A method and system for generating solutions to partial differential equations based on adaptive space frequency sensing

Abstract

This invention relates to a method and system for generating solutions to partial differential equations based on adaptive spatial frequency sensing. The method includes: acquiring the parameter field and noisy solution field of the partial differential equation; performing convolution and nonlinear activation on the parameter field to obtain a latent representation; performing convolution mapping and conditional modulation on the noisy solution field to obtain a feature map; calculating the amplitude spectrum and phase corresponding to the feature map; constructing low-frequency, intermediate-frequency, and high-frequency masks based on the amplitude spectrum; calculating a weighted frequency band based on the low-frequency, intermediate-frequency, and high-frequency masks and the amplitude spectrum; performing learnable amplitude modulation on the weighted frequency band to obtain an updated amplitude; reconstructing the spatial domain based on the phase and the updated amplitude to obtain a frequency branch feature map; fusing the frequency branch feature map and the spatial branch feature map to obtain a fused feature; and outputting a predicted denoised solution field based on the fused feature. Compared with existing technologies, this invention has the advantages of high solution field accuracy and high detail fidelity.

Description

Technical Field

[0001] This invention relates to the field of scientific machine learning technology, and in particular to a method and system for generating solutions to partial differential equations based on adaptive space-frequency sensing. Background Technology

[0002] Physical phenomena can often be modeled using Partial Differential Equations (PDEs). PDEs are equations containing multivariable functions and one or more partial derivatives, and they have wide applications in fluid mechanics, quantum mechanics, biomedicine, and other disciplines. In solving PDEs, PINNs (Problem-Based Networks) are widely used due to their excellent solving capabilities. However, PINNs perform network optimization for a specific equation; if the parameters or initial conditions of the equation are changed, the model needs to be retrained under the new parameter settings. This means that PINNs can only solve one problem at a time, resulting in high computational costs and poor generalization ability. In contrast, operator learning aims to learn the mapping relationship between the parameter space and solution space of PDEs, thereby solving a class of problems. However, existing classical algorithms such as FNO and OFormer learn deterministic mappings, limiting their adaptability to unsteady physical systems. Diffusion models, as probabilistic generative models, can naturally capture multiple possible solutions to a system, showing broad application potential and development prospects in solving fluid dynamics problems under complex conditions. However, current research is mainly limited to feature learning in the spatial domain, while ignoring important frequency domain details in PDEs problems, or ignoring the correlation between the two, thus limiting further improvement in accuracy.

[0003] The invention disclosed in CN121436100B presents a large-scale PDE neural operator pre-training method based on a high-frequency enhancement module. Partial Differential Equation (PDE) data forms a hybrid dataset, and a large-scale PDE neural operator is constructed. A spatiotemporal encoder maps the preprocessed PDE data to a latent representation space. A frequency decomposition module performs frequency space mapping, including parallel high-frequency and low-frequency branches, yielding high-frequency and low-frequency features respectively. A multi-frequency fusion module (GFM) adaptively fuses the high-frequency and low-frequency features through a gating mechanism. Finally, a prediction head processes the fused features to obtain the final output features, i.e., the predicted physical features for the next time step. However, the model designed in this scheme cannot simultaneously capture the global structure, local variation features, and detailed boundary information of the physical field. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the existing technology and provide a method and system for generating partial differential equation solutions based on adaptive space frequency sensing.

[0005] The objective of this invention can be achieved through the following technical solutions: A method for generating solutions to partial differential equations based on adaptive space frequency sensing, comprising: Obtain the parameter field and noisy solution field of the partial differential equation; perform convolution and nonlinear activation on the parameter field to obtain the latent representation; split the latent representation in half along the channel dimension to obtain the scaling parameter and the offset parameter; perform convolution mapping on the noisy solution field to obtain the high-dimensional feature representation; Based on the scaling and offset parameters, the high-dimensional feature representation is conditionally modulated to obtain a feature map; spatial features are then extracted from the feature map using an attention mechanism to obtain spatial features. Calculate the amplitude spectrum and phase corresponding to the feature map; based on the amplitude spectrum, calculate the normalized Euclidean distance of each frequency domain coordinate relative to the DC component; based on the normalized Euclidean distance, construct low-frequency mask, intermediate-frequency mask and high-frequency mask respectively; based on the low-frequency mask, intermediate-frequency mask, high-frequency mask and amplitude spectrum, calculate the weighted frequency band; Learnable amplitude modulation is applied to the weighted frequency band to obtain an updated amplitude; spatial domain reconstruction is performed based on the phase and the updated amplitude to obtain a frequency branch feature map and a spatial branch feature map; the frequency branch feature map and the spatial branch feature map are fused to obtain a fused feature; based on the fused feature, the predicted denoised solution field is output.

[0006] Furthermore, the conditional modulation process specifically includes: modulating the high-dimensional feature representation through learnable scaling and translation operations, with the corresponding modulation expression being: Where γ(a') and β(a') represent the dynamically generated scaling and offset coefficients, respectively, and a' is the latent representation. This represents a high-dimensional feature representation.

[0007] Furthermore, based on the amplitude spectrum, the normalized Euclidean distance of each frequency domain coordinate relative to the DC component is calculated, and the corresponding calculation formula is as follows: in, For the normalized Euclidean distance, h f w represents the height coordinate. f H is the width direction coordinate. f W represents the total height in the frequency domain. f This represents the total bandwidth in the frequency domain.

[0008] Furthermore, based on the normalized Euclidean distance, low-frequency masks, intermediate-frequency masks, and high-frequency masks are constructed respectively; the low-frequency mask, intermediate-frequency mask, and high-frequency mask are respectively expressed as: in, For low-frequency masking, For intermediate frequency mask, For high-frequency masks, σ0, μ1, σ1, α, β, and γ are hyperparameters that control the frequency band boundaries and transition smoothness.

[0009] Furthermore, the low-frequency mask is a Gaussian attenuation mask used to capture the global structure, the mid-frequency mask is a bandpass Gaussian mask used to capture local variations, and the high-frequency mask is a high-pass filter mask used to extract details and boundary information.

[0010] Furthermore, the frequency branch feature map and the spatial branch feature map are fused to obtain the fused feature, and the corresponding fusion expression is: in, For fusion features, V spa For spatial branch feature map, V freq X is the frequency branch feature map.

[0011] Furthermore, the amplitude spectrum is multiplied element-wise with each mask to obtain multiple frequency bands; the amplitude spectrum is then input into a pre-constructed feature extraction network to obtain normalized frequency band weights. The weighted frequency band is obtained by multiplying each weight in the normalized frequency band weight by the multiple frequency bands one by one. The feature extraction network consists of a global average pooling layer, two convolutional layers, an activation function layer, and a softmax normalization layer.

[0012] Furthermore, the weighted frequency band is input into a pre-constructed learnable amplitude processing network to perform the learnable amplitude modulation, and the corresponding modulation expression is: in, The updated amplitude is given by K, where K is the number of frequency bands; g θ For learnable amplitude processing networks, For the i-th frequency band, w i This represents the weight of the i-th frequency band; The learnable amplitude processing network comprises a first convolutional layer, a LeakyReLU activation function, and a second convolutional layer connected in sequence.

[0013] Furthermore, the process of spatial domain reconstruction includes: The phase and updated amplitude are converted from polar coordinates back to rectangular coordinates to obtain the corresponding real and imaginary parts, which are then recombine to form a complex spectrum. Finally, the complex spectrum is restored to the spatial domain using a two-dimensional inverse real-valued fast Fourier transform. The corresponding processing expression is as follows: in, For the frequency branch feature map, X′ freq Let be the complex spectrum, ϕ be the phase, A′ be the updated amplitude, R be the real part, I be the imaginary part, and i be the imaginary unit.

[0014] The present invention also provides a system for generating solutions to partial differential equations based on adaptive space frequency sensing, comprising a memory and a processor, wherein the memory stores a computer program, and the processor calls the computer program to execute the steps of any of the methods described above.

[0015] Compared with the prior art, the present invention has the following advantages: (1) This invention encodes the parameter field into a latent representation through convolution and nonlinear activation, and splits it into scaling parameters and offset parameters along the channel dimension. Then, it performs feature-level affine transformation on the high-dimensional feature representation, which realizes the accurate injection of conditional information. This allows the parameter conditions of the partial differential equation to be continuously and dynamically integrated into the feature expression of the solved physical field in the form of weights and biases, rather than being weakly guided only once in the shallow layer of the network. This enhances the guiding ability for subsequent feature extraction. By introducing learnable affine transformations into the network, the model can flexibly adjust the distribution of intermediate features according to different parameter conditions, thereby more accurately controlling the solution generation process and significantly improving the generation quality and condition consistency of the solution field.

[0016] (2) By constructing three masks with different frequency response characteristics, the present invention employs a low-frequency mask with Gaussian attenuation centered on the DC component, a mid-frequency mask with bandpass Gaussian filtering, and a high-frequency mask with high-pass filtering. This can effectively decouple the multi-scale physical information contained in the solution field of the partial differential equation: the low-frequency component corresponds to the macroscopic trend and overall outline of the physical field, the mid-frequency component corresponds to the transition region and local fluctuations, and the high-frequency component corresponds to the sharp boundary and fine structure. The three work together to ensure that the generated solution field can maintain a high degree of consistency with the real physical field at all levels from coarse-grained to fine-grained. This allows the model to differentiate its processing for different frequency components, overcoming the limitations of fixed-frequency processing methods that cannot flexibly adapt to the frequency distributions of different physical fields. By learning band weights and performing nonlinear mapping on amplitudes, the model can optimize its frequency response for specific instances of partial differential equations, thereby significantly improving its adaptability to different types and parameters of physical systems, and ultimately enhancing the prediction accuracy and detail fidelity of the solution field.

[0017] (3) This invention designs parallel spatial and frequency branches, extracts spatial dependencies through an attention mechanism, and obtains a spatial branch feature map; by transforming the feature map to the frequency domain for processing, a frequency branch feature map is obtained, and finally the two are fused. This allows for the simultaneous capture of local, detailed dependencies in the spatial domain and global, periodic structural information in the frequency domain, achieving complementary modeling of the spatial and frequency domains. Through deep fusion of spatial and frequency features, the model can obtain richer feature information than a single-domain representation, thereby more accurately reconstructing the solution field of partial differential equations, especially when dealing with complex physical systems with multi-scale characteristics or global dependencies. Attached Figure Description

[0018] Figure 1 This is a flowchart of a method for generating solutions to partial differential equations based on adaptive space frequency sensing, provided in an embodiment of the present invention. Detailed Implementation

[0019] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0020] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0021] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0022] Definitions: U-shaped network: This is a convolutional neural network with a symmetric encoder-decoder structure, named for its U-shaped network architecture. Its encoder extracts multi-scale semantic features from the input data through progressive downsampling while continuously expanding the receptive field; the decoder recovers the spatial resolution of the feature map through progressive upsampling, and then concatenates and fuses the high-resolution detail features passed from the corresponding encoder layer through skip connections with the upsampled features, thereby achieving accurate pixel-level or element-level prediction.

[0023] Two-dimensional real-valued fast Fourier transform (FFT): This is an efficient frequency domain transformation algorithm for real-valued input data. It is used to transform two-dimensional real-valued signals in the spatial domain to the frequency domain, obtaining a complex spectral representation.

[0024] Bandpass Gaussian filtering is a filtering method that allows signals within a specific frequency band to pass through while suppressing low-frequency and high-frequency components in the frequency domain. Its transfer function is constructed using a Gaussian function. Specifically, it is usually composed of the difference between two Gaussian functions, or it can be implemented by constructing a Gaussian bandpass mask in the frequency domain with a center frequency as the mean and a specific standard deviation controlling the bandwidth.

[0025] Gaussian attenuation centered on DC: This is a method for constructing a low-pass filter weight mask in the frequency domain. Its core idea is that the weight value gradually decreases in the form of a Gaussian function as the normalized Euclidean distance from the frequency coordinate to DC increases, with the DC component as the center.

[0026] High-pass filtering is a signal processing technique used to allow high-frequency components to pass through while suppressing or attenuating low-frequency components.

[0027] Example 1 like Figure 1 As shown, this embodiment provides a method for generating solutions to partial differential equations based on adaptive space frequency sensing. The method includes the following steps: S1: Obtain the parameter field and noisy solution field of the partial differential equation; perform convolution and nonlinear activation on the parameter field to obtain the latent representation; split the latent representation in half along the channel dimension to obtain the scaling parameter and offset parameter; perform convolution mapping on the noisy solution field to obtain the high-dimensional feature representation; Characteristic Linear Modulation Conditional Injection (FiLM): To achieve better controllable generation, this embodiment introduces FiLM technology into the model, which can be considered a conditional modulation mechanism based on feature-level affine transformation. Previous studies have largely treated the conditions as an additional channel, directly concatenating them with the input data to form a uniform whole. Compared to simple fusion only in the early stages, a more reasonable approach is to continuously retain and utilize the conditional information throughout the entire network.

[0028] Compared to direct concatenation, FiLM modulates intermediate feature representations through learnable scaling and translation operations. Specifically, the parameter field *a* of the partial differential equation is treated as a conditional input and encoded into a latent representation *a'* through convolutional layers and nonlinear activation functions. The encoded result *a'* is then split in two along the channel dimension to obtain scaling and offset parameters. For the noisy solution field *u*... σ Then, through convolution, it is mapped to a high-dimensional feature representation. The FiLM operation is defined as follows: Here, γ(a') and β(a') represent the dynamically generated scaling and offset coefficients, respectively, and ⊙ represents element-wise multiplication. By introducing the FiLM strategy, the parameter information of the PDE is effectively injected into the model, thus providing strong support for the high-quality generation of PDE solutions under this condition.

[0029] S2: Based on scaling and offset parameters, conditional modulation is performed on the high-dimensional feature representation to obtain a feature map; spatial features are extracted from the feature map through an attention mechanism to obtain a spatial branch feature map; Specifically, The feature map is further processed by the U-Net encoder for feature extraction and downsampling before entering the bottleneck layer's dual branches: spatial branch and frequency branch, which are processed in parallel. U-Net consists of three parts: encoder, bottleneck layer, and decoder.

[0030] S3: Calculate the amplitude spectrum and phase corresponding to the feature map; based on the amplitude spectrum, calculate the normalized Euclidean distance of each frequency domain coordinate relative to the DC component; based on the normalized Euclidean distance, construct the low-frequency mask, intermediate-frequency mask, and high-frequency mask respectively; based on the low-frequency mask, intermediate-frequency mask, high-frequency mask, and amplitude spectrum, calculate the weighted frequency band; Adaptive multi-scale frequency processing is employed: different physical fields have varying importance across different frequency ranges, and fixed-frequency processing methods struggle to flexibly adapt to these variations. To overcome this limitation, we propose an adaptive frequency processing module. This module transforms spatial features into frequency features, dividing them into low-frequency, mid-frequency, and high-frequency bands. Then, based on global spectral amplitude information, it learns the importance weights for each band, enhancing or suppressing specific frequency ranges to dynamically or adaptively adjust frequency components, thereby improving the model's adaptability to different physical fields. After band weighting, the model is further optimized through a learnable amplitude modulation network. Finally, the spectrum is reconstructed back into the spatial domain to ensure the consistency of the physical field structure.

[0031] S301: Frequency Domain Transformation: In the frequency domain branch, a two-dimensional real-valued fast Fourier transform is first used to transform the input feature map X∈R. B×C×H×WProcessing is performed to obtain its frequency domain representation: Because real-valued signals have conjugate symmetry, only the non-redundant half of the spectrum is retained, therefore H f = H, W f =W / 2⌋+1. Amplitude and phase are two important components of frequency domain representation, and their calculation methods are as follows: Here, |·| and arg(·) represent the amplitude and phase operators, respectively. The amplitude characterizes the magnitude of each frequency component and determines the distribution of energy at different frequencies. The phase controls the spatial alignment of the frequency components during the inverse transform reconstruction process, playing a crucial role in preserving the geometric structure.

[0032] S302: Adaptive frequency band weighting: To achieve adaptive processing of different frequency components of different physical fields, we divide the frequency domain into three frequency bands and use a network to learn the relative importance weights of each frequency band.

[0033] For each frequency domain coordinate (h) in amplitude spectrum A f , w f ), calculate its normalized Euclidean distance relative to the DC component (Direct Current, DC): Distance function d(h) f , w f The larger the value of d(h), the higher the corresponding frequency. Based on the distance function d(h) f w f Construct K frequency masks. This divides the spectrum into different frequency bands, each covering a different frequency range. When K = 3, corresponding to the low-frequency, mid-frequency, and high-frequency bands, their masks are defined as follows: Where σ0, μ1, σ1, α, β, and γ are hyperparameters controlling the frequency band boundaries and transition smoothness. The indicator function 1{·} is 1 when the condition is met, and 0 otherwise. The low-frequency mask M0 uses a DC-centered Gaussian attenuation form to capture the global structure of the physical field. The mid-frequency mask M1 uses a bandpass Gaussian filter form to capture local variation features. The high-frequency mask M2 uses a high-pass filter form to extract details and boundary information.

[0034] By comparing the amplitude spectrum A with each mask M i By performing element-wise multiplication, we can obtain K frequency bands: To assess the importance of each frequency band, we employ a lightweight network f θ (·), This network takes amplitude A as input and outputs normalized frequency band weights: Specifically, f θ (·) contains a global average pooling layer, two 1×1 convolutional layers and a SiLU activation function, and a softmax normalization layer. Let w i This represents the weight of the i-th frequency band extracted from w.

[0035] S4: Perform learnable amplitude modulation on the weighted frequency band to obtain the updated amplitude; perform spatial domain reconstruction based on the phase and the updated amplitude to obtain the frequency branch feature map; fuse the frequency branch feature map and the spatial branch feature map to obtain the fused feature; output the predicted denoised solution field based on the fused feature.

[0036] S401: Learnable amplitude modulation: To enhance the ability to model nonlinearities in the frequency domain while maintaining parameter efficiency, a shared, learnable amplitude processing network g is used. θ Applied to each weighted frequency band. Network g θ It consists of two 1×1 convolutional layers, with the LeakyReLU activation function in between. Simply put, it performs a non-linear mapping on the features of different frequency bands after scaling them according to importance. The updated amplitude is then: S402: Spatial Domain Reconstruction We convert the phase ϕ and the processed amplitude A′ from polar coordinates back to rectangular coordinates to obtain the corresponding real part R and imaginary part I, and then recombine them to form the complex spectrum X′. freq : Subsequently, the two-dimensional inverse real-valued fast Fourier transform F is applied. -1 The processed frequency domain data is then restored to the spatial domain: Feature map V freq This is the result after frequency branching, which will be further integrated with the spatial branching result.

[0037] S403: Spatial Frequency Dual-Domain Training Architecture The proposed diffusion model uses a U-shaped network as its core architecture. It takes the output of the previous step, the noise level, and the parameters of the partial differential equation as input, and predicts the denoised data for the next step. This iterative update mechanism gradually transforms random noise samples into solutions to the partial differential equations.

[0038] Unlike previous studies, we establish an interaction between the spatial and frequency domains through hybrid domain training, thereby generating more accurate PDE solutions. This hybrid domain module consists of two parallel branches: a spatial branch and a frequency branch. In the spatial branch, spatial dependencies are modeled using an attention mechanism. In the frequency branch, the spatial feature map is converted into a complex frequency domain representation using a two-dimensional real-valued Fast Fourier Transform, and then processed.

[0039] Let the feature map input to the hybrid domain module be X∈R B×C×H×W Where B, C, and H×W represent the batch size, number of channels, and spatial resolution, respectively. Spatial and frequency branches are processed in parallel on X to obtain V. spa and V freq The fusion feature with residual connections is represented as: Here, [·, ·] indicates that two feature maps are concatenated along the channel dimension. 1×1 It is a 1×1 convolutional projection layer used to restore the fused features to their original number of channels. This fusion architecture achieves complementary modeling in the spatial and frequency domains, thereby providing richer feature representations for the generation of solutions to partial differential equations.

[0040] Specifically, The U-Net decoder performs feature fusion and upsampling on the frequency branch feature map and the spatial branch feature map to obtain the fused features.

[0041] Example 2 This embodiment provides a system for generating solutions to partial differential equations based on adaptive space frequency sensing, including a memory and a processor. The memory stores a computer program, and the processor calls the computer program to execute the steps of the method as described in Embodiment 1.

[0042] The improvements of this invention include: Characteristic linear modulation mechanism: Unlike the simple channel splicing method in previous studies, this mechanism incorporates the parameter and other conditional information of the equation into the physical field to be solved in the form of weights and biases, thereby guiding the generation process of the diffusion model more efficiently.

[0043] Hybrid Spatial-Frequency Networks: Inspired by research on multi-domain modeling for low-light image enhancement, we employ a spatial-frequency dual-branch architecture at the neck of U-Net to fully learn complementary frequency domain features that are difficult to capture in the spatial domain. To our knowledge, this is the first meaningful exploration and attempt to integrate multi-domain feature modeling into diffusion neural operators to solve physical systems.

[0044] Adaptive multi-scale frequency domain processing mechanism: The designed mechanism can dynamically enhance or suppress frequency components at different scales, realizing flexible spectrum modulation and multi-scale information learning, thereby improving the adaptability of the model and the solution accuracy.

[0045] 4) To fully verify the effectiveness of the proposed method, experiments were conducted on several representative partial differential equations in this embodiment. Extensive experimental results consistently demonstrate that the proposed model outperforms the classical neural operator baseline model in terms of accuracy, providing new ideas and technical solutions for the further development of diffusion neural operators.

[0046] Considering the limited accuracy of single-domain learning, this embodiment innovatively introduces a space-frequency dual-domain learning architecture into the diffusion model and applies it to the operator learning problem to enhance the perception and learning ability of complementary detail features in the frequency domain. This embodiment further designs an adaptive multi-scale frequency processing mechanism, which dynamically selects and modulates different scale frequency components of different PDEs data, thereby improving the model's learning ability and adaptability. In addition, this embodiment improves the conditional injection strategy by replacing the simple channel splicing method with Feature-wise Linear Modulation (FiLM), thus more effectively guiding the generation process of the diffusion model. Finally, this embodiment conducts numerical experiments on multiple PDEs. Experimental results show that this method can approximate the solution operators of PDEs with varying parameters with higher accuracy and has good robustness.

[0047] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A method for generating solutions to partial differential equations based on adaptive space frequency sensing, characterized in that, include: Obtain the parameter field and noisy solution field of the partial differential equation; The parameter field is convolved and nonlinearly activated to obtain a latent representation; the latent representation is split in two along the channel dimension to obtain scaling parameters and offset parameters; the noisy solution field is convolved and mapped to obtain a high-dimensional feature representation. Based on the scaling and offset parameters, the high-dimensional feature representation is conditionally modulated to obtain a feature map; spatial features are extracted from the feature map through an attention mechanism to obtain a spatial branch feature map. Calculate the amplitude spectrum and phase corresponding to the feature map; Based on the amplitude spectrum, calculate the normalized Euclidean distance of each frequency domain coordinate relative to the DC component; Based on the normalized Euclidean distance, low-frequency masks, mid-frequency masks, and high-frequency masks are constructed respectively. Based on the low-frequency mask, mid-frequency mask, high-frequency mask, and amplitude spectrum, calculate the weighted frequency band; The weighted frequency band is subjected to learnable amplitude modulation to obtain the updated amplitude; Spatial domain reconstruction is performed based on the phase and the updated amplitude to obtain a frequency branch feature map; The frequency branch feature map and the spatial branch feature map are fused to obtain the fused feature; Based on the fusion features, the predicted denoised solution field is output.

2. The method for generating solutions to partial differential equations based on adaptive space frequency sensing according to claim 1, characterized in that, The conditional modulation process specifically includes: modulating the high-dimensional feature representation through learnable scaling and translation operations, with the corresponding modulation expression being: Where γ(a') and β(a') represent the dynamically generated scaling and offset coefficients, respectively, and a' is the latent representation. This represents a high-dimensional feature representation.

3. The method for generating solutions to partial differential equations based on adaptive space frequency sensing according to claim 1, characterized in that, Based on the amplitude spectrum, the normalized Euclidean distance of each frequency domain coordinate relative to the DC component is calculated using the following formula: in, For the normalized Euclidean distance, h f w represents the height coordinate. f H is the width direction coordinate. f W represents the total height in the frequency domain. f This represents the total bandwidth in the frequency domain.

4. The method for generating solutions to partial differential equations based on adaptive space frequency sensing according to claim 3, characterized in that, Based on the normalized Euclidean distance, low-frequency masks, mid-frequency masks, and high-frequency masks are constructed respectively; the low-frequency mask, mid-frequency mask, and high-frequency mask are respectively represented as: in, For low-frequency masking, For intermediate frequency mask, For high-frequency masks, σ0, μ1, σ1, α, β, and γ are hyperparameters that control the frequency band boundaries and transition smoothness.

5. The method for generating solutions to partial differential equations based on adaptive space frequency sensing according to claim 4, characterized in that, The low-frequency mask is a Gaussian attenuation mask used to capture the global structure, the mid-frequency mask is a bandpass Gaussian mask used to capture local variations, and the high-frequency mask is a high-pass filter mask used to extract details and boundary information.

6. The method for generating solutions to partial differential equations based on adaptive space frequency sensing according to claim 1, characterized in that, The frequency branch feature map and the spatial branch feature map are fused to obtain the fused feature, and the corresponding fusion expression is: in, For fusion features, V spa For spatial branch feature map, V freq X is the frequency branch feature map.

7. The method for generating solutions to partial differential equations based on adaptive space frequency sensing according to claim 1, characterized in that, The amplitude spectrum is multiplied element-wise with each mask to obtain multiple frequency bands; the amplitude spectrum is then input into a pre-constructed feature extraction network to obtain normalized frequency band weights. The weighted frequency band is obtained by multiplying each weight in the normalized frequency band weight by the multiple frequency bands one by one. The feature extraction network consists of a global average pooling layer, two convolutional layers, an activation function layer, and a softmax normalization layer.

8. The method for generating solutions to partial differential equations based on adaptive space frequency sensing according to claim 7, characterized in that, The weighted frequency band is input into a pre-constructed learnable amplitude processing network to perform learnable amplitude modulation. The corresponding modulation expression is: in, The updated amplitude is given by K, where K is the number of frequency bands; g θ For learnable amplitude processing networks, For the i-th frequency band, w i This represents the weight of the i-th frequency band; The learnable amplitude processing network comprises a first convolutional layer, a LeakyReLU activation function, and a second convolutional layer connected in sequence.

9. The method for generating solutions to partial differential equations based on adaptive space frequency sensing according to claim 1, characterized in that, The process of reconstructing the spatial domain includes: The phase and updated amplitude are converted from polar coordinates back to rectangular coordinates to obtain the corresponding real and imaginary parts, which are then recombine to form a complex spectrum. Finally, the complex spectrum is restored to the spatial domain using a two-dimensional inverse real-valued fast Fourier transform. The corresponding processing expression is as follows: in, For the frequency branch feature map, X′ freq Let be the complex spectrum, ϕ be the phase, A′ be the updated amplitude, R be the real part, I be the imaginary part, and i be the imaginary unit.

10. A system for generating solutions to partial differential equations based on adaptive space frequency sensing, characterized in that, It includes a memory and a processor, the memory storing a computer program, the processor invoking the computer program to perform the steps of the method as described in any one of claims 1 to 9.

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