A deep learning-based efficient evaluation method for gravitational lens magnification factor
By using the Sinusoidal Representation Network (SIREN) to quickly and accurately calculate the gravitational lensing magnification factor, the problems of high computational cost and insufficient accuracy in traditional methods are solved, enabling real-time analysis and high-precision evaluation of gravitational wave data.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- ZHEJIANG UNIV
- Filing Date
- 2026-04-10
- Publication Date
- 2026-07-03
AI Technical Summary
Traditional methods struggle to quickly and accurately calculate the gravitational wave amplification factor under the gravitational lensing effect, especially in the long wavelength or small lens mass region. Numerical integration is slow to converge, consumes a lot of memory, and deep learning models struggle to fit the high-frequency oscillation characteristics in diffraction integrals.
We employ a sinusoidal representation network (SIREN) that utilizes its periodic activation function to match the oscillatory characteristics of diffraction integrals. Through a multilayer perceptron architecture and adaptive learning rate optimization, combined with a region decomposition strategy, we achieve fast and high-precision computation.
It achieves the real-time processing requirements for gravitational wave data analysis, reaching a relative error on the order of O(n), meeting the accuracy requirements of current and future gravitational wave detectors, and the model file is small and easy to deploy.
Smart Images
Figure CN121996995B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of gravitational wave data processing technology, and more specifically, to a method for efficiently evaluating gravitational lensing magnification factors based on deep learning. Background Technology
[0002] In gravitational wave astronomy, gravitational lensing occurs when gravitational waves pass near massive objects (such as black holes and galaxies). As ripples in spacetime, the propagation paths of gravitational waves are bent by the gravitational field, causing signals from the same source to reach Earth along multiple paths, creating interference. To accurately analyze this effect, especially in the "wave optics" region with long wavelengths or small lens masses, the diffraction integral must be calculated to obtain the magnification factor F.
[0003] However, directly calculating this diffraction integral faces significant challenges: the integral kernel contains highly oscillating complex terms, resulting in extremely slow numerical integration convergence and extremely high computational costs; for gravitational wave data analysis tasks that require processing massive parameter spaces, traditional numerical integration methods are insufficient to meet real-time processing requirements; traditional lookup table methods experience exponential growth in memory consumption as the parameter dimension increases, lacking scalability; and ordinary deep learning models (such as MLPs using the ReLU activation function) struggle to fit the high-frequency oscillation characteristics in the diffraction integral due to the "spectral bias" problem, leading to insufficient accuracy.
[0004] To address the aforementioned problems, this invention proposes a deep learning method based on a sinusoidal representation network, which utilizes the periodic activation function to match the oscillatory characteristics of the diffraction integral, thereby achieving fast and high-precision calculations. Summary of the Invention
[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a high-efficiency evaluation method for gravitational lensing magnification factor based on deep learning. This method utilizes the periodic activation function characteristics of the sinusoidal representation network to match the inherent oscillation characteristics of the diffraction integral kernel, thereby achieving fast and high-precision calculation of the magnification factor.
[0006] To achieve the above objectives, the present invention adopts the following technical solution:
[0007] A deep learning-based method for efficiently evaluating gravitational lensing magnification factors includes the following steps:
[0008] Step S1: Establish the diffraction integral model of the gravitational lensing system, define the dimensionless frequency and dimensionless source position, and transform the physical magnification factor into a universal magnification factor that depends only on the dimensionless parameters in order to achieve the scale invariance of the model.
[0009] Step S2: Obtain the model training dataset containing dimensionless parameters and their corresponding amplification factor true values;
[0010] Step S3: Construct a sinusoidal representation network as the core computational model, and use the dataset to learn and train the sinusoidal representation network;
[0011] Step S4: The frequency and parameters of the gravitational wave signal to be analyzed are converted into a dimensionless form, input into the trained sine representation network, and the corresponding complex amplification factor is directly output for the modulation and analysis of the gravitational wave waveform.
[0012] Furthermore, in step S2, for a specific lens model, the corresponding lens quality distribution function is substituted into the established diffraction integral model, and the true value dataset of the magnification factor is generated by numerical integration.
[0013] Furthermore, when the lens model is a point-mass lens model, the dimensionless frequency... Defined as Universal magnification factor It is obtained by integrating the geometric time delay function, and its integral expression is:
[0014]
[0015] In the formula, represents the dimensionless coordinates on the source plane; The dimensionless position on the source plane; The imaginary unit; For the redshift quality of the lens; This is the frequency of the gravitational wave.
[0016] Furthermore, in step S3, the sinusoidal representation network adopts a multilayer perceptron architecture, and the activation function of each neuron in each layer is a periodic sinusoidal function.
[0017] Furthermore, the sinusoidal activation function takes the form of: , This is the weight matrix. For bias vectors, This is the input vector of the neuron. This is a hyperparameter.
[0018] Furthermore, in step S3, the dimensionless parameters in the dataset are used as inputs, the corresponding amplification factors are used as labels, and the sinusoidal representation network is trained. The mean squared error is used as the loss function, and the network weights are optimized by combining an adaptive learning rate strategy.
[0019] Furthermore, it also includes a region decomposition strategy: dividing the parameter space into low-frequency and high-frequency regions;
[0020] In the low-frequency region, a sinusoidal representation network is used for full-wave optical calculations;
[0021] In the high-frequency region, if the prediction error of the sine representation network exceeds a preset threshold, it will automatically switch to the geometric optics approximation formula or the semiclassical approximation formula for calculation.
[0022] Furthermore, in step S4, to process the complex amplification factor, the output layer of the sine representation network is set to two channels to predict the real and imaginary parts of the amplification factor, or the amplitude and phase, respectively.
[0023] This invention also provides a high-efficiency evaluation system for gravitational lensing magnification factor based on deep learning, the system comprising:
[0024] The preprocessing module is used to convert the physical parameters in the gravitational wave detection data into dimensionless parameters;
[0025] The inference engine module is used to load the pre-trained sinusoidal representation model, perform millisecond-level forward inference on the input dimensionless parameters, and calculate the amplification factor.
[0026] The waveform generation module applies the frequency domain sequence of the amplification factor obtained through inference to the original gravitational wave waveform to generate a template waveform with lens effect, which is used for matched filter search or parameter estimation.
[0027] The beneficial effects of this invention are:
[0028] 1. Compared with traditional numerical integration, the neural network inference speed of this invention is improved by several orders of magnitude, which can meet the needs of real-time gravitational wave data analysis.
[0029] 2. On point-mass lenses and singular isothermal sphere models, the method of this invention can achieve O( The relative error is on the order of magnitude 1.5, which is sufficient to meet the accuracy requirements of current and future gravitational wave detectors.
[0030] 3. Compared to the large interpolation table, the neural network model file of this invention is extremely small, making it easy to distribute and deploy. Attached Figure Description
[0031] Figure 1 This is a flowchart of a deep learning-based method for efficiently evaluating the gravitational lensing magnification factor in this embodiment.
[0032] Figure 2 This is an architecture diagram of a sine representation network in this embodiment;
[0033] Figure 3 This is a structural framework diagram of a deep learning-based high-efficiency evaluation system for gravitational lensing magnification factor in this embodiment;
[0034] Figure 4 This is a prediction graph of the sine representation network in this embodiment;
[0035] Figure 5 This is a relative error distribution diagram of a model in the dimensionless frequency and source location parameter space in this embodiment.
[0036] Figure label: Preprocessing module 1, Inference engine module 2, Waveform generation module 3. Detailed Implementation
[0037] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0038] Example: An efficient evaluation method for gravitational lensing magnification factor based on deep learning, such as... Figure 1 As shown, it includes the following steps:
[0039] Step S1, Physical Model Construction and Dimensionlessness: Establish the diffraction integral model of the gravitational lensing system, define the dimensionless frequency and dimensionless source position, and transform the physical magnification factor into a universal magnification factor that depends only on the dimensionless parameters to achieve the scale invariance of the model.
[0040] This step aims to establish a diffraction integral model describing the gravitational lensing effect and to transform the magnification factor, which depends on physical parameters, into a general form that depends only on dimensionless parameters, thereby achieving scale invariance of the model and making it applicable to lenses of different masses and gravitational waves of different frequencies.
[0041] Specifically, within the framework of wave optics, the magnification factor of gravitational lensing. By defining Kirchhoff's diffraction integral, its general form is:
[0042]
[0043] In the formula, The frequency of gravitational waves; This is for lens redshift; For arrival time delay function; represents the dimensionless coordinates on the source plane; The dimensionless position on the source plane; It is the imaginary unit.
[0044] To achieve scale invariance of the model, the physical parameters are transformed into dimensionless parameters, and dimensionless frequencies are defined. and dimensionless source position Physical amplification factor Transformed into a universal amplification factor that depends only on dimensionless parameters .
[0045] Taking the point mass lens model as an example, dimensionless frequency Defined as ,in, The redshift mass of the lens is expressed as:
[0046]
[0047] In the formula, This refers to the inherent mass of the lens.
[0048] At this time, the amplification factor It can be obtained by integrating the geometric time delay function, and its integral expression is:
[0049]
[0050] By converting physical parameters (frequency, lens quality, position) into dimensionless parameters This allows the trained model to have scale invariance, meaning that a model can be applied to lenses of arbitrary mass and gravitational waves of arbitrary frequency, greatly improving the model's versatility.
[0051] Step S2, Data Acquisition: Acquire the dataset for model training. The dataset contains dimensionless parameters and their corresponding amplification factor ground truth values.
[0052] This step aims to generate data. For a specific lens model (such as a point mass lens (PML) or a singular isothermal sphere (SIS), the corresponding lens mass distribution function is substituted into the diffraction integral model established in step S1. Using a high-precision numerical integration method, a large number of sample points are generated in the dimensionless parameter space and a dataset is constructed, providing a foundation for subsequent network training.
[0053] Step S3, Neural Network Construction and Training: Construct a Sinusoidal Representation (SIREN) network as the core computational model, and use the dataset to learn and train the Sinusoidal Representation network.
[0054] Unlike traditional ReLU networks, this embodiment uses a sine function as the activation function for neurons. Since the diffraction integral is essentially the superposition of wave interference, its mathematical form has a high degree of periodicity and oscillation. The sine function, representing the periodic activation function of the network, perfectly matches this physical characteristic and can accurately capture high-frequency diffraction fringe features.
[0055] Specifically, such as Figure 2As shown, the sine representation network adopts a multilayer perceptron architecture, including an input layer, multiple hidden layers, and an output layer. The activation function of each neuron in each layer is a periodic sine function. For the output of the l-th layer, its expression is:
[0056]
[0057] In the formula, Here is the weight matrix for the l-th layer; Let l be the bias vector of the l-th layer; The input vector of the neuron, i.e., the dimensionless frequency. and dimensionless position ; This is a hyperparameter (e.g., set to 30) used to control the frequency of the sine function so that it can match the high-frequency oscillation characteristics in the diffraction integral.
[0058] By utilizing a multilayer perceptron architecture, the network possesses a strong inductive bias, enabling it to effectively fit the high-frequency phase interference characteristics in diffraction integrals.
[0059] During model training, the dimensionless parameters in the generated dataset are used as inputs, and the corresponding amplification factors are used as labels to supervise the training of the sinusoidal representation network. The mean squared error is used as the loss function, and the network weights are optimized by combining an adaptive learning rate strategy.
[0060] Step S4, Rapid Evaluation: The frequency and parameters of the gravitational wave signal to be analyzed are converted into dimensionless form and input into a trained sine wave representation network. The network directly outputs the corresponding complex amplification factor. This complex amplification factor can be used for subsequent modulation and analysis of the gravitational wave waveform, such as generating template waveforms with lensing effects for matched filter search or parameter estimation.
[0061] Meanwhile, to handle complex amplification factors, the sinusoidal representation network output layer is set to two channels to predict the real and imaginary parts of the amplification factor, or the amplitude and phase, respectively.
[0062] Furthermore, to improve the prediction accuracy in the high-frequency region, the method proposed in this embodiment also introduces a region decomposition strategy: dividing the parameter space into low-frequency and high-frequency regions;
[0063] In the low-frequency region, a sinusoidal representation network is used for full-wave optical calculations;
[0064] In the high-frequency region (i.e., dimensionless frequency) If the prediction error of the sine representation network exceeds the preset threshold, it will automatically switch to the geometric optics approximation formula or the semiclassical approximation formula for calculation in order to suppress extrapolation errors at high frequencies.
[0065] Figure 4 This demonstrates the amplification factor predicted by the sine representation network and the degree of agreement between it and its theoretical value, as well as the error level. Figure 4 It can be seen that the dimensionless frequency Amplification factor ranging from 0.01 to 5.0 The predicted values remained stable, with the relative error remaining stable. In all All points remain at extremely low levels. With other parameters fixed (such as source location)... When a certain value is taken, the network can accurately predict the amplification factor over a wide frequency range. Since the error between the predicted value and the theoretical value is extremely small, the network has good frequency generalization ability and numerical accuracy. It can effectively fit the high-frequency oscillation characteristics in the diffraction integral and solve the problem of the decrease in accuracy of traditional methods in the high-frequency region.
[0066] This implementation also provides a high-efficiency evaluation system for gravitational lensing magnification factors based on deep learning, such as... Figure 3 As shown, the system includes a preprocessing module 1, an inference engine module 2, and a waveform generation module 3.
[0067] Among them, the preprocessing module 1 is responsible for converting the physical parameters in the gravitational wave detection data into dimensionless parameters;
[0068] The inference engine module 2 is responsible for loading the pre-trained sinusoidal representation model, performing millisecond-level forward inference on the input dimensionless parameters, and calculating the amplification factor.
[0069] Waveform generation module 3 is responsible for applying the frequency domain sequence of the amplification factor obtained from the inference to the original gravitational wave waveform to generate a template waveform with lens effect, which is used for matched filter search or parameter estimation.
[0070] The following example uses a point mass lens (PML) model to verify the effectiveness of the method in this embodiment:
[0071] Data preparation: at dimensionless frequencies and source location Within the range, generate using a high-precision numerical integration library 100 sample points were used as the training set.
[0072] Network setup: Construct a sinusoidal representation network with 5 hidden layers and 256 neurons per layer. The input layer of this network is 2D. The output layer is 2-dimensional (real part and imaginary part).
[0073] Training process: Using the Adam optimizer, the learning rate is set to... , 100 rounds of training.
[0074] Result validation: Evaluate on the test set, such as... Figure 5 As shown, in and Within the parameter space, the sine indicates that the relative error distribution of the network is uniform and remains at a low level.
[0075] The above description is merely a preferred embodiment of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should also be considered within the scope of protection of the present invention.
Claims
1. A method for efficiently evaluating the gravitational lensing magnification factor based on deep learning, characterized in that, Includes the following steps: Step S1: Establish the diffraction integral model of the gravitational lensing system, define the dimensionless frequency and dimensionless source position, and transform the physical magnification factor into a universal magnification factor that depends only on the dimensionless parameters in order to achieve the scale invariance of the model. Step S2: Obtain the model training dataset containing dimensionless parameters and their corresponding amplification factor true values; Step S3: Construct a sinusoidal representation network as the core computational model, and use the dataset to learn and train the sinusoidal representation network; Step S4: Convert the frequency and parameters of the gravitational wave signal to be analyzed into a dimensionless form, input it into the trained sine representation network, and directly output the corresponding complex amplification factor for the modulation and analysis of the gravitational wave waveform. In step S1, the lens model is a point mass lens model with dimensionless frequency. Defined as , For the redshift quality of the lens; The frequency of gravitational waves; In step S3, the sinusoidal representation network adopts a multilayer perceptron architecture, and the activation function of each neuron in each layer is a periodic sinusoidal function; The form of the sinusoidal activation function is: , This is the weight matrix. For bias vectors, This is the input vector of the neuron. This is a hyperparameter.
2. The efficient evaluation method for gravitational lensing magnification factor based on deep learning according to claim 1, characterized in that, In step S2, for a specific lens model, the corresponding lens quality distribution function is substituted into the established diffraction integral model, and the true value dataset of the magnification factor is generated by numerical integration.
3. The efficient evaluation method for gravitational lensing magnification factor based on deep learning according to claim 2, characterized in that, Universal magnification factor of point mass lens model It is obtained by integrating the geometric time delay function, and its integral expression is: In the formula, represents the dimensionless coordinates on the source plane; The dimensionless position on the source plane; The imaginary unit; For the redshift quality of the lens; This is the frequency of the gravitational wave.
4. The efficient evaluation method for gravitational lensing magnification factor based on deep learning according to claim 1, characterized in that, In step S3, the dimensionless parameters in the dataset are used as inputs, the corresponding amplification factors are used as labels, and the sinusoidal representation network is trained. The mean squared error is used as the loss function, and the network weights are optimized by combining an adaptive learning rate strategy.
5. The efficient evaluation method for gravitational lensing magnification factor based on deep learning according to claim 1, characterized in that, It also includes a region decomposition strategy: dividing the parameter space into low-frequency and high-frequency regions; In the low-frequency region, a sinusoidal representation network is used for full-wave optical calculations; In the high-frequency region, if the prediction error of the sinusoidal representation network exceeds a preset threshold, it will automatically switch to the geometrical optics approximation formula or the semiclassical approximation formula for calculation.
6. The efficient evaluation method for gravitational lensing magnification factor based on deep learning according to claim 1, characterized in that, In step S4, to process the complex amplification factor, the output layer of the sine representation network is set to two channels to predict the real and imaginary parts of the amplification factor, or the amplitude and phase, respectively.
7. A high-efficiency evaluation system for gravitational lensing magnification factor based on deep learning for implementing the method of claim 1, characterized in that, The system includes: The preprocessing module (1) is used to convert the physical parameters in the gravitational wave detection data into dimensionless parameters; The inference engine module (2) is used to load the pre-trained sinusoidal representation model, perform millisecond-level forward inference on the input dimensionless parameters, and calculate the amplification factor. The waveform generation module (3) applies the frequency domain sequence of the amplification factor obtained by reasoning to the original gravitational wave waveform to generate a template waveform with lens effect, which is used for matched filter search or parameter estimation.