A Deep Learning-Based Super-Resolution Topological Method for Sparse Holographic Sound Pressure Fields

Abstract

The application discloses a kind of based on deep learning's sparse holographic surface sound pressure field super-resolution topology method, for the pain point problem of insufficient holographic surface sound pressure resolution in the actual sparse sampling test of large structure acoustics and low precision of traditional interpolation reconstruction, with low-resolution holographic surface sound pressure under sparse sampling as input, high-resolution sound pressure field is obtained by feature construction, depth network topology;At the same time, the physical consistency loss based on surface Helmholtz equation is introduced, so that the network balances between numerical accuracy and physical rationality.The application improves the accuracy and robustness of holographic surface sound pressure topology compared with the traditional interpolation method, and reduces the implementation cost of large structure sound holographic measurement.

Description

Technical Field

[0001] This invention belongs to the field of acoustic measurement and sound field reconstruction technology, and more specifically, it is a super-resolution topological method for sparse holographic surface sound pressure field based on deep learning, which is particularly suitable for high-precision reconstruction of sound field in acoustic measurement of large structures. Background Technology

[0002] In the field of modern acoustic engineering, achieving high-precision reconstruction of radiated sound fields is crucial for noise control of large structures such as submarine hulls, aircraft fuselages, and high-speed train bodies. This is of great significance for reducing environmental noise pollution, improving the stealth performance of military equipment, and ensuring structural health monitoring. However, with the increasing scale of engineering objects and the rise in analysis frequencies, acoustic measurement-based sound field reconstruction techniques, such as classical near-field acoustic holography, face severe physical and engineering bottlenecks.

[0003] In acoustic measurement, the spatial Nyquist sampling theorem requires that the spacing between measurement points be less than half a wavelength. For large structures tens of meters long, meeting this condition in the mid-to-high frequency band means deploying tens of thousands or even hundreds of thousands of microphone measurement points. Constructing such a massive array presents extremely high challenges in terms of hardware cost, channel calibration, data transmission, and on-site implementation. Therefore, in actual measurements of large structures, holographic surfaces can usually only be sparsely arranged, leading to high-energy artifacts in the subsequent reconstruction results and severely reducing the accuracy of sound source identification and sound field reconstruction.

[0004] To address the insufficient resolution of holographic surfaces under sparse measurement conditions, existing techniques typically employ linear interpolation, least-squares fitting, and spline interpolation to perform topological completion on sparse measurement data. These methods are applicable to low-frequency ranges or situations where sound field spatial variations are relatively gentle. However, as frequency increases, sound field spatial oscillations intensify, and the ability of sparse measurement points to represent the details of the true sound field decreases sharply. In this case, traditional interpolation methods rely solely on the smoothing assumption between adjacent measurement points for numerical fitting, failing to incorporate the physical mechanisms of sound field propagation and spatial structural characteristics. This makes them more prone to problems such as over-smoothing of the reconstructed sound field, amplification of local artifacts, and unstable results, thus limiting the accuracy and robustness of sound field reconstruction under sparse measurement conditions for large structures. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, this invention provides a deep learning-based super-resolution topology method for sparse holographic surface sound pressure fields. Addressing the pain points of insufficient holographic surface sound pressure resolution and low accuracy of traditional interpolation reconstruction in practical sparse sampling tests of large-structure acoustics, this invention uses low-resolution holographic surface sound pressure as input, and obtains a high-resolution sound pressure field through feature construction and deep network topology. Simultaneously, a physical consistency loss based on the surface Helmholtz equation is introduced to achieve a balance between numerical accuracy and physical plausibility in the network.

[0006] The present invention adopts the following technical solution to solve the technical problem: The features of the deep learning-based super-resolution topology method for sparse holographic surface sound pressure fields in this invention are as follows: For the acquisition of sound pressure from large-structure holographic surfaces, it takes low-resolution holographic surface sound pressure under sparse sampling as input, captures local gradient details of the sound field through geometrically perceptual convolution, performs global modeling in the frequency domain using Fourier neural operators, and introduces physical consistency loss based on the surface Helmholtz equation to achieve deep integration of data-driven and physical layers, so that the network achieves a balance between numerical accuracy and physical rationality, and realizes super-resolution topology of sound pressure fields from large-structure holographic surfaces.

[0007] The super-resolution topology method for sparse holographic acoustic pressure fields based on deep learning in this invention is also characterized by the following: the large-scale acoustic holographic surface is formed by circular array scanning, and the holographic surface is cylindrical in shape. The super-resolution topology method for acoustic pressure fields of cylindrical holographic surfaces is carried out according to the following steps: Step a: Apply low-resolution complex sound pressure levels to the cylindrical holographic surface. and high-resolution complex sound pressure Represented by equation (1): (1); In formula (1): by Represents the field of complex numbers; by and These represent low resolution and high resolution, respectively. by and These represent the number of discrete sampling points along the axial and circumferential directions of the cylinder, respectively. and These represent the number of discrete sampling points along the axial and circumferential directions of the cylinder at low resolution, respectively. and These represent the number of discrete sampling points along the axial and circumferential directions of the cylinder under high resolution, respectively. Normalization is performed according to equation (2) to obtain the normalized low-resolution complex sound pressure levels. and normalized high-resolution complex sound pressure ; (2); In equation (2): normalized scaling factor Calculated from equation (3): (3); In formula (3): Represents the 95th percentile; The normalized low-resolution complex sound pressure level and normalized high-resolution complex sound pressure The dual-channel low-resolution sound pressure and dual-channel high-resolution sound pressure are represented by equation (4) in the form of real and imaginary parts: (4); In equation (4): by Dual-channel low-resolution sound pressure level, representing both real and imaginary parts; by Dual-channel high-resolution sound pressure level displaying real and imaginary parts; by Indicate the real part, with Indicates the imaginary part; Step b: To simultaneously characterize the periodic geometric features, local detailed structure, long-distance spatial coupling relationships, and response changes under different operating conditions of the cylindrical unfolded surface, local convolutional feature encoding and frequency domain global modeling are performed on a sparse sampling grid; the dual-channel low-resolution sound pressure level is then converted into a single signal. and dual-channel high-resolution sound pressure Unified as input features ; In the local convolutional feature encoding module, the input features are... Convolutional feature extraction is performed, and local convolutional features are calculated using equation (5). : (5); In equation (5): by Represents geometrically perceptual convolution The local enhancement module consists of GELU activation, channel attention, and residual connections; the geometrically perceptual convolution... Circular filling is used in the circumferential direction, and reflective filling is used in the axial direction, as characterized by equation (6): (6); In formula (6): by Indicates circumferential expansion; with Indicates axial reflection extension; with Represents convolution; In the frequency domain global modeling module, a frequency domain spectral convolution branch is introduced to enhance the global modeling capability, and the frequency domain features are calculated by equation (7). : (7); In formula (7): Represents low-modal spectral convolution in a two-dimensional Fourier domain, with Represents a pointwise linear mapping; The fusion features are obtained by adaptive fusion using equation (8). : (8); In formula (8): This represents element-wise multiplication. It is obtained by calculation from equation (9): (9); In equation (9): Indicates by Gated mappings constructed using convolutions; Indicates Sigmoid activation; Step c: Introduce a conditional feature modulation mechanism for encoding: For the input features Global average pooling is performed, and the style vector represented by Equation (10) is extracted via a multilayer perceptron. : (10); In formula (10): This represents the style extraction mapping function composed of a multilayer perceptron; The normalized physical frequency is calculated using equation (11). Physical frequency The normalized physical frequency is calculated using equation (11). , (11); In formula (11): Representing physical frequency, in Represents the minimum physical frequency, in Indicates the maximum physical frequency; The normalized scaling factor obtained from equation (3) Mapping to the logarithmic field and constructing the external condition vector represented by equation (12). c : (12); In equation (12): The scaling factor is the logarithmic normalization factor. The style vector With external condition vector By splicing the equations together, we obtain the complete condition represented by equation (13). : (13); Complete conditions Channel-by-channel modulation parameters are generated by mapping according to equation (14). : (14); In formula (14): This indicates that the conditional mapping function consists of three fully connected layers and two GELU activation functions; Utilizing fusion features The low-resolution feature map is obtained from equation (15). , (15); To enable the network to explicitly perceive the geometric topology of the cylindrical surface, coordinate features are constructed on the low-resolution and high-resolution meshes respectively according to equation (16). : (16); In formula (16): Represents the normalized axial coordinates, with Represents the circumferential angular coordinates; The low-resolution feature map Flatten into a token sequence and extract coordinate features After linear mapping, the result is superimposed onto the representation of the token sequence to obtain the initial token sequence represented by equation (17). : (17); In equation (17): by This represents flattening a two-dimensional feature map and mapping it to a sequence representation; by This represents the coordinate feature embedding mapping function consisting of a single fully connected layer; Global feature updates are performed via multi-head self-attention and a feedforward network according to equations (18) and (19): (18); (19); In equations (18) and (19): by Representation layer normalization, to This represents multi-head self-attention computation; by This represents a feedforward network; T1 and T2 represent the transition and output token sequences, respectively. The result obtained from equation (19) Rearranged into two-dimensional feature maps As in equation (20): (20); In formula (20): This represents restoring the sequence form to a two-dimensional feature map; Step d: Obtain the high-resolution complex acoustic pressure field of a large-scale holographic surface through high-resolution decoding; Step e: Introduce physical consistency loss based on the surface Helmholtz equation to update the high-resolution sound pressure field, and complete the super-resolution topology of the sparse holographic surface sound pressure field.

[0008] The feature of the deep learning-based sparse holographic surface acoustic pressure field super-resolution topology method of this invention is that, in step d, high-resolution decoding is performed as follows: First, the low-resolution two-dimensional feature map Interpolate to the target high-resolution mesh to obtain high-resolution global features. As shown in equation (21): (twenty one); In equation (21): This indicates an interpolation upsampling operation; Then normalize the low-resolution complex sound pressure level. Direct interpolation to a high-resolution mesh as a high-resolution substrate B The coordinate features on the high-resolution grid are denoted as... The complete condition derived from equation (13) Equation (22) is mapped to a constant condition graph. And extend to high-resolution meshes; (twenty two); In equation (22): by This represents a conditional graph mapping function consisting of two fully connected layers and a GELU activation function; by This indicates replication and expansion along the axial and circumferential directions of the high-resolution mesh; Then, the decoder's initial input Characterized by equation (23): (twenty three); Initialize the decoder input Input geometry-aware convolution Together with the residual refinement module, the decoding features are obtained. As shown in equation (24): (twenty four); In formula (24): This represents a series of geometrically perceptive convolutional layers. High-resolution decoding mapping consisting of residual refinement modules and nonlinear activation functions; The decoded features After passing through the residual prediction head and the gated prediction head respectively, the residual terms characterized by equations (25) and (26) are obtained. and gated graph : (25); (26); In equations (25) and (26): Representing the residual prediction mapping, in Represents the gating prediction mapping; Gating diagram For residual terms Perform element-wise multiplication and with a high-resolution substrate Adding them together yields the high-resolution complex acoustic pressure field prediction result. As shown in equation (27): (27); This enables high-resolution complex acoustic pressure field decoding of large-scale structural holographic surfaces.

[0009] The characteristic of the deep learning-based super-resolution topological method for sparse holographic surface acoustic pressure fields in this invention also lies in the fact that step e is performed in the following manner: For a cylindrical holographic surface, the coordinates of the circumferential arc length are denoted as... , mark the axial coordinates as In the passive region, the surface complex sound pressure Satisfying equation (28): (28); In equation (28): k For wave number, , Speed ​​of sound; In discrete implementation, the circumferential second derivative Using the cyclic difference shown in equation (29), the axial second derivative The reflection boundary difference is adopted as shown in equation (30): (29); (30); in: by and These represent discrete measurement points along the circumferential and axial directions of the cylindrical surface, respectively. by Indicates circumference Point and axis Complex sound pressure level at the point; by Indicates circumference Point and axis Complex sound pressure level at the point; by Indicates circumference Point and axis Complex sound pressure level at the point; by Indicates circumference Point and axis Complex sound pressure level at the point; by Indicates circumference Point and axis Complex sound pressure level at the point; Based on this, a discrete residual is constructed. As in equation (31): (31); To balance the differences in residual magnitudes at different frequencies and grid scales, normalization is performed using equation (32) to obtain the normalized discrete residuals. : (32); Then define physical loss As in equation (33): (33); In formula (32): Indicates the total number of discrete points; Ultimately utilize physical loss Update the high-resolution sound pressure field and complete the super-resolution topology of the sparse holographic surface sound pressure field.

[0010] Compared with existing technologies, the beneficial effects of this invention are reflected in: 1. This invention takes the sound pressure of a low-resolution cylindrical holographic surface under sparse sampling as input, and uses deep learning holographic surface sound pressure topology technology to combine the circumferential periodicity, axial boundary features and broadband condition information of the cylindrical surface to achieve accurate completion of the high-resolution holographic surface sound pressure field. Compared with traditional linear interpolation, least squares interpolation and spline interpolation methods, it can more effectively restore the spatial details and phase distribution of the sound field and improve the accuracy of holographic surface reconstruction under complex sound field conditions.

[0011] 2. Based on preserving the original observation information, this invention introduces the surface Helmholtz equation constraint to impose physical consistency constraints on the reconstruction process, effectively reducing non-physical problems, local artifacts and high-frequency distortion caused by sparse measurements, and improving the stability and reliability of high-resolution sound pressure field reconstruction results. Attached Figure Description

[0012] Figure 1 This invention relates to the spatial distribution of the cylindrical holographic surface and the object under test.

[0013] Figure 2 This invention illustrates the spatial distribution of the cylindrical holographic surface at different resolutions.

[0014] Figure 3 The image shows a comparison of the reconstruction results of the holographic surface sound pressure by the present invention and three traditional interpolation methods at 700 Hz.

[0015] Figure 4 The image shows a comparison of the reconstruction results of the holographic surface sound pressure by the present invention and three traditional interpolation methods at 1000 Hz.

[0016] Figure 5 The relative error curves are for the reconstruction results of each method within the frequency band from 100 Hz to 1000 Hz.

[0017] Figure 6 The absolute error curves for the reconstruction results of each method within the frequency band from 100 Hz to 1000 Hz are shown. Detailed Implementation

[0018] In this embodiment, the super-resolution topology method for sparse holographic surface sound pressure field based on deep learning is as follows: For the acquisition of sound pressure from large-structure holographic surfaces, the low-resolution holographic surface sound pressure under sparse sampling is used as input. Geometric perceptual convolution is used to capture local gradient details of the sound field, Fourier neural operators are used for frequency domain global modeling, and physical consistency loss based on the surface Helmholtz equation is introduced to achieve deep integration of data-driven and physical layers. This allows the network to achieve a balance between numerical accuracy and physical rationality, thus realizing the super-resolution topology of the sound pressure field of large-structure holographic surfaces.

[0019] In this embodiment, the large-scale acoustic holographic surface is formed by circular array scanning, and the holographic surface is cylindrical in shape. The super-resolution topological method for the sound pressure field of the cylindrical holographic surface is performed according to the following steps: Step a: Apply low-resolution complex sound pressure levels to the cylindrical holographic surface. and high-resolution complex sound pressure Represented by equation (1): (1); In formula (1): by Represents the field of complex numbers; and These represent low resolution and high resolution, respectively. by and These represent the number of discrete sampling points along the axial and circumferential directions of the cylinder, respectively. and These represent the number of discrete sampling points along the axial and circumferential directions of the cylinder at low resolution, respectively. and These represent the number of discrete sampling points along the axial and circumferential directions of the cylinder, respectively, at high resolution.

[0020] Since the sound pressure amplitude varies greatly at different excitation locations and frequencies, this embodiment normalizes the sound pressure according to equation (2) to obtain normalized low-resolution complex sound pressure levels. and normalized high-resolution complex sound pressure ; (2); In equation (2): normalized scaling factor Calculated from equation (3): (3); In formula (3): Represents the 95th percentile; Normalized low-resolution complex sound pressure and normalized high-resolution complex sound pressure The dual-channel low-resolution sound pressure and dual-channel high-resolution sound pressure are represented by equation (4) in the form of real and imaginary parts: (4); In formula (4): Represents dual-channel low-resolution sound pressure levels in both real and imaginary form; Representing dual-channel high-resolution sound pressure levels in both real and imaginary form; Indicate the real part, with Indicates the imaginary part.

[0021] Step b: To simultaneously characterize the periodic geometric features, local detailed structure, long-distance spatial coupling relationships, and response changes under different operating conditions of the cylindrical unfolded surface, local convolutional feature encoding and frequency domain global modeling are performed on a sparse sampling grid; the dual-channel low-resolution sound pressure level is then converted into a single signal. and dual-channel high-resolution sound pressure Unified as input features .

[0022] In the local convolutional feature encoding module, the input features are... Convolutional feature extraction is performed, and local convolutional features are calculated using equation (5). : (5); In formula (5): Represents geometrically perceptual convolution Local enhancement modules consisting of GELU activation, channel attention, and residual connections; geometrically perceptive convolution. Circular filling is used in the circumferential direction, and reflective filling is used in the axial direction, as characterized by equation (6): (6); In formula (6): by Indicates circumferential expansion; with Indicates axial reflection extension; with This indicates convolution; this branch is mainly responsible for extracting high-frequency details such as interference fringes, local energy abrupt changes, and rapid phase changes.

[0023] In the frequency domain global modeling module, a frequency domain spectral convolution branch is introduced to enhance the global modeling capability, and the frequency domain features are calculated by equation (7). : (7); In formula (7): Represents low-modal spectral convolution in a two-dimensional Fourier domain, with Represents a pointwise linear mapping; The fusion features are obtained by adaptive fusion using equation (8). : (8); In formula (8): This represents element-wise multiplication. It is obtained by calculation from equation (9): (9); In equation (9): Indicates by Gated mappings constructed using convolutions; This indicates Sigmoid activation; through this gating fusion, the network can achieve an adaptive trade-off between local detail modeling and global frequency domain modeling based on spatial location.

[0024] Step c: Introduce a conditional feature modulation mechanism for encoding: For input features Global average pooling is performed, and the style vector represented by Equation (10) is extracted via a multilayer perceptron. : (10); In formula (10): This represents the style extraction mapping function composed of a multilayer perceptron; The normalized physical frequency is calculated using equation (11). Physical frequency The normalized physical frequency is calculated using equation (11). : (11); In formula (11): Representing physical frequency, in Represents the minimum physical frequency, in Indicates the maximum physical frequency; The normalized scaling factor obtained from equation (3) Mapping to the logarithmic field and constructing the external condition vector represented by equation (12). c : (12); In equation (12): The scaling factor is the logarithmic normalization factor. style vectors With external condition vector By splicing the equations together, we obtain the complete condition represented by equation (13). : (13); Complete conditions Channel-by-channel modulation parameters are generated by mapping according to equation (14). : (14); In formula (14): This indicates that the conditional mapping function consists of three fully connected layers and two GELU activation functions; Utilizing fusion features The low-resolution feature map is obtained from equation (15). , (15); To enable the network to explicitly perceive the geometric topology of the cylindrical surface, coordinate features are constructed on the low-resolution and high-resolution meshes respectively according to equation (16). : (16); In formula (16): Represents the normalized axial coordinates, with Represents the circumferential angular coordinates; low-resolution feature maps Flatten into a token sequence and extract coordinate features After linear mapping, the result is superimposed onto the representation of the token sequence to obtain the initial token sequence represented by equation (17). : (17); In formula (17): This represents flattening a two-dimensional feature map and mapping it to a sequence representation; by This represents the coordinate feature embedding mapping function consisting of a single fully connected layer; Global feature updates are performed via multi-head self-attention and a feedforward network according to equations (18) and (19): (18); (19); In equations (18) and (19): by Representation layer normalization, to Indicates multi-head self-attention operation, Indicates a feedforward network; Let T1 and T2 represent the token sequences for the transition and output, respectively; The result obtained from equation (19) Rearranged into two-dimensional feature maps As in equation (20): (20); In formula (20): This means restoring the sequence to a two-dimensional feature map.

[0025] Step d: Perform high-resolution decoding as follows to obtain the high-resolution complex acoustic pressure field of the large-scale holographic surface; Low-resolution two-dimensional feature maps Interpolate to the target high-resolution mesh to obtain high-resolution global features. As shown in equation (21): (twenty one); In equation (21): This indicates an interpolation upsampling operation; Normalized low-resolution complex sound pressure Direct interpolation to a high-resolution mesh as a high-resolution substrate B The coordinate features on the high-resolution grid are denoted as... The complete condition derived from equation (13) Equation (22) is mapped to a constant condition graph. And extend to high-resolution meshes; (twenty two); In equation (22): by This represents a conditional graph mapping function consisting of two fully connected layers and a GELU activation function; by This indicates replication and expansion along the axial and circumferential directions of the high-resolution mesh; Then, the decoder's initial input Characterized by equation (23): (twenty three); Then initialize the decoder input. Input geometry-aware convolution Together with the residual refinement module, the decoding features are obtained. As shown in equation (24): (twenty four); In formula (24): This represents a series of geometrically perceptive convolutional layers. High-resolution decoding mapping is composed of residual refinement modules and nonlinear activation functions.

[0026] Decoding features After passing through the residual prediction head and the gated prediction head respectively, the residual terms characterized by equations (25) and (26) are obtained. and gated graph : (25); (26); In equations (25) and (26): Representing the residual prediction mapping, in Represents the gating prediction mapping; Gating diagram For residual terms Perform element-wise multiplication and with a high-resolution substrate Adding them together yields the high-resolution complex acoustic pressure field prediction result. As shown in equation (27): (27); This enables high-resolution complex acoustic pressure field decoding of large-scale structural holographic surfaces.

[0027] Step e: Introduce the physical consistency loss based on the surface Helmholtz equation to update the high-resolution acoustic pressure field in the following manner to complete the super-resolution topology of the sparse holographic surface acoustic pressure field.

[0028] To ensure that the network output not only approximates the real high-resolution sound field in terms of data but also satisfies the basic wave laws, this embodiment introduces physical consistency constraints based on the surface Helmholtz equation during the training process. For a cylindrical holographic surface, the coordinates of the circumferential arc length are denoted as... , mark the axial coordinates as In the passive region, the surface complex sound pressure Satisfying equation (28): (28); In equation (28): k For wave number, , Speed ​​of sound; In discrete implementation, the circumferential second derivative Using the cyclic difference shown in equation (29), the axial second derivative The reflection boundary difference is adopted as shown in equation (30): (29); (30); In equations (29) and (30): and These represent discrete measurement points on the circumferential and axial directions of the cylindrical surface, respectively; with Indicates circumference Point and axis Complex sound pressure level at point; Indicates circumference Point and axis Complex sound pressure level at point; Indicates circumference Point and axis Complex sound pressure level at point; Indicates circumference Point and axis Complex sound pressure level at point; Indicates circumference Point and axis Complex sound pressure level at the point; Based on this, a discrete residual is constructed. As in equation (31): (31); To balance the differences in residual magnitudes at different frequencies and grid scales, normalization is performed using equation (32) to obtain the normalized discrete residuals. : (32); Then define physical loss As in equation (33): (33); In formula (32): Indicates the total number of discrete points; Ultimately utilize physical loss Update the high-resolution sound pressure field and complete the super-resolution topology of the sparse holographic surface sound pressure field.

[0029] The super-resolution topological method for sparse holographic acoustic pressure fields based on deep learning in this invention is verified as follows: Simulation Verification Design: To verify the generalization ability of the proposed method under unknown excitation conditions, a cylindrical shell radiating sound field model was established in the air domain using COMSOL, designing six sets of conditions with different excitation positions. Five of these conditions were used to generate the model training set, and the remaining set served as the verification set. During the verification phase, only the low-resolution (LR) holographic sound pressure data of the verification condition was input, and the trained deep network model predicted the corresponding high-resolution (HR) holographic sound pressure field. The simulation frequency range was 100 Hz–1000 Hz, with a frequency step size of 1 Hz.

[0030] Simulation Model: The cylindrical shell model used in the simulation has a radius of 0.15 m, a length of 1.5 m, a wall thickness of 0.0025 m, and is made of Q235 structural steel with a density of 7850 kg / m³, a Young's modulus of 200 GPa, and a Poisson's ratio of 0.28. The cylindrical shell model is established in a rectangular coordinate system, with the origin located at the center of the shell, and the cylinder axis coinciding with the x-axis. Figure 1 As shown. To obtain model training data, boundary loads with an amplitude of 1 N were applied at five different locations on the shell, with excitation frequencies ranging from 100 Hz to 1000 Hz. The same excitation signal was also applied at a different location on the shell for model generalization verification.

[0031] Figure 2 The figure shows the spatial distribution of the cylindrical holographic surface under different resolutions of the present invention. The holographic surface parameters are as follows: the holographic surface is set as a cylindrical surface coaxial with the cylindrical shell, with a radius of 0.185 m and a length of 1.7 m. Among them, the low-resolution holographic array (sparse sampling) shown in Figure (2a) has a total of 96 measurement points, of which 12 measurement points are evenly distributed in the circumference with an angular interval of 30°, and 8 measurement points are evenly distributed in the axis with a spacing of 0.2429 m. The high-resolution holographic array (non-sparse sampling) shown in Figure (2b) has a total of 768 measurement points, of which 24 measurement points are evenly distributed in the circumference with an angular interval of 15°, and 32 measurement points are evenly distributed in the axis with a spacing of 0.0548 m.

[0032] To conduct quantitative evaluation, both relative error and absolute error indicators are used. The amplitudes of both predicted and actual sound pressure levels are converted into sound pressure levels, which are defined as: ; in, This is the complex sound pressure level. The reference sound pressure is used. For air, the standard reference sound pressure is typically taken as 20 μPa. The average relative error between the predicted and actual complex sound pressure fields is defined as: ; in, The predicted sound pressure level is obtained from a model or interpolation method. This corresponds to the actual sound pressure level. This represents the L2 norm of the complex sound pressure level calculation for all discrete measurement points. This dimensionless index reflects the proportion of the prediction error to the total energy of the true sound field; a smaller value indicates that the reconstructed result is closer to the true sound field. Furthermore, to characterize the sound pressure level deviation from an engineering application perspective, the mean absolute error is defined as: ; Figure 3 The figure shows a comparison of the reconstruction results of the holographic surface sound pressure by the present invention and three traditional interpolation methods at 700 Hz. Among them, Figure (3a) shows the low-resolution holographic surface sound pressure distribution, Figure (3b) shows the true value distribution of holographic surface sound pressure under non-sparse sampling, and Figures (3c), (3d), (3e) and (3f) show the holographic surface sound pressure data reconstructed by the present invention, linear interpolation, least squares fitting and spline interpolation, respectively.

[0033] Figure 4 The figure shows a comparison of the reconstruction results of the holographic surface sound pressure by the present invention and three traditional interpolation methods at 1000 Hz. Among them, Figure (4a) shows the low-resolution holographic surface sound pressure distribution, Figure (4b) shows the true value distribution of holographic surface sound pressure under non-sparse sampling, and Figures (4c), (4d), (4e) and (4f) show the holographic surface sound pressure data reconstructed by the present invention, linear interpolation, least squares fitting and spline interpolation, respectively.

[0034] Figure 5 The relative error curves of the reconstruction results of each method are shown in the frequency band from 100 Hz to 1000 Hz. Figure 6 The absolute error curves of the reconstruction results of each method in the frequency band from 100 Hz to 1000 Hz are shown. Figure 5 and Figure 6 In the diagram, curves a1, a2, a3, and a4 correspond one-to-one to linear interpolation, least squares method, spline interpolation, and the method proposed in this invention.

[0035] The results show that traditional interpolation methods are ineffective in recovering high-resolution details of complex sound fields: linear interpolation has the most significant error; while least squares reconstruction results are smoother, they exhibit significant distortion in rapidly changing local regions of the sound field; spline interpolation has some fitting ability in the low-frequency range, but the error gradually increases with frequency. This is because the spatial changes of the sound field intensify with increasing frequency, sparse measurement points cannot fully characterize the sound field details, and traditional methods lack global feature modeling and physical constraints, making accurate completion difficult. In contrast, the method of this invention maintains a relative reconstruction error within 10% and an absolute error within 1 dB across the entire frequency range of 100 Hz-1000 Hz, demonstrating significantly superior high-resolution reconstruction accuracy compared to traditional baseline methods.

[0036] The simulation results fully verify that the super-resolution topological method for sparse holographic acoustic pressure field based on deep learning in this invention can achieve accurate reconstruction of high-resolution cylindrical holographic acoustic pressure field under sparse sampling, effectively solving the core technical problem of acoustic holographic measurement under sparse sampling of large cylindrical structures.

Claims

1. A super-resolution topological method for sparse holographic surface acoustic pressure fields based on deep learning, characterized by: To acquire the sound pressure level of a large-scale holographic surface, this paper takes the low-resolution holographic surface sound pressure level under sparse sampling as input, captures the local gradient details of the sound field through geometrically perceptual convolution, performs global modeling in the frequency domain using Fourier neural operators, and introduces a physical consistency loss based on the surface Helmholtz equation to achieve a deep integration of data-driven and physical layers. This allows the network to achieve a balance between numerical accuracy and physical rationality, thus realizing a super-resolution topology of the sound pressure field of a large-scale holographic surface.

2. The deep learning-based super-resolution topological method for sparse holographic surface acoustic pressure fields according to claim 1, characterized in that: The large-scale acoustic holographic surface is formed by circular array scanning, and the holographic surface is cylindrical in shape. The super-resolution topological method for the sound pressure field of the cylindrical holographic surface is performed according to the following steps: Step a: Apply low-resolution complex sound pressure levels to the cylindrical holographic surface. and high-resolution complex sound pressure Represented by equation (1): （1）； In formula (1): by Represents the field of complex numbers; and These represent low resolution and high resolution, respectively. by and These represent the number of discrete sampling points along the axial and circumferential directions of the cylinder, respectively. and These represent the number of discrete sampling points along the axial and circumferential directions of the cylinder at low resolution, respectively. and These represent the number of discrete sampling points along the axial and circumferential directions of the cylinder under high resolution, respectively. Normalization is performed according to equation (2) to obtain the normalized low-resolution complex sound pressure levels. and normalized high-resolution complex sound pressure ; （2）； In equation (2): normalized scaling factor Calculated from equation (3): （3）； In formula (3): Represents the 95th percentile; The normalized low-resolution complex sound pressure level and normalized high-resolution complex sound pressure The dual-channel low-resolution sound pressure and dual-channel high-resolution sound pressure are represented by equation (4) in the form of real and imaginary parts: （4）； In equation (4): by Dual-channel low-resolution sound pressure level, representing both real and imaginary parts; by Representing dual-channel high-resolution sound pressure levels in both real and imaginary form; Indicate the real part, with Indicates the imaginary part; Step b: To simultaneously characterize the periodic geometric features, local detailed structure, long-distance spatial coupling relationships, and response changes under different operating conditions of the cylindrical unfolded surface, local convolutional feature encoding and frequency domain global modeling are performed on a sparse sampling grid; the dual-channel low-resolution sound pressure level is then converted into a single signal. and dual-channel high-resolution sound pressure They are collectively referred to as input features. ; In the local convolutional feature encoding module, the input features are... Convolutional feature extraction is performed, and local convolutional features are calculated using equation (5). : （5）； In formula (5): Represents geometrically perceptual convolution The local enhancement module consists of GELU activation, channel attention, and residual connections; the geometrically perceptual convolution... Circular filling is used in the circumferential direction, and reflective filling is used in the axial direction, as characterized by equation (6): （6）； In formula (6): by Indicates circumferential expansion; by Indicates axial reflection extension; with Represents convolution; In the frequency domain global modeling module, a frequency domain spectral convolution branch is introduced to enhance the global modeling capability, and the frequency domain features are calculated by equation (7). : （7）； In formula (7): Represents low-modal spectral convolution in a two-dimensional Fourier domain, with Represents a pointwise linear mapping; The fusion features are obtained by adaptive fusion using equation (8). : （8）； In formula (8): This represents element-wise multiplication. It is obtained by calculation from equation (9): （9）； In formula (9): Indicates by Gated mappings constructed using convolutions; Indicates Sigmoid activation; Step c: Introduce a conditional feature modulation mechanism for encoding: For the input features Global average pooling is performed, and the style vector represented by Equation (10) is extracted via a multilayer perceptron. : （10）； In formula (10): This represents the style extraction mapping function composed of a multilayer perceptron; The normalized physical frequency is calculated using equation (11). ; physical frequency The normalized physical frequency is calculated using equation (11). : （11）； In equation (11): by Representing physical frequency, in Represents the minimum physical frequency, in Indicates the maximum physical frequency; The normalized scaling factor obtained from equation (3) Mapping to the logarithmic field and constructing the external condition vector represented by equation (12). c : （12）； In equation (12): The scaling factor is the logarithmic normalization factor. The style vector With external condition vector By splicing the equations together, we obtain the complete condition represented by equation (13). : （13）； Complete conditions Channel-by-channel modulation parameters are generated by mapping according to equation (14). : （14）； In formula (14): This indicates that the conditional mapping function consists of three fully connected layers and two GELU activation functions; Utilizing fusion features Low-resolution feature maps are obtained from equation (15). , （15）； To enable the network to explicitly perceive the geometric topology of the cylindrical surface, coordinate features are constructed on the low-resolution and high-resolution meshes respectively according to equation (16). : （16）； In formula (16): Represents the normalized axial coordinates, with Represents the circumferential angular coordinates; The low-resolution feature map Flatten into a token sequence and extract coordinate features After linear mapping, the result is superimposed onto the representation of the token sequence to obtain the initial token sequence represented by equation (17). : （17）； In equation (17): by This represents flattening a two-dimensional feature map and mapping it to a sequence representation; by This represents the coordinate feature embedding mapping function consisting of a single fully connected layer; Global feature updates are performed via multi-head self-attention and a feedforward network according to equations (18) and (19): （18）； （19）； In equations (18) and (19): by Representation layer normalization, to This indicates multi-head self-attention computation. by This represents a feedforward network; T1 and T2 represent the transition and output token sequences, respectively. The result obtained from equation (19) Rearranged into two-dimensional feature maps As in equation (20): （20）； In formula (20): This represents restoring the sequence form to a two-dimensional feature map; Step d: Obtain the high-resolution complex acoustic pressure field of a large-scale holographic surface through high-resolution decoding; Step e: Introduce physical consistency loss based on the surface Helmholtz equation to update the high-resolution sound pressure field, and complete the super-resolution topology of the sparse holographic surface sound pressure field.

3. The deep learning-based super-resolution topological method for sparse holographic surface acoustic pressure fields according to claim 2, characterized in that: in In step d, high-resolution decoding is performed as follows: First, the low-resolution two-dimensional feature map Interpolate to the target high-resolution mesh; Obtain high-resolution global features As shown in equation (21): （21）； In formula (21): This indicates an interpolation upsampling operation; Then normalize the low-resolution complex sound pressure level. Direct interpolation to a high-resolution mesh as a high-resolution substrate B The coordinate features on the high-resolution grid are denoted as... The complete condition derived from equation (13) Equation (22) is mapped to a constant condition graph. And extend to high-resolution grids; （22）； In equation (22): by This represents a conditional graph mapping function consisting of two fully connected layers and a GELU activation function; by This indicates replication and expansion along the axial and circumferential directions of the high-resolution mesh; Then, the decoder's initial input As represented by equation (23): （23）； Initialize the decoder input Input geometry-aware convolution Together with the residual refinement module, the decoding features are obtained. As shown in equation (24): （24）； In formula (24): This represents a series of geometrically perceptive convolutional layers. High-resolution decoding mapping consisting of residual refinement modules and nonlinear activation functions; The decoded features After passing through the residual prediction head and the gated prediction head respectively, the residual terms characterized by equations (25) and (26) are obtained. and gated graph : （25）； （26）； In equations (25) and (26): Representing the residual prediction mapping, in Represents the gating prediction mapping; Gating diagram For residual terms Perform element-wise multiplication and with a high-resolution substrate Adding them together yields the high-resolution complex acoustic pressure field prediction result. As shown in equation (27): （27）； This enables high-resolution complex acoustic pressure field decoding of large-scale structural holographic surfaces.

4. The deep learning-based super-resolution topological method for sparse holographic surface acoustic pressure fields according to claim 2, characterized in that: Step e is performed as follows: For a cylindrical holographic surface, the coordinates of the circumferential arc length are denoted as... , mark the axial coordinates as In the passive region, the surface complex sound pressure Satisfying equation (28): （28）； In equation (28): k For wave number, , Speed ​​of sound; In discrete implementation, the circumferential second derivative Using the cyclic difference shown in equation (29), the axial second derivative The reflection boundary difference is adopted as shown in equation (30): （29）； （30）； In equations (29) and (30): by and These represent discrete measurement points along the circumferential and axial directions of the cylindrical surface, respectively. by Indicates circumference Point and axis Complex sound pressure level at the point; by Indicates circumference Point and axis Complex sound pressure level at the point; by Indicates circumference Point and axis Complex sound pressure level at the point; by Indicates circumference Point and axis Complex sound pressure level at the point; by Indicates circumference Point and axis Complex sound pressure level at the point; Based on this, a discrete residual is constructed. As in equation (31): （31）； To balance the differences in residual magnitudes at different frequencies and grid scales, normalization is performed using equation (32) to obtain the normalized discrete residuals. : （32）； Then define physical loss As in equation (33): （33）； In formula (32): Indicates the total number of discrete points; Ultimately utilize physical loss Update the high-resolution sound pressure field and complete the super-resolution topology of the sparse holographic surface sound pressure field.

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