A meta-learning-based method for nonlinear flow reduction modeling and prediction

Abstract

The present invention provides a method for nonlinear flow reduction modeling and prediction based on meta-learning. The method proposed in the present invention first establishes a flow prediction model with an autoencoder configuration suitable for the current flow problem; secondly, the relationship between flow fields with different parameters is learned through the MAML algorithm, and a meta-model is established; then, the established meta-model is used as the initialization model under new physical parameters, and fine-tuned with a small number of training data points; finally, the flow prediction model after small sample training is used to perform reduction analysis and prediction research on the flow field under the new parameters. By introducing a meta-learning strategy, this method proposes a new solution to the problems of poor generalization ability and long training time of the current model. In addition, the method is simple to implement, has high accuracy, and strong versatility. It can be widely used in the reduction modeling and prediction of complex parameterized flow systems.

Description

Technical Field

[0001] The present invention relates to a meta-learning-based unsteady flow order reduction and prediction method, which can robustly and accurately perform fast nonlinear order reduction modeling and prediction on complex unsteady systems with different parameters, and belongs to the field of complex system modeling and prediction. Background Art

[0002] Fluid flows are typically described using the Navier-Stokes (NS) partial differential equations. These equations are often highly complex due to the presence of high-dimensional, highly nonlinear, and multi-scale flow phenomena. This inevitably requires extensive computation to process these complex flow phenomena. However, many complex fluid flows can be characterized by their dominant spatiotemporal coherent structures, allowing them to be decomposed into lower-dimensional spaces. Therefore, analyzing the key physical characteristics of unsteady fluid flows and establishing reduced-order models of complex systems based on these important physical characteristics is a promising approach.

[0003] Reduced-order models are generally categorized into two types: embedded projection-based models and non-embedded data-driven models. Embedded projection-based models compute and project high-dimensional Navier-Stokes partial differential equations (PDEs) to obtain an approximate low-dimensional ODE model. These models require an explicit understanding of the governing physical equations and are coupled to a Navier-Stokes solver. In contrast, non-embedded data-driven models are derived directly from measured data without a complete understanding of the governing equations. This type of reduced-order model is becoming increasingly popular and has addressed many problems in fluid dynamics, including proper orthogonal decomposition (POD), spectral proper orthogonal decomposition (SPOD), and dynamic mode decomposition (DMD). POD is a widely used reduced-order model in fluid dynamics. It represents the high-dimensional system through the modes with the highest energy content, thereby capturing the most important flow features, known as POD modes. These POD modes are used to represent the physical structure of the POD modes for fluid dynamics analysis. Compared to the POD method, the DMD method is an ideal combination of spatial dimensionality reduction techniques and temporal Fourier transforms, describing dynamic modes based on given frequencies and growth or decay rates. However, these traditional reduced-order models are linear models, which may not be suitable for dealing with highly nonlinear flow problems, especially for strongly coupled fluid-structure interaction systems or high Reynolds number turbulent flows.

[0004] In recent years, machine learning algorithms have become a very promising tool in fluid dynamics due to their strong ability to handle nonlinear flow phenomena and flexible structure. In the field of reduced-order modeling, the recently popular autoencoder model, in particular, leverages its powerful nonlinear processing capabilities to achieve nonlinear order reduction of complex flows and, based on the reduced-order results, can quickly and accurately predict complex flow fields. However, this model currently faces a significant challenge: the model requires multiple iterations of training on a GPU, resulting in very slow model building. For parameterized flow problems, the cost of retraining every physical parameter is prohibitive. Summary of the Invention

[0005] In response to the shortcomings of the existing technology, the present invention proposes a method for nonlinear flow reduction modeling and prediction based on meta-learning. The invention adopts a meta-learning-based method (MAML) to learn the parameter relationship of flow under different parameters, and then obtains a meta-model that can quickly adapt to flows with different motion parameters. The method is simple to implement, has high accuracy, strong versatility, and is more suitable for engineering applications. The method proposed in the present invention provides guidance for the large-scale application of the current deep learning-based autoencoder flow modeling method, and also has good versatility and scalability.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A meta-learning-based nonlinear flow reduction modeling method, the method specifically includes:

[0008] Step S1: constructing a flow field dataset X; the flow field dataset X includes flow field data under different motion parameters, and the flow field data under each motion parameter includes flow snapshots at different moments;

[0009] Step S2: Construct a nonlinear flow reduction model with an autoencoder structure, and use the constructed flow field dataset X, with the flow field snapshot at time t as input and the flow field snapshot at time t+1 as output. Therefore, the output of this model can also achieve the ability of flow prediction. Training is based on meta-learning, as follows:

[0010] Step S2.1: Extract a set of training samples X from the constructed flow field dataset X (b) ;

[0011] Step S2.2: Divide the extracted training sample set into the support set and Inquiry Set

[0012] Step S2.3: The parameter values ​​of the flow nonlinear reduced-order element model obtained in the previous iteration are As the initial value of the flow nonlinear reduced order model parameters, calculate the gradient of the flow nonlinear reduced order model under the support set where f θ represents the nonlinear reduced-order model of the flow with weight parameter θ; Support set The loss function is calculated by taking the sample of as input, wherein the loss function includes the error between the predicted value output by the flow nonlinear reduction model and the true value;

[0013] Step S2.4: Calculate the adaptive parameter values ​​of the flow nonlinear reduced-order model by n-fold gradient descent: Among them, θ n-1 represents the model parameter value of the n-1th gradient descent, and β represents the learning rate;

[0014] Step S2.5: Update the flow nonlinear reduced-order model parameters based on the adaptive parameter values: in, represents the parameter value of the nonlinear reduced-order element model of the flow after this iteration update, represents the parameter value of the flow nonlinear reduced-order meta-model obtained in the previous iteration, and η represents the meta-learning rate; Indicates the query set The loss function is calculated by taking the sample of as input;

[0015] Repeat steps S2.1-S2.5 for multiple iterations to update the flow nonlinear reduced-order model parameters to obtain the flow nonlinear reduced-order element model;

[0016] Step S3: Construct flow field dataset X new ; The flow field data set X new Only the flow field data under the same motion parameters are included. The flow field data under each motion parameter contains flow snapshots at different times.

[0017] Using the constructed flow field dataset X new , taking the flow field snapshot at time t as input and the flow field snapshot at time t+1 as output, the obtained flow nonlinear reduced-order element model is trained and fine-tuned to obtain the adapted flow field dataset X new A nonlinear reduced-order model of the flow under corresponding motion parameters is developed, and the flow at future moments can be directly predicted through this model.

[0018] Furthermore, the flow field data set X includes flapping wing flow field data under different motion parameters, and the motion parameters include flapping frequency and flapping wing wake vortex pattern.

[0019] Furthermore, the flow field dataset X is obtained through CFD numerical simulation.

[0020] Furthermore, the flow nonlinear reduction model of the autoencoder structure includes an encoder and a decoder, wherein the encoder is composed of an input layer, multiple groups of convolutional layers-maximum pooling layers, and fully connected layers connected in sequence; and the decoder is composed of an input layer, a fully connected layer, and multiple groups of upsampling layers-convolutional layers connected in sequence.

[0021] Furthermore, the loss function of the flow nonlinear reduced-order model is:

[0022]

[0023] Where x t+1 and x t are the flow snapshots at time t+1 and t respectively; represents the square of L2 norm, N b represents the number of samples, Ψ(*) represents the output of the encoder, and Φ(*) represents the output of the decoder.

[0024] A flow prediction method, specifically comprising:

[0025] The flow snapshot at time t is input into the flow nonlinear order reduction model constructed by the meta-learning-based flow nonlinear order reduction modeling method, and the flow snapshot at time t+1 can be immediately obtained.

[0026] The present invention provides a meta-learning-based nonlinear flow reduction modeling method. By learning the flow relationships between different physical parameters, a meta-model is established. This meta-model is then used to rapidly adapt parameterized flow problems using a small amount of data, enabling rapid modeling and flow prediction under new physical parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 is a flow chart of the method of the present invention;

[0028] Figure 2 The figure is a comparison of the training speed of the method proposed by the present invention and the traditional method under different motion parameters f;

[0029] Figure 3 Comparison chart of the prediction results between the proposed method and the traditional method. DETAILED DESCRIPTION

[0030] The present invention provides a meta-learning-based nonlinear reduced-order modeling method for flow, which uses the meta-learning method to accelerate the training speed of the autoencoder network, so that the model can quickly adapt to the flow under new physical parameters with a small amount of data.

[0031] The present invention will be described in detail below with reference to the accompanying drawings.

[0032] Figure 1The flow chart of a method for nonlinear flow reduction modeling based on meta-learning provided by the present invention is as follows: Figure 1 As shown, the method of the present invention comprises the following steps:

[0033] Step S1: constructing a flow field dataset X; the flow field dataset X includes flow field data under different motion parameters, and the flow field data under each motion parameter includes flow snapshots at different moments;

[0034] This example is a parametric flapping wing flow problem under Reynolds number Re = 100. The chord length of the flapping wing is 1m, and the incoming flow condition is velocity U ∞ =1m / s uniform incoming flow, the flapping wing moves in a sinusoidal motion centered around the leading edge, as follows:

[0035] α=α max sin(2πft)

[0036] Where f is the flapping frequency, α is the flapping angle amplitude, and α max is the maximum flapping angle amplitude. In this embodiment, only the flapping frequency is changed, and the maximum flapping angle amplitude is a certain value, set to α max =0.09948. In this embodiment, 15 parameter values ​​are selected as training and test data sets, as shown in Table 1:

[0037] Table 1. Flapping wing motion parameter settings

[0038] Frequency / f 0.320 0.330 0.340 0.350 0.360 0.528 0.534 0.538 Vortex mode 2P 2P 2P 2P 2P 2P 2P 2P→2S Frequency / f 0.542 0.546 0.600 0.610 0.620 0.630 0.640 Vortex mode 2P→2S 2P→2S 2S 2S 2S 2S 2S

[0039] In order to make the problem more representative, the motion parameters selected in this embodiment are distributed in different flapping wake vortex modes.

[0040] All the unsteady flow field data shown in Table 1 were obtained by CFD numerical simulation. The flow field data were mapped to a uniform 200×200 Cartesian coordinate in the range of x-axis [-2, 12] and y-axis [-2, 2]. The time step of data collection was τ = 1 / (f×N t ), where N t =100 represents the number of samples, thus constructing the flow field dataset X;

[0041] Step S2: Construct a flow nonlinearity reduction model with an autoencoder structure. Generally, an autoencoder consists of an encoder Ψ and a decoder Φ. The encoder is used to reduce the dimensionality of high-dimensional flow field data to obtain its low-dimensional representation z, and the decoder is used to project the low-dimensional representation back to the high-dimensional flow field. The input of the network is the flow field snapshot at time t, and the output is the flow field snapshot at time t+1. Therefore, the model can predict the flow field snapshot at time t+1 by inputting the flow field snapshot at time t. Table 2 shows the specific parameters of an autoencoder model.

[0042] Table 2 Specific parameters of an autoencoder model

[0043]

[0044]

[0045] Using the prepared flow field dataset X, with the flow field snapshot at time t as input and the flow field snapshot at time t+1 as output, training is performed based on meta-learning as follows:

[0046] Step S2.1: Extract a set of samples from the constructed flow field dataset X to form the training sample set X (b) ;

[0047] Step S2.2: Randomly divide the extracted training sample set into support sets and Inquiry Set

[0048] Step S2.3: The parameter values ​​of the flow nonlinear reduced-order element model obtained in the previous iteration are As the initial value of the flow nonlinear reduced-order model parameters, that is, For the first iteration, the initial values ​​of the flow nonlinear reduced order model parameters can be random values, and the gradient of the flow nonlinear reduced order model under the support set is calculated. where f θ represents the nonlinear reduced-order model of the flow with weight parameter θ; Support set The sample is used as the input for the loss function, which includes the error between the predicted value output by the flow prediction model and the true value. In this embodiment, the loss function is specifically:

[0049]

[0050] Where x t+1 and x t are the flow snapshots at time t+1 and t respectively; Ψ(*) represents the output of the encoder, Φ(*) represents the output of the decoder, N b Indicates the number of samples. represents the square of L2 norm.

[0051] Step S2.4: Calculate the adaptive parameter values ​​of the flow nonlinear reduced-order model by n-fold gradient descent: Among them, θ n-1 represents the model parameter value of the n-1th gradient descent, β represents the learning rate; in this embodiment, β = 1×10 -2 ;

[0052] Step S2.5: Update the flow nonlinear reduced-order model parameters based on the adaptive parameter values: η represents the meta-learning rate. In this embodiment, η=1×10 -3 ; Indicates the query set The loss function is calculated by taking the sample of as input;

[0053] Repeat steps S2.1-S2.5 multiple times to obtain the flow nonlinear reduced-order element model;

[0054] Step S4: Construct flow field dataset X new ; The flow field data set X new Only the flow field data under the same motion parameters are included. The flow field data under each motion parameter contains flow snapshots at different times. Under the new motion parameters, the same method as step S1 is used to collect a small amount of simulation data to construct the flow field dataset X. new ;

[0055] Using the constructed flow field dataset X new , taking the flow field snapshot at time t as input and the flow field snapshot at time t+1 as output, the obtained flow nonlinear reduced-order element model is trained and fine-tuned to obtain the adapted flow field dataset X new A nonlinear reduced-order model of the flow under corresponding motion parameters is obtained, and the output of the model can be directly used to predict the flow at future moments.

[0056] Figure 2 The results of the comparison of model training speed under different motion parameters using the method of the present invention are as follows: Figure 2 As shown in the figure, Random initialization is a traditional method, Reptile is a mainstream meta-learning acceleration algorithm, and MAML represents the method proposed by the present invention. It can be seen that the method proposed by the present invention has the highest accuracy and the fastest speed.

[0057] The nonlinear reduced-order flow model constructed by the present invention can be used for flow prediction. The flow prediction model under the obtained new motion parameters can directly predict the flow snapshot at each moment under the motion parameters.

[0058] Figure 3 The specific flow field prediction results are given. It can be seen that the prediction accuracy of the method proposed in the present invention is very consistent with the actual flow field state.

[0059] Obviously, the above embodiments are merely examples for clarity of explanation and are not intended to limit the implementation methods. Those skilled in the art will appreciate that other variations or modifications can be made based on the above description. It is not necessary and impossible to enumerate all implementation methods here. Obvious variations or modifications derived therefrom remain within the scope of protection of the present invention.

Claims

1. A method for flow nonlinear order reduction modeling based on meta-learning, characterized by: The method specifically includes: Step S1: constructing a flow field dataset X; the flow field dataset X includes flow field data under different motion parameters, and the flow field data under each motion parameter includes flow snapshots at different moments; Step S2: Construct a flow nonlinear reduction model with an autoencoder structure, and use the constructed flow field dataset X, with the flow field snapshot at time t as input and the flow field snapshot at time t+1 as output, to perform training based on meta-learning, as follows: Step S2.1: Extract a set of training samples X from the constructed flow field dataset X (b) ; Step S2.2: Divide the extracted training sample set into the support set and Inquiry Set Step S2.3: The parameter values ​​of the flow nonlinear reduced-order element model obtained in the previous iteration are As the initial value of the flow nonlinear reduced order model parameters, calculate the gradient of the flow nonlinear reduced order model under the support set where f θ represents the nonlinear reduced-order model of the flow with weight parameter θ; Support set The loss function is calculated by taking the sample of as input, wherein the loss function includes the error between the predicted value output by the flow nonlinear reduction model and the true value; Step S2.4: Calculate the adaptive parameter values ​​of the flow nonlinear reduced-order model by n-fold gradient descent: Among them, θ n-1 represents the model parameter value of the n-1th gradient descent, and β represents the learning rate; Step S2.5: Update the flow nonlinear reduced-order model parameters based on the adaptive parameter values: in, represents the parameter value of the nonlinear reduced-order element model of the flow after this iteration update, represents the parameter value of the flow nonlinear reduced-order meta-model obtained in the previous iteration, and η represents the meta-learning rate; Indicates the query set The loss function is calculated by taking the sample of as input; Repeat steps S2.1-S2.5 for multiple iterations to update the flow nonlinear reduced-order model parameters to obtain the flow nonlinear reduced-order element model; Step S3: Construct flow field dataset X new ; The flow field data set X new Only the flow field data under the same motion parameters are included. The flow field data under each motion parameter contains flow snapshots at different times. Using the constructed flow field dataset X new , taking the flow field snapshot at time t as input and the flow field snapshot at time t+1 as output, the obtained flow nonlinear reduced-order element model is trained and fine-tuned to obtain the adapted flow field dataset X new Nonlinear reduced-order model of flow under corresponding motion parameters.

2. The method according to claim 1, characterized in that The flow field data set X contains flapping wing flow field data under different motion parameters, and the motion parameters include flapping frequency and flapping wing wake vortex pattern.

3. The method according to claim 1, characterized in that The flow field dataset X is obtained through CFD numerical simulation.

4. The method according to claim 1, wherein The flow nonlinear reduction model of the autoencoder structure includes an encoder and a decoder, wherein the encoder is composed of an input layer, multiple groups of convolutional layers-maximum pooling layers, and fully connected layers connected in sequence; the decoder is composed of an input layer, a fully connected layer, and multiple groups of upsampling layers-convolutional layers connected in sequence.

5. The method according to claim 1, wherein The loss function of the flow nonlinear reduced-order model is: Where x t+1 and x t are the flow snapshots at time t+1 and t respectively; represents the square of L2 norm, N b represents the number of samples, Ψ(*) represents the output of the encoder, and Φ(*) represents the output of the decoder.

6. A flow prediction method, characterized in that: Specifically: The flow snapshot at time t is input into the flow nonlinear reduced-order model constructed by the meta-learning-based flow nonlinear reduced-order modeling method described in any one of claims 1 to 5, that is, the flow snapshot at time t+1 is obtained.

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