior knowledge of flow velocity changes represented by the partial differential equations related to flow velocity, and trains the physical information neural network to be trained. Based on the predicted flow velocity output by the physical information neural network to be trained for spatiotemporal sample points and the partial differential equations, the physical residual loss is determined. Based on the physical residual loss, the network weights of the physical information neural network to be trained are updated. The trained physical information neural network is used to predict flow velocity for target spatiotemporal points.
[0007] Optionally, the physics category identification module identifies the physics category of the partial differential equations related to flow velocity, including: The operator structure of the partial differential equations related to flow velocity is analyzed. When the partial differential equation related to the flow velocity contains higher-order diffusion terms, the physical category of the partial differential equation is determined to be diffusion-dominant. When the partial differential equation related to flow rate contains strongly nonlinear reaction terms, the physical category of the partial differential equation is determined to be reaction-dominant. When the boundary conditions of the partial differential equation related to flow velocity are periodic boundary conditions, or when the oscillation source term of the partial differential equation related to flow velocity is present, it is identified as an oscillation-dominant type. If the partial differential equation related to the flow velocity contains a first-order convection term, the physical category of the partial differential equation is determined to be convection-dominated.
[0008] Optionally, the activation function set includes at least: exponential activation function, sine activation function, hyperbolic tangent activation function, Softplus activation function, and Swish activation function; the combination coefficient initialization module generates initial combination coefficients for each activation function in the activation function set based on the recognition results from the physical category recognition module, including: When the identification result from the physical category identification module is a diffusion-dominant type, the initial combination coefficient generated for the exponential activation function is higher than the initial combination coefficient of any other activation function in the activation function set. When the identification result from the physical category identification module is a reaction-dominant type, the initial combination coefficient generated for the hyperbolic tangent activation function is higher than the initial combination coefficient of any other activation function in the activation function set; If the identification result from the physical category identification module is an oscillation-dominant type, or if the identification result from the physical category identification module is a convection-dominant type, the initial combination coefficient generated for the exponential activation function is higher than the initial combination coefficient of any other activation function in the activation function set.
[0009] Optionally, it also includes: The time-space control parameter configuration module, through the weight mapping network to be trained, configures the spatiotemporal sample points. The time-limited control parameter mapped to the initial combination coefficients of the i-th activation function in the set of activation functions. ; For spatiotemporal sample points The target activation function is determined according to the following formula. : , Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The target activation function is applied through the physical information network to be trained. For spatiotemporal sample points Output predicted flow rate; The training module updates the network weights of the physical information neural network to be trained and the weight mapping network to be trained, based on the physical residual loss. The trained physical information neural network and the trained weight mapping network are used to predict flow velocity for the target spatiotemporal point.
[0010] Optionally, the system further includes: The flow velocity prediction module, through the trained weight mapping network, determines the target spatiotemporal point. The time-limited control parameter mapped to the initial combination coefficients of each activation function in the activation function set. ; For the target spatiotemporal point The target activation function is determined according to the following formula. : , Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The trained physical information network is then used with the target activation function. Targeting the spatiotemporal point Output the predicted flow rate.
[0011] A second aspect of this application discloses a method for training a flow velocity prediction model based on a physical information neural network, the method comprising: Identify the physical categories of partial differential equations related to flow velocity, which characterize the physical prior knowledge of flow velocity changes; Based on the recognition results, generate the initial combination coefficients of each activation function in the activation function set; Based on the initial combination coefficients, the target activation function is obtained, and the physical information neural network to be trained is constructed according to the target activation function. With the goal of learning the physical prior knowledge of flow velocity changes represented by the partial differential equations related to flow velocity, the physical information neural network to be trained is trained; based on the predicted flow velocity output by the physical information neural network to be trained for spatiotemporal sample points and the partial differential equations, the physical residual loss is determined; based on the physical residual loss, the network weights of the physical information neural network to be trained are updated; the trained physical information neural network is used to predict flow velocity for target spatiotemporal points.
[0012] Optionally, the method further includes: The spatiotemporal sample points are mapped using a weight mapping network to be trained. The time-limited control parameter mapped to the initial combination coefficients of the i-th activation function in the set of activation functions. ; For spatiotemporal sample points The target activation function is determined according to the following formula. : , Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The target activation function is applied through the physical information network to be trained. For spatiotemporal sample points Output predicted flow rate; Based on the physical residual loss, in addition to updating the network weights of the physical information neural network to be trained, the network weights of the weight mapping network to be trained are also updated. The trained physical information neural network and the trained weight mapping network are used to predict the flow velocity for the target spatiotemporal point.
[0013] Optionally, the method further includes: The trained weight mapping network is used to map the target spatiotemporal point. The time-limited control parameter mapped to the initial combination coefficients of each activation function in the activation function set. ; For the target spatiotemporal point The target activation function is determined according to the following formula. : , Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The trained physical information network is then used with the target activation function. Targeting the spatiotemporal point Output the predicted flow rate.
[0014] A third aspect of this application discloses an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the flow rate prediction model training method based on physical information neural network described in the first aspect of this application.
[0015] A fourth aspect of this application discloses a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the steps of the flow velocity prediction model training method based on a physical information neural network as described in the first aspect of this application.
[0016] A fifth aspect of this application discloses a computer program product, including a computer program that, when executed by a processor, implements the steps of the flow velocity prediction model training method based on a physical information neural network as described in the first aspect of this application.
[0017] The embodiments of this application have the following advantages: By using a physical category identification module to perform structural analysis on partial differential equations related to flow velocity and identify their physical categories, the construction of activation functions can be tailored to the physical characteristics of the equations themselves. This avoids the blindness of traditional methods where activation function selection relies on human experience or random initialization, and provides an optimized starting point for network training that matches the physical structure.
[0018] The combination coefficient initialization module generates the initial combination coefficients of each activation function in the activation function set based on the recognition results. This enables the target activation function to have an expression tendency that is compatible with the physical behavior dominated by partial differential equations in the initial training stage, reducing the uncertainty of the optimization direction caused by random initialization and improving the convergence efficiency in the early training stage.
[0019] The activation function configuration module constructs the target activation function according to the initial combination coefficients and builds the physical information neural network to be trained. This enables the neural network to acquire the representation ability that is compatible with the physical mechanism dominated by the equation in the initial training stage, thereby improving the fitting accuracy of flow velocity fields with different physical characteristics. Combined with the training module, the network weights are updated with physical residual loss as the guide, so that the entire training process is carried out under the guidance of physical prior knowledge. This enhances the network's ability to fit the flow velocity change law and improves the accuracy and stability of the trained physical information neural network in predicting flow velocity at the target spatiotemporal point.
[0020] Thus, while maintaining the basic training framework of the physical information neural network, this system achieves the automatic construction of physical perception activation functions in a modular way, without introducing additional discrete optimization steps. It features simple implementation, easy integration, and wide applicability. Attached Figure Description
[0021] To more clearly illustrate the technical solutions of the embodiments of this application, the drawings used in the description of the embodiments of this application will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0022] Figure 1 This is a schematic diagram of a flow velocity prediction model training system based on a physical information neural network provided in an embodiment of this application; Figure 2 This is a schematic diagram illustrating the generation process of a target activation function provided in an embodiment of this application; Figure 3 This is a flowchart illustrating the steps of a flow velocity prediction model training method based on a physical information neural network, as provided in an embodiment of this application. Figure 4This is a schematic diagram of the structure of an electronic device provided in an embodiment of this application. Detailed Implementation
[0023] To make the above-mentioned objectives, features, and advantages of this application more apparent and understandable, the technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.
[0024] This application addresses the problems in physical information neural networks for modeling physical fields such as flow velocity prediction, where the activation function lacks the ability to perceive physical structure, leading to uncertainty in the optimization direction during the initial training phase, and the difficulty in differentiated modeling of complex local regions due to the global sharing of weights in the activation function combination. It proposes a flow velocity prediction model training system, method, device, and medium based on physical information neural networks.
[0025] The core inventive concept lies in: using a physical category identification module to perform structural analysis on partial differential equations related to flow velocity, identifying their dominant physical type, and generating initial combination coefficients for each activation function in the activation function set. This ensures that the target activation function possesses an expressive tendency that matches the physical characteristics of the equation from the initial training stage. Based on this, the training module updates the network weights using physical residual loss as a guide, ensuring the entire training process is guided by prior physical knowledge. This scheme realizes a shift in activation function construction from empirical selection to physical structure awareness, improving the convergence efficiency and prediction accuracy of physical information neural networks in flow velocity prediction tasks.
[0026] The following description, in conjunction with the accompanying drawings, details the training system, method, device, and medium for flow velocity prediction models based on physical information neural networks provided in this application, through specific embodiments and application scenarios.
[0027] Reference Figure 1 As shown, Figure 1 This is a schematic diagram of a flow velocity prediction model training system based on a physical information neural network, provided in an embodiment of this application. Figure 1 As shown, the system may include: The physical category identification module identifies the physical category of the partial differential equations related to flow velocity, which characterize the physical prior knowledge of flow velocity changes.
[0028] The combination coefficient initialization module generates the initial combination coefficients of each activation function in the activation function set based on the recognition results from the physical category recognition module. The activation function configuration module obtains the target activation function based on the initial combination coefficients from the combination coefficient initialization module, and constructs the physical information neural network to be trained according to the target activation function. The training module aims to learn the physical prior knowledge of flow velocity changes represented by the partial differential equations related to flow velocity, and trains the physical information neural network to be trained. Based on the predicted flow velocity output by the physical information neural network to be trained for spatiotemporal sample points and the partial differential equations, the physical residual loss is determined. Based on the physical residual loss, the network weights of the physical information neural network to be trained are updated. The trained physical information neural network is used to predict flow velocity for target spatiotemporal points.
[0029] In this embodiment, the physics category identification module is used to identify the physics category of partial differential equations related to flow velocity. For example, the Navier-Stokes equations and convection-diffusion equations can be used to characterize the physical prior knowledge that governs changes in flow velocity. In real fluid systems, flow fields dominated by different physical mechanisms have different mathematical characteristics: diffusion-dominated flow fields typically exhibit a smooth decay trend, oscillation-dominated flow fields show periodic fluctuations, and reaction-dominated flow fields may exhibit local abrupt changes or peak structures. This module identifies the dominant physical characteristics by analyzing the operator structure of the partial differential equations, providing a basis for the adaptive construction of subsequent activation functions.
[0030] The combination coefficient initialization module receives the recognition results output by the physical category recognition module and generates initial combination coefficients for each activation function in the activation function set based on these results. The activation function set contains multiple candidate functions with different mathematical properties; for example, some functions excel at characterizing smooth changes, some at capturing high-frequency oscillations, and some are suitable for representing step or abrupt changes. By setting differentiated initial combination coefficients according to the physical category, the activation functions possess an expressive tendency that matches the physical behavior dominated by partial differential equations from the initial training stage, thus avoiding the uncertainty in optimization direction caused by random initialization in traditional methods.
[0031] The activation function configuration module constructs a target activation function based on the initial combination coefficients provided by the combination coefficient initialization module, and then builds the physical information neural network to be trained according to this target activation function. Specifically, the target activation function can be represented as a weighted combination of multiple candidate activation functions, where the initial combination coefficients determine the weight of each candidate function in the combination. The physical information neural network constructed in this way acquires a representational capability adapted to the physical characteristics of the partial differential equation to be solved before training begins.
[0032] The training module aims to learn the prior physical knowledge represented by partial differential equations and trains the physical information neural network to be trained. During training, the module acquires the coordinates of sample points in the spatiotemporal domain (including internal residual points, boundary points, and initial points) and calculates the predicted flow velocity corresponding to each spatiotemporal sample point using the current network. Based on this, the training module substitutes the predicted flow velocity into the partial differential equation and calculates the equation residual as the physical residual loss. This physical residual loss reflects the degree to which the network output deviates from the physical constraints. The training module aims to minimize the physical residual loss by updating the network weights of the physical information neural network to be trained using the backpropagation algorithm, enabling the network to gradually learn the physical laws described by the partial differential equations. After training, the physical information neural network can be used to predict flow velocity for any target spatiotemporal point.
[0033] This embodiment achieves a complete process from physical category identification to adaptive activation function construction and physical constraint training through the collaborative work of the four modules mentioned above. The introduction of the physical category identification module and the combination coefficient initialization module imbues the activation function construction with physical structure awareness, providing an optimized starting point that matches the dominant characteristics of partial differential equations for training. The cooperation between the activation function configuration module and the training module enables the network to continuously optimize under the guidance of prior physical knowledge, learning a mapping relationship that conforms to the flow velocity change law. Compared with traditional solutions, this embodiment avoids the blind selection of activation functions, improves the convergence efficiency in the early stages of training, and endows the trained network with higher flow velocity prediction accuracy and stability.
[0034] In one optional embodiment, the physical category identification module identifies the physical category of the partial differential equation related to the flow velocity, specifically including the following steps A1 to A5: Step A1: Analyze the operator structure of the partial differential equations related to flow velocity.
[0035] Specifically, the physics category identification module extracts the components of partial differential equations, including time derivative terms, spatial derivative terms (first, second, and higher orders), nonlinear terms, source terms, and boundary conditions. For example, the Navier-Stokes equations describing fluid flow include convection terms, diffusion terms, and pressure gradient terms; the convection-diffusion equations include first-order convection terms and second-order diffusion terms. By analyzing the structures of these operators, the module can identify the existence form and relative importance of each term in the partial differential equations, providing a basis for subsequent physics category determination.
[0036] Step A2: If the partial differential equation related to the flow velocity contains higher-order diffusion terms, determine that the physical category of the partial differential equation is diffusion-dominant.
[0037] In this context, higher-order diffusion terms typically refer to second-order or higher spatial derivative terms. In practical velocity prediction scenarios, diffusion-dominated flows usually exhibit a smooth decay of the velocity field over time and space, such as the fully developed flow of viscous fluids in a pipe. After identifying a flow as diffusion-dominated, the subsequent combination coefficient initialization module will assign higher initial weights to activation functions (such as exponential functions) that are adept at characterizing smooth decay.
[0038] Step A3: If the partial differential equation related to the flow rate contains strongly nonlinear reaction terms, determine that the physical category of the partial differential equation is reaction-dominant.
[0039] Among them, strongly nonlinear response terms usually refer to nonlinear function terms related to the unknown function u, such as... , , In the form of [variable name], and this type of term dominates the equation. In flow rate prediction scenarios, reaction-dominated types may appear in systems involving chemical reaction flows. Reaction-dominated flow fields exhibit local abrupt changes, interface formation, or spike structures, requiring activation functions (such as hyperbolic tangent functions) capable of characterizing steep changes.
[0040] Step A4: If the boundary conditions of the partial differential equation related to the flow velocity are periodic boundary conditions, or if the partial differential equation related to the flow velocity has an oscillation source term, then it is identified as an oscillation-dominant type.
[0041] Specifically, in flow velocity prediction scenarios, oscillation-dominated flow types are commonly found in pulsating flow, wave propagation, or periodically driven flow systems. Oscillation-dominated flow fields require activation functions to capture periodic fluctuations; therefore, subsequent initialization will assign higher initial weights to oscillatory activation functions such as sine functions.
[0042] Step A5: If the partial differential equation related to the flow velocity contains a first-order convection term, determine that the physical category of the partial differential equation is convection-dominated.
[0043] The first-order convection term typically manifests as In flow velocity prediction, the physical meaning of flow velocity is the transport of physical quantities along with fluid motion. Convection-dominated flow fields are often accompanied by wavefront propagation, shock waves, or boundary layers, placing special requirements on the gradient propagation characteristics of the activation function. In flow velocity prediction, convection-dominated types are widely found in engineering scenarios such as high-speed flows and atmospheric boundary layer flows.
[0044] It should be noted that the above identification rules are not mutually exclusive. In practical applications, partial differential equations may contain multiple types of operator terms simultaneously. In this case, the physics category identification module can make a comprehensive judgment based on the relative strength of each term or a pre-set priority, or identify it as a mixed dominant type, thereby generating a more refined initialization strategy.
[0045] This embodiment, through the aforementioned identification process, determines the physical category of partial differential equations, enabling the construction of activation functions to be specifically designed based on the mathematical structure of the equation itself, rather than relying on manual experience or random initialization. This identification process does not depend on additional manual annotation or external prior knowledge, and is characterized by a high degree of automation and wide applicability. Through this embodiment, the system can obtain activation function initialization tendencies that match the physical characteristics of the partial differential equation to be solved before training begins, providing accurate and reliable input for the subsequent combination coefficient initialization module, thereby improving the optimization efficiency and prediction accuracy of the entire flow velocity prediction model training system.
[0046] In one optional embodiment, the activation function set includes at least: exponential activation function, sine activation function, hyperbolic tangent activation function, Softplus activation function, and Swish activation function.
[0047] These five activation functions are complementary in their mathematical properties. Through weighted combination, the target activation function can possess multiple representational capabilities, thus better adapting to the modeling needs of complex physical fields. The activation function set in this embodiment is not limited to the above five types. In practical applications, other candidate functions can be added or replaced according to the specific characteristics of the problem.
[0048] Furthermore, the combination coefficient initialization module generates initial combination coefficients for each activation function in the activation function set based on the recognition results from the physical category recognition module, specifically including the following steps B1 to B3: Step B1: If the identification result from the physical category identification module is a diffusion-dominant type, the initial combination coefficient generated for the exponential activation function is higher than the initial combination coefficient of any other activation function in the activation function set.
[0049] Specifically, solutions to diffusion-dominated partial differential equations typically exhibit exponential decay or exponential growth characteristics. Exponential activation functions, mathematically similar to these solutions, can be maximized by setting their initial combination coefficients to the highest level. This allows the target activation function to possess an expressive tendency that matches the diffusion behavior from the initial training phase. This strategy helps the network quickly capture the smooth decay trend in the initial stage.
[0050] Step B2: If the identification result from the physical category identification module is a reaction-dominant type, the initial combination coefficient generated for the hyperbolic tangent activation function is higher than the initial combination coefficient of any other activation function in the activation function set.
[0051] Specifically, solutions to reaction-dominated partial differential equations often exhibit steep interfaces or sharp peak structures, and the hyperbolic tangent activation function can characterize such discontinuous or drastically changing physical behavior. Setting its initial combination coefficients to the maximum allows the network to possess sensitivity to local abrupt changes from the initial stage.
[0052] Step B3: If the identification result from the physical category identification module is oscillation-dominant type, or if the identification result from the physical category identification module is convection-dominant type, the initial combination coefficient generated for the exponential activation function is higher than the initial combination coefficient of any other activation function in the activation function set.
[0053] Specifically, this initialization strategy is based on the following considerations: oscillatory and convection-dominated physical fields often experience energy decay or dissipation during propagation, and exponential activation functions can effectively characterize this decay. For oscillatory-dominated fields, this strategy helps the network prioritize capturing the overall decay envelope, and then gradually enhances the weight of the oscillatory component through dynamic adjustment of the combination coefficients, reflecting a progressive learning approach from global features to local details.
[0054] This embodiment employs the aforementioned differentiated initialization strategy to establish a correlation between the combined weights of the activation function and the dominant physical properties of the partial differential equations before training begins. This provides a more reasonable starting point for subsequent training, thereby improving convergence efficiency in the early stages of training and laying the foundation for improving the final flow velocity prediction accuracy.
[0055] In an optional embodiment, the system further includes: The time-space control parameter configuration module, through the weight mapping network to be trained, configures the spatiotemporal sample points. The time-limited control parameter mapped to the initial combination coefficients of the i-th activation function in the set of activation functions. ; For spatiotemporal sample points The target activation function is determined according to the following formula. : (Formula 1) Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The target activation function is applied through the physical information network to be trained. For spatiotemporal sample points Output predicted flow rate; The training module updates the network weights of the physical information neural network to be trained and the weight mapping network to be trained, based on the physical residual loss. The trained physical information neural network and the trained weight mapping network are used to predict flow velocity for the target spatiotemporal point.
[0056] In this embodiment, considering that different spatial regions or different time stages in real fluid systems often exhibit different dominant physical behaviors, a spatiotemporal control parameter configuration module is introduced to address the difficulty of achieving local differentiated representation through global shared activation combinations. The core of this module is a weight mapping network to be trained, which maps spatiotemporal sample points... As input, the temporal control parameters of each activation function in the set of activation functions are output. The modulation parameters and the initial combination coefficients Combining these elements, the activation function is constructed at spatiotemporal sample points. The final combination coefficient at the location ,Right now:
[0057] Based on this, for spatiotemporal sample points Target activation function Defined as a weighted combination of candidate activation functions, the weights are the values of the final combination coefficients after normalization mapping, as shown in formula (1) above.
[0058] During the training process, the physical information neural network to be trained adopts the aforementioned target activation function. spatiotemporal sample points Forward propagation is performed to output the predicted flow velocity. The training module calculates the physical residual loss based on the predicted flow velocity and jointly optimizes the network parameters using the backpropagation algorithm. The trainable parameters at this point consist of three parts: the network weights of the physical information neural network, the network weights of the weight mapping network, and the initial combination coefficients of each activation function. The training module synchronously updates all the above parameters based on the physical residual loss, enabling the entire system to achieve collaborative optimization.
[0059] With this structure, the combination coefficients of the activation function are no longer globally shared constants, but dynamically change with spatial location and time. When the physical residual in a certain region is large, the weight mapping network can learn modulation parameters that enhance the expressive power of that region through backpropagation, automatically increasing the combination weights of the activation function suitable for characterizing the local complex structure. This mechanism achieves an organic combination of local error feedback and spatiotemporal adaptive operator scheduling.
[0060] Thus, this embodiment achieves spatial partitioning adaptive scheduling of activation function combination coefficients by introducing a temporal control parameter configuration module. This system can dynamically adjust the activation function expression form of different regions according to the distribution of physical residuals in the spatiotemporal domain, enabling the network to obtain stronger characterization capabilities in complex regions while maintaining efficient and stable fitting characteristics in smooth regions.
[0061] like Figure 2 As shown, Figure 2 This is a schematic diagram illustrating the generation process of a target activation function according to an embodiment of this application. Specifically, firstly, the partial differential equations related to flow velocity are analyzed by a physical category identification module to identify their dominant physical category, including diffusion-dominated, convection-dominated, oscillation-dominated, or reaction-dominated types. Based on the identification results, a combination coefficient initialization module assigns differentiated initial combination coefficients to each candidate activation function in the activation function set, enabling the target activation function to have an expression tendency that matches the dominant physical characteristics of the equation from the early stages of training. On this basis, a weight mapping network is introduced, taking spatiotemporal coordinates as input and outputting spatiotemporal control parameters for each candidate activation function. The initial combination coefficients are superimposed with the corresponding spatiotemporal control parameters to obtain the final combination coefficients that dynamically change with spatial location and time. Each candidate activation function is weighted and fused according to these dynamic combination coefficients to generate a spatiotemporally adaptive target activation function. Throughout the process, physical category identification ensures global-level physical structure perception, while the weight mapping network achieves differentiated scheduling in local regions through residual feedback. The final activation function can automatically adjust its expression form according to the physical characteristics of different spatiotemporal regions, providing a more accurate representation capability for the physical information neural network.
[0062] In an optional embodiment, the system further includes: The flow velocity prediction module, through the trained weight mapping network, determines the target spatiotemporal point. The time-limited control parameter mapped to the initial combination coefficients of each activation function in the activation function set. ; For the target spatiotemporal point The target activation function is determined according to the following formula. : (Formula 2) Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The trained physical information network is then used with the target activation function. Targeting the spatiotemporal point Output the predicted flow rate.
[0063] In this embodiment, the flow velocity prediction module is used for deployment and invocation during the practical application phase, that is, outputting the corresponding predicted flow velocity for the target spatiotemporal point (i.e., the spatial location and time point of the flow velocity to be predicted). During the application phase, the flow velocity prediction module outputs the coordinates of the target spatiotemporal point... The input is fed into the trained weight mapping network, which outputs the spatiotemporal control parameters for the target point. The modulation parameters are related to the initial combination coefficients determined and saved during the training phase. These factors combine to form the final combination coefficients of the activation function at the target spatiotemporal point. .
[0064] Based on this, the flow velocity prediction module determines the target activation function at the target spatiotemporal point according to the above formula (2). After determining the target activation function at the target spatiotemporal point, the flow velocity prediction module uses the above target activation function through the trained physical information neural network. Perform forward propagation calculations and finally output the result for the target spatiotemporal point. The predicted flow rate.
[0065] In this embodiment, since the trained weight mapping network has learned to adaptively adjust the combination coefficients of the activation function according to the spatiotemporal location, the system can use an activation function expression that matches the local physical characteristics of any target spatiotemporal point for prediction, thus ensuring the accuracy of the prediction results. The application phase only performs forward computation, ensuring the computational efficiency and real-time performance of the prediction process.
[0066] This application also provides a method for training a flow velocity prediction model based on a physical information neural network, referring to... Figure 3 As shown, Figure 3 This is a flowchart illustrating the steps of a flow velocity prediction model training method based on a physical information neural network, as provided in this application embodiment. The method includes the following steps S310 to S340: Step S310: Identify the physical category of the partial differential equations related to flow velocity, which characterize the physical prior knowledge of flow velocity changes.
[0067] In this step, the system first obtains partial differential equations related to flow velocity, such as the Navier-Stokes equations or convection-diffusion equations, which describe the physical laws governing flow velocity changes. By analyzing the operator structure of the partial differential equations, the dominant physical category is identified, including diffusion-dominated, convection-dominated, oscillation-dominated, or reaction-dominated types. This identification process provides physical guidance for the subsequent adaptive construction of activation functions.
[0068] Step S320: Based on the recognition results, generate the initial combination coefficients of each activation function in the activation function set.
[0069] In this step, after obtaining the physical category identification results, initial combination coefficients for each candidate activation function in the activation function set are generated based on the identification results. The activation function set contains multiple candidate functions with different mathematical properties, such as exponential activation functions, sine activation functions, hyperbolic tangent activation functions, etc. By setting differentiated initial combination coefficients for different physical categories, the target activation function has an expression tendency that matches the physical behavior dominated by partial differential equations in the initial training stage, thereby avoiding the problem of uncertainty in the optimization direction caused by random initialization.
[0070] Step S330: Obtain the target activation function based on the initial combination coefficients, and construct the physical information neural network to be trained according to the target activation function.
[0071] In this step, the initial combination coefficients generated in step S320 are weighted and combined with each candidate activation function to construct the target activation function. Specifically, the target activation function can be expressed as a weighted sum of multiple candidate activation functions, where the weights are the initial combination coefficients. Subsequently, a physical information neural network to be trained is constructed according to this target activation function, so that the network acquires a representation capability adapted to the physical characteristics of the partial differential equation to be solved before training begins.
[0072] Step S340: With the goal of learning the physical prior knowledge of the partial differential equation related to flow velocity that characterizes the change in flow velocity, the physical information neural network to be trained is trained; based on the predicted flow velocity output by the physical information neural network to be trained for spatiotemporal sample points and the partial differential equation, the physical residual loss is determined; based on the physical residual loss, the network weights of the physical information neural network to be trained are updated; the trained physical information neural network is used to predict the flow velocity for the target spatiotemporal point.
[0073] In this step, the system aims to learn the prior physical knowledge represented by partial differential equations by training a physical information neural network. During training, the coordinates of sample points in the spatiotemporal domain (including internal residual points, boundary points, and initial points) are acquired, and the predicted flow velocity corresponding to each spatiotemporal sample point is calculated using the current network. Subsequently, the predicted flow velocity is substituted into the partial differential equation, and the equation residual is calculated as the physical residual loss, which reflects the degree to which the network output deviates from the physical constraints. The system aims to minimize the physical residual loss by updating the network weights of the physical information neural network through the backpropagation algorithm, enabling the network to gradually learn the physical laws described by the partial differential equations. After training, the physical information neural network can be used to predict flow velocity for any target spatiotemporal point.
[0074] This method embodiment shares the same technical concept as the aforementioned system embodiment. By recognizing physical categories and constructing adaptive activation functions, it achieves a shift from experience-based selection to physical structure perception, thereby improving the convergence efficiency and prediction accuracy of physical information neural networks in flow velocity prediction tasks.
[0075] In an optional embodiment, the method further includes steps D1 to D4 to implement time-varying control of the activation function combination coefficients.
[0076] Step D1: Using the weight mapping network to be trained, the spatiotemporal sample points are... The time-limited control parameter mapped to the initial combination coefficients of the i-th activation function in the set of activation functions. .
[0077] In this step, considering that different spatial regions or time stages in real fluid systems often exhibit different dominant physical behaviors—for example, the velocity gradient in the boundary layer region changes drastically, while the distribution in the mainstream region is relatively smooth—a weight mapping network to be trained is introduced. This network maps the coordinates of spatiotemporal sample points... As input, the time-limited control parameter of the i-th activation function in the set of activation functions is output. This is used to dynamically adjust the combined weights of each candidate activation function in subsequent steps.
[0078] Step D2: For spatiotemporal sample points The target activation function is determined according to the following formula. :
[0079] Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions.
[0080] Step D3: Using the target activation function through the physical information network to be trained. For spatiotemporal sample points Output the predicted flow rate.
[0081] Step D4: Based on the physical residual loss, in addition to updating the network weights of the physical information neural network to be trained, the network weights of the weight mapping network to be trained are also updated. The trained physical information neural network and the trained weight mapping network are used to predict the flow velocity for the target spatiotemporal point.
[0082] In this step, the system calculates the physical residual loss based on the predicted flow velocity and jointly optimizes the network parameters using the backpropagation algorithm. The trainable parameters consist of three parts: the network weights of the physical information neural network, the network weights of the weight mapping network, and the initial combination coefficients of each activation function. The system synchronously updates all of these parameters based on the physical residual loss, enabling collaborative optimization throughout the training process.
[0083] Thus, by introducing a weighted mapping network, the combination coefficients of the activation functions are no longer globally shared constants, but dynamically change with spatial location and time. When the physical residual in a certain region is large, the weighted mapping network can learn modulation parameters that enhance the expressive power of that region through backpropagation, automatically increasing the combination weights of activation functions suitable for characterizing local complex structures. When the solution structure in a region is relatively smooth, the proportion of smoothing basis functions can be increased accordingly, thereby avoiding excessive oscillation or overfitting. After training, the physical information neural network and the weighted mapping network together constitute a complete flow velocity prediction model, capable of high-precision flow velocity prediction for any target spatiotemporal point.
[0084] In an optional embodiment, the method further includes steps E1 to E3 to enable the practical application of the trained model.
[0085] Step E1: Using the trained weight mapping network, the target spatiotemporal point is... The time-limited control parameter mapped to the initial combination coefficients of each activation function in the activation function set. .
[0086] In this step, the flow velocity prediction module calls the trained weight mapping network to map the coordinates of the target spatiotemporal point. The input is fed into the network, and the network outputs the time-space control parameters for the target spatiotemporal point. During the training phase, this weighted mapping network has learned how to generate appropriate spatiotemporal control parameters based on spatiotemporal coordinates.
[0087] Step E2: Targeting the spatiotemporal point The target activation function is determined according to the following formula. :
[0088] Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions.
[0089] Step E3: Using the trained physical information network, apply the target activation function. Targeting the spatiotemporal point Output the predicted flow rate.
[0090] In this step, the trained physical information neural network adopts the target activation function determined in step E2. Targeting the spatiotemporal point Forward propagation calculations are performed to ultimately output the predicted flow velocity at the target spatiotemporal point. In the application phase, the parameters of both the weight mapping network and the physical information neural network are fixed and no further backpropagation updates are performed; therefore, only forward computation is executed, without the need for gradient calculations or parameter optimization.
[0091] This embodiment achieves the deployment and invocation of the trained model in a real-world application scenario through the steps described above. Since the trained weight mapping network has learned to adaptively adjust the activation function combination coefficients based on spatiotemporal location, the system can predict any target spatiotemporal point using an activation function expression that matches the local physical characteristics of that point, ensuring the accuracy of the prediction results. Furthermore, the application phase only requires one forward propagation calculation, resulting in low computational overhead and meeting the needs of real-time prediction.
[0092] This application also provides an electronic device, see embodiments thereof. Figure 4 , Figure 4 This is a schematic diagram of the structure of an electronic device provided in an embodiment of this application. For example... Figure 4 As shown, the electronic device 400 includes a memory 410 and a processor 420. The memory 410 and the processor 420 are connected via a bus for communication. The memory 410 stores a computer program that can run on the processor 420 to implement the steps of the flow velocity prediction model training method based on physical information neural network described in the embodiments of this application.
[0093] This application also provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the steps of the flow velocity prediction model training method based on physical information neural network described in this application.
[0094] This application also provides a computer program product, including a computer program that, when executed by a processor, implements the steps of the flow velocity prediction model training method based on physical information neural networks described in this application.
[0095] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.
[0096] This application describes embodiments of methods and apparatus according to flowchart illustrations and / or block diagrams. It should be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing terminal device to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing terminal device, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.
[0097] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing terminal device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.
[0098] These computer program instructions can also be loaded onto a computer or other programmable data processing terminal equipment, causing a series of operational steps to be performed on the computer or other programmable terminal equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable terminal equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.
[0099] Although preferred embodiments of the present application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of the embodiments of the present application.
[0100] Finally, it should be noted that in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or terminal device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or terminal device. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or terminal device that includes said element.
[0101] The above provides a detailed description of the flow velocity prediction model training system, method, device, and medium based on physical information neural networks provided in this application. Specific examples have been used to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of this application. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this application. Therefore, the content of this specification should not be construed as a limitation of this application.
Claims
1. A training system for a flow velocity prediction model based on a physical information neural network, characterized in that, include: The physical category identification module identifies the physical category of the partial differential equations related to flow velocity, wherein the partial differential equations related to flow velocity characterize the physical prior knowledge of flow velocity changes; The combination coefficient initialization module generates the initial combination coefficients of each activation function in the activation function set based on the recognition results from the physical category recognition module. The activation function configuration module obtains the target activation function based on the initial combination coefficients from the combination coefficient initialization module, and constructs the physical information neural network to be trained according to the target activation function. The training module aims to learn the physical prior knowledge of flow velocity changes represented by the partial differential equations related to flow velocity, and trains the physical information neural network to be trained. Based on the predicted flow velocity output by the physical information neural network to be trained for spatiotemporal sample points and the partial differential equations, the physical residual loss is determined. Based on the physical residual loss, the network weights of the physical information neural network to be trained are updated. The trained physical information neural network is used to predict flow velocity for target spatiotemporal points.
2. The flow velocity prediction model training system based on physical information neural network according to claim 1, characterized in that, The physics category identification module identifies the physics category of partial differential equations related to flow velocity, including: The operator structure of the partial differential equations related to flow velocity is analyzed. When the partial differential equation related to the flow velocity contains higher-order diffusion terms, the physical category of the partial differential equation is determined to be diffusion-dominant. When the partial differential equation related to flow rate contains strongly nonlinear reaction terms, the physical category of the partial differential equation is determined to be reaction-dominant. When the boundary conditions of the partial differential equation related to flow velocity are periodic boundary conditions, or when the oscillation source term of the partial differential equation related to flow velocity is present, it is identified as an oscillation-dominant type. If the partial differential equation related to the flow velocity contains a first-order convection term, the physical category of the partial differential equation is determined to be convection-dominated.
3. The flow velocity prediction model training system based on physical information neural network according to claim 2, characterized in that, The activation function set includes at least: exponential activation function, sine activation function, hyperbolic tangent activation function, Softplus activation function, and Swish activation function; the combination coefficient initialization module generates initial combination coefficients for each activation function in the activation function set based on the recognition results from the physical category recognition module, including: When the identification result from the physical category identification module is a diffusion-dominant type, the initial combination coefficient generated for the exponential activation function is higher than the initial combination coefficient of any other activation function in the activation function set. When the identification result from the physical category identification module is a reaction-dominant type, the initial combination coefficient generated for the hyperbolic tangent activation function is higher than the initial combination coefficient of any other activation function in the activation function set; If the identification result from the physical category identification module is an oscillation-dominant type, or if the identification result from the physical category identification module is a convection-dominant type, the initial combination coefficient generated for the exponential activation function is higher than the initial combination coefficient of any other activation function in the activation function set.
4. The flow velocity prediction model training system based on physical information neural network according to claim 1, characterized in that, Also includes: The time-space control parameter configuration module, through the weight mapping network to be trained, configures the spatiotemporal sample points. The time-limited control parameter mapped to the initial combination coefficients of the i-th activation function in the set of activation functions. ; For spatiotemporal sample points The target activation function is determined according to the following formula. : ， Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The target activation function is applied through the physical information network to be trained. For spatiotemporal sample points Output predicted flow rate; The training module updates the network weights of the physical information neural network to be trained and the weight mapping network to be trained, based on the physical residual loss. The trained physical information neural network and the trained weight mapping network are used to predict flow velocity for the target spatiotemporal point.
5. The flow velocity prediction model training system based on physical information neural network according to claim 4, characterized in that, The system also includes: The flow velocity prediction module, through the trained weight mapping network, determines the target spatiotemporal point. The time-varying control parameter mapped to the initial combination coefficients of each activation function in the activation function set. ; For the target spatiotemporal point The target activation function is determined according to the following formula. : ， Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The trained physical information network is then used with the target activation function. Targeting the spatiotemporal point Output the predicted flow rate.
6. A method for training a flow velocity prediction model based on a physical information neural network, characterized in that, include: Identify the physical categories of partial differential equations related to flow velocity, which characterize the physical prior knowledge of flow velocity changes; Based on the recognition results, generate the initial combination coefficients of each activation function in the activation function set; Based on the initial combination coefficients, the target activation function is obtained, and the physical information neural network to be trained is constructed according to the target activation function. With the goal of learning the physical prior knowledge of flow velocity changes represented by the partial differential equations related to flow velocity, the physical information neural network to be trained is trained; based on the predicted flow velocity output by the physical information neural network to be trained for spatiotemporal sample points and the partial differential equations, the physical residual loss is determined; based on the physical residual loss, the network weights of the physical information neural network to be trained are updated; the trained physical information neural network is used to predict flow velocity for target spatiotemporal points.
7. The method for training a flow velocity prediction model based on a physical information neural network according to claim 6, characterized in that, Also includes: The spatiotemporal sample points are mapped using a weight mapping network to be trained. The time-limited control parameter mapped to the initial combination coefficients of the i-th activation function in the set of activation functions. ; For spatiotemporal sample points The target activation function is determined according to the following formula. : ， Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The target activation function is applied through the physical information network to be trained. For spatiotemporal sample points Output predicted flow rate; Based on the physical residual loss, in addition to updating the network weights of the physical information neural network to be trained, the network weights of the weight mapping network to be trained are also updated. The trained physical information neural network and the trained weight mapping network are used to predict the flow velocity for the target spatiotemporal point.
8. The method for training a flow velocity prediction model based on a physical information neural network according to claim 6, characterized in that, Also includes: The trained weight mapping network is used to map the target spatiotemporal point. The time-varying control parameter mapped to the initial combination coefficients of each activation function in the activation function set. ; For the target spatiotemporal point The target activation function is determined according to the following formula. : ， Where N represents the total number of activation functions in the activation function set. This represents the i-th activation function in the set of activation functions; This represents the initial combination coefficient of the i-th activation function in the set of activation functions; The trained physical information network is then used with the target activation function. Targeting the spatiotemporal point Output the predicted flow rate.
9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the flow velocity prediction model training method based on physical information neural network as described in any one of claims 6-8.
10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the computer program implements the steps of the flow rate prediction model training method based on a physical information neural network as described in any one of claims 6-8.