A method for predicting multi-working-condition transient physical field of fluid equipment based on residual distillation

Abstract

The application discloses a kind of fluid equipment multi-working condition transient physical field prediction methods based on residual distillation. The method first utilizes high-fidelity data to construct teacher model, obtains the accurate approximation of partial differential equation solution and its residual distribution;Subsequently, a larger student model is constructed, with boundary conditions, initial conditions as physical constraints, while introducing the KL divergence distillation loss between the probability distribution of physical residual of teacher model and the probability distribution of physical residual of student model for joint optimization, using dynamic sampling based on residual distribution, learning rate scheduling and loss weighting strategy.The method of the application can effectively improve the solving accuracy and generalization ability of partial differential equation under different working conditions while ensuring physical consistency, and when predicting flow field, temperature field and pressure field, it ensures stable working condition switching, reduces the cost of repeated simulation time, and improves design control response.

Description

Technical Field

[0001] This invention belongs to the field of physical field prediction and artificial intelligence technology based on neural networks. Specifically, it relates to a method for predicting transient physical fields of fluid equipment under multiple operating conditions based on residual distillation. It can serve as a rapid replacement for traditional simulations such as finite element / finite volume in fluid and heat transfer, supporting real-time monitoring, operating condition switching and optimized control. Background Technology

[0002] Partial differential equations (PDEs) describe the changes of physical quantities in the objective world with time and spatial location, and are widely found in fields such as fluid mechanics, thermodynamics, and quantum mechanics. In industrial engineering fields such as aerospace, automotive, energy, and chemical engineering, iterative solutions to PDEs are needed to obtain flow fields, temperature fields, and stress fields. Traditional solvers are costly, necessitating online, near-real-time methods that can output physical field distributions at the millisecond to second level to replace some batch simulations, shorten iteration cycles, and reduce computational dependence. With the development of computing power and deep learning, neural network-based PDE solution methods have become important tools in scientific research and engineering. Physical information neural networks (PINNs) encode physical conservation laws into the network loss by using PDE residuals, boundary conditions, and initial conditions as training constraints, thereby approximating PDE solutions under weakly supervised or even unsupervised conditions.

[0003] However, PINN still faces training and generalization challenges in practical applications, limiting its widespread adoption in industrial and multi-condition problems. On the one hand, the PDE residual term and the boundary / initial condition term often differ significantly in gradient scale, causing some loss terms to be suppressed or over-amplified during backpropagation, resulting in training instability. On the other hand, multiple coupling constraints in the high-dimensional parameter space will produce a complex optimization landscape (including saddle points and local minima), making the optimization method prone to getting stuck in bad critical points and causing the solution to not converge or be physically inconsistent.

[0004] In addition, traditional PINN is often trained for a single operating condition (specific inlet / outlet velocity or wall temperature, etc.). If multiple operating conditions need to be covered, repeated training or complex transfer strategies are usually required, which makes it difficult to meet the needs of real-time, multi-condition inference in actual engineering. Summary of the Invention

[0005] To address the aforementioned problems in existing technologies, this invention proposes a multi-condition transient physics prediction method for fluid equipment based on residual distillation. This method constructs a teacher model using limited high-fidelity data, accurately constructs PDE residuals through automatic differentiation, and incorporates the probability distribution of these residuals, along with boundary and initial conditions, as physical constraints into the training loss of the student model. This ensures that the student model is always constrained by physical conservation laws during parameter updates. Unlike traditional knowledge distillation, which often targets "soft labels" or intermediate network representations, the core innovation of this invention lies in using the probability distribution of the PDE residuals encoded by the teacher model as the distillation object. Through probability distribution alignment (based on KL divergence with a temperature factor), the physical structure of the teacher model is injected into the PINN student model. This joint training strategy retains PINN's ability to directly characterize physical constraints while importing information about the higher-order physical structure and smooth optimization landscape provided by the teacher into the student model through residual distribution alignment. This achieves accelerated convergence, reduces the risk of getting trapped in saddle points / sharp minima, and improves generalization ability under variable initial / boundary conditions without sacrificing physical consistency.

[0006] This invention provides the following technical solution: A method for predicting the transient physical field of a fluid equipment under multiple operating conditions based on residual distillation, the specific steps of which are as follows: S1: Determine the specific form of the objective partial differential equation (PDE), determine the computational domain according to requirements, and select multiple different initial conditions or boundary condition sets.

[0007] Furthermore, the PDE form in S1 is shown in equation (1): (1); in It is a PDE solution. Represents a nonlinear differential operator. It is the computational spatial domain. This refers to the computation time domain. Furthermore, the boundary conditions can be Dirichlet or Neumann boundary conditions, depending on the specific circumstances.

[0008] S2: Divide multiple initial conditions or boundary conditions into training and testing groups using appropriate sampling methods, and construct a high-fidelity dataset through numerical simulation.

[0009] Furthermore, the sampling method in S2 can be random sampling, Latin hypercube sampling, etc. High-fidelity data can be calculated using a high-precision numerical solver, or experimentally measured if conditions permit. The dataset is required to include the following parameters: a set of spatial coordinates. Time coordinate set High-fidelity solution and the encoded initial / boundary conditions .

[0010] S3: Construct and train a teacher model. The inputs to the teacher model are spatial coordinates, time coordinates, inlet flow velocity conditions, and wall boundary conditions. The output is the corresponding predicted physical field value. The teacher model is trained by minimizing the difference between the predicted values ​​and the high-fidelity data values. Specifically: A teacher model network is constructed, using a multilayer feedforward neural network (MLP) as the backbone network, with the network input being... The output is First, determine the model structure, activation function, optimizer, and learning rate, then construct the loss function and begin supervised training.

[0011] Furthermore, the MLP network described in S3 includes an input layer, multiple hidden layers, and an output layer. Here, the total number of hidden layers in the neural network is preferably 4-5, and the number of neurons in each layer is preferably 64-128. The Adam optimizer is preferred, and commonly used activation functions include tanh, gelu, and swish. Tanh is often used for physical problems that require nonlinear mapping, gelu is suitable for deep networks, and swish can improve convergence speed and model performance to a certain extent and alleviate gradient divergence. The initial learning rate is preferably set to 0.001, and the minimum value can be reduced to 0.0001. A learning rate scheduling strategy (such as learning rate decay) is used to avoid premature and excessive updates, ensuring stable training and gradual convergence. The loss function of the teacher model is shown in equation (2): (2); Where N represents the number of data points. This represents the teacher model's predicted value for the current data point under certain initial / boundary conditions. This represents the high-fidelity data value of the current data point.

[0012] S4: The student model takes spatial coordinates, time coordinates, inlet flow velocity conditions, and wall boundary conditions as input, and outputs the corresponding physical field predictions. The student model is trained by minimizing its loss function using an optimization algorithm. The loss function measures the difference between the student model's predictions and the high-fidelity data values, as well as the difference between the student model's physical residual probability distribution and the teacher model's physical residual probability distribution. Specifically: The student model network is constructed, also using an MLP as the backbone network, with the network input being... The output is The model structure, activation function, optimizer, and learning rate are determined. The student model output is automatically differentiated to calculate the temporal and spatial derivatives, which are then substituted into the PDE to be solved, yielding the student model's residual values. These residual values ​​are normalized at the sampling points to form the student model's residual probability distribution. The residual probability distribution at the same sampling points is obtained from the trained teacher model. A loss function for the student model is constructed, consisting of three parts: a PDE residual distribution consistency term and boundary / initial condition constraint terms, achieving distillation of the physical prior.

[0013] Furthermore, the neural network in S4 is larger than that in S3, with a total of 6-8 layers and 64-128 neurons per layer. The Adam optimizer is selected, and activation functions such as tanh, gelu, and swish are optional. The initial learning rate is initially set to 0.001, and a learning rate decay mechanism is used during training, with the minimum value decreasing to 0.0001. The loss function of the student model is shown in equation (3): (3); in and These are the root mean square errors between the predicted values ​​and high-fidelity values ​​of the student model at different boundaries and initial positions. The residual probability distribution is calculated by equation (1) from the student model. The residual probability distribution of the approximation teacher model is calculated using equation (1). The KL divergence information loss, where T represents the distillation temperature. These are the weight coefficients of different loss terms. An adaptive loss weighting mechanism is adopted. During the training process, the weights of each part in the loss function are dynamically adjusted for different working conditions and different gradient scales. In the early stage of training, the student model mainly obtains a good solution quickly by fitting data and learning the constraints of physical equations. During the training process, we adopt a phased loss weight scheduling strategy. The entire training process is divided into an "initial stage" and a "late stage" on the timeline: In the initial stage (usually the first 20%-30% of training rounds, or until the ratio of the validation set error to the gradient of each loss term enters a stable range), the weights of the data fitting term and the basic physical residual term are kept relatively large, while the weights of the distillation term are relatively small. This allows the student model to first obtain a reasonable initial solution space position through data fitting and basic physical constraints. When the total loss decreases at a slower pace and the gradient norm ratio approaches equilibrium, the process enters the late stage. The proportion of residual alignment and distillation loss in the total loss is gradually increased (e.g., by increasing their weights linearly or according to the GradNorm rule). This guides the student model to pay more attention to the physical residual structure consistent with the teacher model in the later training stages, gradually approaching the higher-order physical patterns encoded by the teacher.

[0014] S5: Sample data within the computational domain using the appropriate sampling method as input values ​​for the student model, and minimize the value of the loss function through an optimization algorithm to obtain the target student model; use the trained student model to predict the conditions in the test group to obtain physical field distribution data.

[0015] Furthermore, the sampling methods described in S5 include random sampling, Latin hypercube sampling, etc. To further improve training efficiency and solution accuracy, this invention adopts an adaptive sampling strategy. In the early stages of training, dense sampling is mainly performed in more important regions (such as near the boundary and in areas with large physical residuals); as training progresses, the sampling density in these regions is gradually reduced.

[0016] Increasing sampling over a wider region helps student models converge faster and maintain good generalization ability across the entire solution domain.

[0017] S6: Input the physical field distribution data into the control system of the fluid equipment to generate control signals for adjusting the operating parameters of the equipment, or design parameters for optimizing the equipment structure.

[0018] Furthermore, the model performance is compared, and the specific steps are as follows: Step 1: Construct the PINN network, also using an MLP as the backbone network. The network input is... The output is The model structure, activation function, optimizer, and learning rate are determined, and the model structure and learning strategy should be consistent with those in step S4. The loss function of PINN consists of PDE residuals, boundary conditions, and initial conditions. The PDE residual term measures the error of the model in solving physical equations, while the boundary conditions and initial conditions ensure the physical consistency of the model under specified constraints. The design of the loss function is shown in equation (4). Step S5 is repeated for training until the loss function converges, thus obtaining the PINN model. This model serves as the basis for comparative experiments and will be compared with the model proposed in this invention in terms of performance. (4).

[0019] Step 2: Use the trained student model and PINN model to predict the values ​​in the test group under different initial / boundary conditions. Simultaneously, calculate the root mean square difference between the predicted values ​​and the corresponding high-fidelity values. Repeat this process several times and take the average value to evaluate the accuracy of this training.

[0020] Furthermore, the inlet flow velocity condition is a time series of pipe inlet flow velocities collected by a flow sensor, and the wall boundary condition is the wall temperature distribution collected by a temperature sensor.

[0021] Furthermore, in S6, the control system is a PID controller, and the control signal is used to adjust the valve opening to maintain the pressure of the pipeline node within a preset safe range.

[0022] Furthermore, the fluid device includes at least one of pipes, valves, heat exchangers, or reactors; the physical field is at least one of velocity field, temperature field, and pressure field.

[0023] By employing the above-described technology, the beneficial effects of the present invention compared to the prior art are as follows: 1) This invention introduces the residual information of the high-fidelity teacher model and uses KL divergence to constrain the distribution of the student model residuals, so that the student model can obtain global physical understanding guidance from the teacher model during the optimization process, thus avoiding the gradient oscillation and convergence instability problems that are easy to occur in traditional PINN when multiple loss couplings occur. 2) In this invention, the residual of the teacher model is directly derived from the PDE approximation solution driven by high-fidelity data, while the student model is simultaneously driven by both PDE constraints and teacher residual distribution distillation. Therefore, while maintaining strict physical constraints, it can learn higher-order structural features that are more in line with actual physics, and can better predict phenomena such as shock waves in the flow field. 3) Based on the distillation mechanism adopted in this invention, not only is the solution accuracy improved under specific initial and boundary conditions, but the student model is also able to quickly adapt and transfer to different operating conditions (variable initial conditions, boundary conditions, etc.), thereby improving PINN's cross-condition generalization ability. It can ensure stable switching of operating conditions, reduce the time cost of repeated simulation, and improve the design control response when predicting flow field, temperature field, and pressure field based on an integrated industrial simulation and digital twin system. Attached Figure Description

[0024] Figure 1 This is a schematic flowchart of the method of the present invention; Figure 2 This is a training diagram of the solution method of the present invention; Figure 3 In an embodiment of the present invention, The residual distribution of the equation predicted by the teacher model; Figure 4 In an embodiment of the present invention, The prediction results and errors of the PINN model in solving the Burgers equation; Figure 5 In an embodiment of the present invention, The prediction results and errors of the student model in solving the Burgers equation. Detailed Implementation

[0025] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0026] Conversely, this invention encompasses any substitutions, modifications, equivalent methods, and solutions made within the spirit and scope of the invention as defined in the claims. Furthermore, to provide a better understanding of the invention, certain specific details are described in detail below. However, those skilled in the art will fully understand the invention even without these detailed descriptions.

[0027] A method for predicting transient physical fields under multiple operating conditions in fluid equipment based on residual distillation, such as... Figure 1-2 As shown, the specific steps are as follows: S1: The PDE solved in this example is the one-dimensional Burgers equation. Burgers represents viscous convection-diffusion, and it is often used in practice for transient flows such as pipes and valve throttling zones. The core problem is that when the initial flow velocity u changes, the flow field solution will show a significant gradient increase, and even form a shock wave transition layer. Neural networks usually have difficulty learning in these high gradient regions, and the output flow field results are prone to numerical oscillation or over-smoothing. The specific equation form and initial / boundary conditions are as shown in equation (5): (5); Among them, fluid kinematic viscosity .

[0028] S2: This embodiment uses the change of initial conditions as an example to illustrate the implementation of multiple working conditions; at the same time, this method is also applicable to multiple working condition scenarios caused by changes in boundary conditions or other parameters.

[0029] Specifically: the coefficients of the initial flow velocity u-function Ten different initial conditions were randomly sampled within the range [-1, 0], corresponding to different working conditions. The flow velocity ranged from -1 m / s to 1 m / s sinusoidal fluctuation. The training and test groups were divided in a 7:3 ratio. Periodic boundary conditions were applied. The model accepts any initial flow velocity curve as input and outputs the flow velocity field of the problem in the entire spatiotemporal region. The classic fourth-order Runge-Kutta method was used for the solution. (6); After obtaining the high-fidelity flow velocity solutions corresponding to all initial conditions, collect the set of spatial coordinates. Time coordinate set High-fidelity flow rate solution and the corresponding initial flow velocity Build training and test datasets.

[0030] S3: Construct the teacher model network using an MLP network structure. The network has 4 hidden layers, with 64 neurons per layer. The network input is... The output is The Adam optimizer and tanh activation function were selected, with a total of 10,000 training epochs. The initial learning rate was set to 0.001, and a learning rate decay mechanism was used, reducing it to 0.0001 after 8,000 steps. The network output value was then constructed. and high fidelity The root mean square error between the two is used as the loss function to begin training.

[0031] S4: Construct the student model network using an MLP network structure. The network has 6 hidden layers, with 64 neurons per layer. The network input is... The output is The Adam optimizer was selected. Because higher-order derivatives are involved in the training process, Swish was chosen as the activation function for the student model. The initial learning rate was set to 0.001, and a loss function containing the residual probability distribution, boundary conditions, and initial conditions was constructed using equation (3). The distillation temperature was set to 10.

[0032] S5: The Latin hypercube sampling method is used to sample time and spatial coordinates within the computational domain, with intensified sampling in boundary regions, regions with abrupt velocity gradient changes, and regions with large physical residuals, serving as input values ​​for the student model. For PDE equations of different dimensions (e.g., one-dimensional, two-dimensional, three-dimensional), the number of sampling points and the total number of training epochs need to be dynamically adjusted according to the complexity of the equation. For example, for one-dimensional PDE equations, the number of sampling points per epoch should be no less than 4000, and the number of training epochs should be no less than 20000; for two-dimensional PDE equations, the number of sampling points per epoch should be no less than 20000, and the number of training epochs should be no less than 200000; for three-dimensional PDE equations, the number of sampling points per epoch should be no less than 50000, and the number of training epochs should be no less than 500000. Since this embodiment uses a one-dimensional partial differential equation, the total number of training epochs is set to 20000, and the number of sampling points per epoch is 5000. A learning rate decay mechanism is used, reducing the learning rate to 0.0001 after 15000 epochs. By continuously iterating through the optimization algorithm to minimize the value of the loss function, the model backpropagates to update the parameters, thus obtaining the solution over the entire computational domain.

[0033] S6: Input the physical field distribution data into the control system of the fluid equipment to generate control signals for adjusting the operating parameters of the equipment, or design parameters for optimizing the equipment structure.

[0034] The following is a comparison of model performance: Step 1: Select the same model structure and training strategy as in step S4 to build the PINN model. Repeat step S5 until the training is complete to obtain the PINN comparison model.

[0035] Step 2: Using the trained student model and PINN model, predict the values ​​in the test group under different initial conditions, and calculate the root mean square error between the predicted values ​​and the corresponding high-fidelity values ​​over the entire computational domain. The experiment was repeated 10 times, and the average value was used to evaluate the accuracy of this training. The experimental results are shown in Table 1 below. The accuracy is more than one order of magnitude higher than that of PINN. Table 1: Comparison of Root Mean Square Error of Prediction in Test Group Models

[0036] in, Figure 3 Showing The residual distribution of the equation predicted by the teacher's model is used as prior knowledge to guide the student's model training. Figure 4 and Figure 5 They were shown respectively The prediction results and error distributions of the PINN model and the student model in solving the Burgers equation are compared. It can be clearly seen that the prediction error of the student model proposed in this embodiment is significantly lower than that of the traditional PINN model, and it can predict the shock transition layer very well.

[0037] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for predicting the transient physical field of a fluid equipment under multiple operating conditions based on residual distillation, characterized in that, Includes the following steps: S1. For transient flow problems in fluid equipment, determine the computational domain, time domain, and multiple sets of different initial or boundary conditions for the partial differential equation to be solved; S2. Divide multiple sets of initial conditions or boundary conditions into training groups and test groups, and generate corresponding high-fidelity datasets through numerical simulation; S3. Construct and train a teacher model. The input to the teacher model is spatial coordinates, time coordinates, inlet flow velocity conditions, and wall boundary conditions. The output is the corresponding physical field prediction value. The teacher model is trained by minimizing the difference between the teacher model's prediction value and the high-fidelity data value. S4. Construct a student model. The inputs to the student model are spatial coordinates, time coordinates, inlet flow velocity conditions, and wall boundary conditions. The output is the corresponding physical field prediction value. The student model is trained by minimizing its loss function using an optimization algorithm. The loss function measures the difference between the student model's prediction value and the high-fidelity data value, as well as the difference between the student model's physical residual probability distribution and the teacher model's physical residual probability distribution. The specific process is as follows: S4.

1. Calculate the derivative of the student model output by automatic differentiation and substitute it into the target partial differential equation (PDE) to obtain the residual, generating the student model residual probability distribution; and obtain the corresponding teacher model residual probability distribution from the trained teacher model. S4.

2. Construct a loss function for the student model, wherein the loss function includes at least: a data fitting term for constraining the student model output to match high-fidelity data, and a distribution consistency term for driving the student model residual probability distribution to be consistent with the teacher model residual probability distribution; The distribution consistency term is achieved by calculating the KL divergence between the residual probability distribution of the student model and the residual probability distribution of the teacher model. The loss function of the student model is specifically expressed as follows: ； Where Ls is the total loss of the student model. These are the initial and boundary condition constraints. For the student model residual probability distribution P s The residual probability distribution P of the teacher model t KL divergence between them, λ icbc and λ distill These are the weighting coefficients for each loss term; An adaptive loss weighting mechanism is used to dynamically adjust the weights in the loss function, as follows: During training, the weights of each term in the loss function are dynamically adjusted for different working conditions and gradient scales. In the early stage of training, the weights of the data fitting term and the basic physical residual term are kept relatively large, while the weight of the distillation term is relatively small, so that the student model can obtain an initial solution space position through data fitting and physical constraints. In the later stage of training, the proportion of residual alignment and distillation loss in the total loss is increased, guiding the student model to pay more attention to the physical residual structure consistent with the teacher model in the later training, and to approximate the high-order physical mode encoded by the teacher. S5. Use the trained student model to predict the conditions in the test group to obtain physical field distribution data; S6. Input the physical field distribution data into the control system of the fluid equipment to generate control signals for adjusting the operating parameters of the equipment, or design parameters for optimizing the equipment structure.

2. The method for predicting the transient physical field of a fluid equipment under multiple operating conditions based on residual distillation according to claim 1, characterized in that, In step S5, an adaptive sampling strategy is used to sample training data, as follows: In each training cycle, sampling points are allocated according to the PDE residual size of the current student model solution. Higher sampling probabilities are set in regions where the residual value is greater than the average value, and denser sampling is performed in the boundary regions of the fluid device or regions where the physical field gradient changes abruptly.

3. The method for predicting the transient physical field of a fluid device under multiple operating conditions based on residual distillation according to claim 1, characterized in that, Both the teacher model and the student model use a multilayer perceptron (MLP) as the backbone network, and the student model has more network layers than the teacher model.

4. The method for predicting the transient physical field of a fluid device under multiple operating conditions based on residual distillation according to claim 1, characterized in that, The inlet flow velocity condition is the time series of the pipe inlet flow velocity collected by the flow sensor, and the wall boundary condition is the wall temperature distribution collected by the temperature sensor.

5. The method for predicting the transient physical field of a fluid equipment under multiple operating conditions based on residual distillation according to claim 1, characterized in that, In S6, the control system is a PID controller, and the control signal is used to adjust the valve opening to maintain the pressure of the pipeline node within a preset safe range.

6. The method for predicting the transient physical field of a fluid equipment under multiple operating conditions based on residual distillation according to claim 1, characterized in that, The fluid equipment includes at least one of pipes, valves, heat exchangers, or reactors; the physical field is at least one of velocity field, temperature field, and pressure field.

Images