Genomics cross-modal condition generation method and system based on potential diffusion model
Through the genomic cross-modal condition generation method based on the potential diffusion model, the problem of cross-modal missing data generation is solved, efficient and high-quality data generation is achieved, and the effect of multi-omic joint analysis is improved.
Patent Information
- Application Number
- CN202510618755.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-05-14
- Publication Date
- 2025-06-20
- Estimated Expiration
- 2045-05-14
AI Technical Summary
The existing technology is difficult to effectively generate cross-modal missing data, resulting in incomplete data in the multiomics joint analysis of genomics and transcriptomics, affecting the training and reasoning capabilities of the model.
A cross-modal condition generation method of genomics based on the latent diffusion model is adopted, and a cross-modal feature alignment network is constructed by constructing a variable autoencoder and a latent diffusion model, combining the attention mechanism to achieve feature alignment and data generation between gene expression data and gene mutation data.
It realizes efficient generation of high-quality prediction data in the absence of mutation data, improves the utilization efficiency of data, makes up for the lack of experimental data, and optimizes the fusion quality of multimodal data.
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Figure CN120183507A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the field of bioinformatics, relates to the generation of multi-modal missing data, and particularly relates to a genomics cross-modal conditional generation method and system based on a Latent Diffusion Model (LDM). Background Art
[0002] The integrated analysis of multiple omics such as genomics and transcriptomics is one of the core methods in modern biological and medical research, and is particularly significant in cancer research. Genomics mainly studies the genomic DNA of organisms, including gene sequences, gene mutations, structural variations, and genetic variations. Genomic data provides the potential genetic background of tumorigenesis, helping to understand how cancer drives abnormal cell proliferation and metastasis through gene mutations. Gene mutations are the key driving factors in cancer development. Mutations may change the function of genes, leading to uncontrolled cell growth or escape from immune surveillance. Transcriptomics, on the other hand, focuses on the level of gene expression, analyzes the transcription of RNA, and reveals the expression patterns of genes in different cells or tissues. Transcriptomic data can reflect how genes respond to various physiological and pathological signals in the tumor microenvironment, and how these changes drive the occurrence and development of tumors. The integrated analysis of genomic and transcriptomic data can provide more comprehensive and profound cancer biology information. By analyzing the association between gene mutations and gene expression, researchers can reveal the molecular mechanisms of cancer development, and deeply understand how different gene mutations affect the proliferation, migration, and drug resistance of cancer cells by changing the gene expression profile. For example, certain driver mutations may lead to the overexpression or silencing of specific genes, thereby affecting the cell cycle, DNA repair mechanisms, or cell death pathways. This integrated analysis helps to identify tumor-related driver genes, mutation hotspots, and potential therapeutic targets.
[0003] As a technology that integrates multiple data sources, multi-modal learning can effectively integrate data from different omics levels, such as genomic data, transcriptomic data, clinical data, etc., thereby improving the prediction ability and interpretability of the model. In the field of cancer, if multi-modal learning is applied, it can achieve in-depth mining of complex biological data, not only capturing the impact of gene mutations on gene expression, but also learning potential biomarkers from different data sources, and then guiding the implementation of precision medicine. This cross-domain data fusion method not only helps in the early detection of cancer, but also promotes the formulation of individualized treatment plans, improves treatment effects, and reduces side effects. Therefore, the integrated analysis of multiple omics of genomics and transcriptomics, combined with multi-modal learning, can provide strong support for cancer research and clinical applications.
[0004] Multi-omics joint analysis or multi-modal learning based on omics data is of great value in biomedical research. However, one of the key problems is the missing of sample modal data. The reasons for data missing may include experimental technical limitations, high sequencing costs, uneven sample quality, different acquisition times of different modal data, and difficulty in obtaining some omics data. For example, in cancer research, some patients may only have gene expression data but lack gene mutation or proteomics data, which leads to incomplete data and affects the training and inference ability of the model. Current research mainly focuses on the generation of single-modal data, such as synthesizing missing single-omics data through generative adversarial networks (GANs) or diffusion models. However, this method cannot capture the corresponding relationship between samples in different modalities, resulting in difficulty in maintaining cross-modal consistency during joint analysis. Therefore, it is of great significance to study cross-modal data generation. First, it can complement the missing modal data, making multi-omics joint analysis more comprehensive and improving the robustness of the model. Second, cross-modal data generation can be used for data augmentation to expand the training dataset and improve the generalization ability of the model. In addition, research based on cross-modal generation can explore the potential relationships between different omics levels and promote the development of precision medicine. Therefore, exploring effective cross-modal data generation methods will provide new ideas for multi-omics analysis and multi-modal learning. Summary of the Invention
[0005] The purpose of the present invention is to provide a genomics cross-modal conditional generation method and system based on a latent diffusion model in view of the deficiencies of the prior art.
[0006] The purpose of the present invention is achieved through the following technical solutions: In the first aspect, a genomics cross-modal conditional generation method based on a latent diffusion model is provided, including the following steps: Construct variational autoencoders for gene expression data and gene mutation data respectively for training to obtain a first encoder, a first decoder based on gene expression data, a second encoder, and a second decoder based on gene mutation data; construct a latent feature space according to the second encoder, freeze the parameters of the second encoder and the second decoder, and train an unconditional-guided latent diffusion model; Construct a cross-modal feature alignment network based on the attention mechanism, freeze the parameters of the first encoder and the second encoder, input the latent representation obtained from the gene expression data based on the first encoder into the cross-modal feature alignment network, and perform feature alignment with the latent representation obtained from the gene mutation data based on the second encoder to train the cross-modal feature alignment network; Jointly fine-tune the unconditional-guided latent diffusion model and the cross-modal feature alignment network, and use the output of the cross-modal feature alignment network as the guiding condition of the latent diffusion model; When generating corresponding gene mutation data using gene expression data, the gene expression data is sequentially passed through a trained first encoder and a cross-modal feature alignment network, and then input into a latent diffusion model for initial noise addition and iterative denoising, and decoded by a second decoder to generate gene mutation data.
[0007] Further, the gene expression data is calculated using the TPM method and subjected to logarithmic transformation; the gene mutation data takes values of 0 or 1, where 0 indicates no mutation and 1 indicates mutation. Preferably, before constructing and training the variational autoencoder, the gene expression data is also normalized.
[0008] In some embodiments, the variational autoencoder includes an encoder and a decoder. The encoder includes multiple fully connected layers with gradually decreasing dimensions, LeakyReLU activation function layers and dropout layers between layers, and the decoder includes multiple fully connected layers with gradually increasing dimensions and LeakyReLU activation function layers between layers.
[0009] In some embodiments, when training the variational autoencoder, the reconstruction loss function and the KL divergence loss function are added together as the loss function, and the reconstruction loss function uses the mean squared error.
[0010] Further, the latent diffusion model includes gradually adding noise to the latent representation of the gene mutation data obtained based on the second encoder until it is transformed into pure noise, and using a denoising neural network to gradually denoise the pure noise to obtain a clean latent representation.
[0011] Preferably, the denoising neural network adopts a U-Net structure, including a self-attention layer, a 1D convolutional layer and layer normalization.
[0012] In some embodiments, when training the latent diffusion model, the loss function is the square of the L2 norm of the difference between the real noise added at each step and the predicted noise of the denoising neural network.
[0013] In some embodiments, in the cross-modal feature alignment network constructed based on the attention mechanism, the input features are respectively mapped to Q, K, and V through a linear layer, capture multi-level features through a multi-head attention mechanism, and are output after passing through a first normalization layer, a fully connected layer, and a second normalization layer in sequence.
[0014] In some embodiments, when training the cross-modal feature alignment network, the alignment loss function and the cosine similarity loss function are added together as the loss function, and the alignment loss function uses the mean squared error.
[0015] Preferably, during the joint fine-tuning, the guiding condition and the denoising neural network use a cross-attention mechanism for feature fusion.
[0016] In some embodiments, the loss function used in the joint fine-tuning is the square of the L2 norm of the difference between the real noise under guidance and the predicted noise of the denoising neural network.
[0017] Preferably, the initial noise is Gaussian noise, and the iterative denoising gradually denoises the Gaussian noise using the trained denoising neural network.
[0018] As a second aspect, the present invention also provides a genomics cross-modal conditional generation system based on a latent diffusion model, including one or more processors for implementing the above-mentioned genomics cross-modal conditional generation method based on a latent diffusion model.
[0019] As a third aspect, the present invention also provides a computer-readable storage medium with a program stored thereon, which, when executed by a processor, is used to implement the above-mentioned genomics cross-modal conditional generation method based on a latent diffusion model.
[0020] Compared with the prior art, the beneficial effects of the present invention are as follows: The present invention can fully utilize the pre-trained encoder model to extract multi-level gene expression and gene mutation characterizations in the cross-modal data generation task, and through cross-modal feature alignment, the potential association between gene expression data and gene mutation data can be mined, promoting information interaction between different modalities. Using the latent diffusion model, high-quality prediction data can be efficiently generated in the case of missing mutation data. The genomics cross-modal conditional generation system proposed by the present invention not only improves the utilization efficiency of data but also makes up for the lack of experimental data, thereby optimizing the fusion quality of multi-modal data. BRIEF DESCRIPTION OF THE DRAWINGS
[0021] Figure 1 It is a schematic flowchart of the genomics cross-modal conditional generation method based on a latent diffusion model provided by an embodiment of the present invention; Figure 2 It is a structural diagram of an autoencoder model for gene expression and gene mutation data respectively provided by an embodiment of the present invention; Figure 3 It is a structural diagram of an unconditional guided latent diffusion model provided by an embodiment of the present invention; Figure 4 It is a structural diagram of a cross-modal feature alignment network provided by an embodiment of the present invention; Figure 5 It is a structural diagram of a latent diffusion model with a cross-modal feature alignment network as a conditional guidance provided by an embodiment of the present invention; Figure 6 It is a structural diagram of a denoising U-Net network provided by an embodiment of the present invention; Figure 7 It is a schematic diagram of a hardware structure provided by an embodiment of the present invention. Detailed implementation manners
[0022] The present invention will be described in detail below with reference to the accompanying drawings. Without conflict, the features in the following embodiments and implementation manners can be combined with each other. In the following embodiments, if there is no specified method, the conventional methods in the art can be adopted.
[0023] The genomics cross-modal conditional generation method based on the latent diffusion model proposed by the present invention, as Figure 1 shown, includes the following steps: (1) First, preprocess the gene expression data B and gene mutation data A (sample missing modality data). For the preprocessed gene expression data and gene mutation data (sample missing modality data, where m < n, representing the number of samples of gene mutation data and gene expression data respectively, and representing the data dimensions of gene expression data and gene mutation data respectively), design and construct autoencoders and perform model training, namely and .
[0024] Specifically, the preprocessed gene expression data adopts the TPM (Transcripts Per Million) calculation method and performs a logarithmic transformation log2(TPM + 1). The gene mutation data takes values of 0 or 1, where 0 represents no mutation and 1 represents mutation. When the above gene expression data is input into the model for training, a normalization operation is performed.
[0025] Specifically, as Figure 2 shown, design and construct an autoencoder for the gene expression data B based on the variational autoencoder architecture. The encoder mainly includes three fully connected layers, gradually reducing the dimension ( gradually reducing to 512, 256, and 128 dimensions in sequence), and using LeakyReLU as the activation function to avoid gradient disappearance and relieve the activation problem of sparse input. And dropout is adopted to prevent overfitting of high-dimensional small-sample data; the decoder uses three fully connected layers, gradually increasing the dimension (increasing to 128, 256, 512 dimensions in sequence, and finally increasing to dimension ), and using LeakyReLU as the activation function. Through the encoder, the gene expression data can be well feature-extracted, and then the gene expression data is reconstructed through the latent features to train the autoencoder .
[0026] Specifically, as Figure 2 shown, design and construct an autoencoder , also based on the variational autoencoder architecture, where the encoder mainly consists of three fully connected layers, gradually reducing the dimension ( gradually reducing to 512, 256, and 128 dimensions in sequence), and using LeakyReLU as the activation function to avoid gradient vanishing, alleviate the activation problem of sparse inputs, and using dropout to prevent overfitting of high-dimensional small-sample data; the decoder uses three fully connected layers to gradually increase the dimension (increasing to 128, 256, 512 dimensions in sequence, and finally increasing to the dimension ), and using LeakyReLU as the activation function. Through the encoder, the gene mutation data can be well feature-extracted, and through the decoder, the gene mutation data can be well decoded for data reconstruction, thereby training the autoencoder .
[0027] During model training, the loss function used for autoencoder training both uses the reconstruction loss function and the KL divergence loss function , where the reconstruction loss uses the mean squared error loss (MSE) to measure the similarity between the original data x and the reconstructed data , represents the number of samples, and the KL divergence loss encourages the posterior distribution output by the encoder to be close to the prior distribution, avoiding over-dispersion of the latent representation space and promoting good latent structure learning, where, represents the posterior distribution output by the encoder, z is the latent representation output by the encoder, represents the prior distribution, represents the calculation function of the KL divergence: ; ; ; (2) Construct a latent feature space based on the encoder of the gene mutation data A and train an unconditional guided latent diffusion model LDM using the frozen parameters.
[0028] Specifically, as Figure 3 shown, LDM operates on the latent feature space constructed according to , including the forward diffusion process and the reverse denoising process, where is the gene mutation data before input , and the forward diffusion process gradually adds Gaussian noise to the clean data in the latent space, and finally transforms it into pure Gaussian noise ; where the reverse denoising process is carried out through a denoising neural network gradually denoises from the noise to recover the clean data in the latent space, and finally generates data through the decoder .
[0029] Specifically, the denoising U-Net network structure is as Figure 6 shown. The denoising U-Net network mainly consists of a self-attention layer (Self-Attention), 1D convolution (Conv1d), and layer normalization. The activation function uses LeakyReLU. This network uses 1D convolution, which is suitable for processing genomic data or time series data and can retain local dependencies. The self-attention layer allows the model to focus on global dependencies, compensating for the limitations of the local receptive field of one-dimensional convolution and being suitable for processing cross-gene or cross-feature dependencies. Adding layer normalization (LayerNorm) after the Conv1d layer can stabilize model training. Feature compression is performed on the left side (downsampling path), and feature recovery is performed on the right side (upsampling path). Skip connections are made from left to right, allowing features of different layers to be directly transmitted to retain key information and prevent information loss.
[0030] Specifically, when training the unconditional guided latent diffusion model LDM, the model parameters of the encoder and the decoder are both frozen during the training process. The loss function is the square of the L2 norm of the difference between the real noise (the noise added at each step of the forward diffusion process) and the predicted noise: ; where represents the real noise, represents the noise predicted by the denoising neural network at the t-th step based on , represents the latent representation of the data at the t-th step, is the noise predicted by the denoising neural network at the t-th step under the given observation , and is implemented through a time-conditioned U-Net model.
[0031] (3) Based on the encoder trained with gene expression data and the encoder trained with gene mutation data, design and construct a cross-modal feature alignment network , and freeze the two encoders to train the cross-modal feature alignment network .
[0032] Specifically, as Figure 4 shown, the cross-modal feature alignment network The model is implemented using a neural network based on the attention mechanism. First, the input features are mapped to Q (Query), K (Key), and V (Value) through three linear layers in this network structure. Then, multi-head attention mechanism is used to capture multi-level features. Next, layer normalization is performed, and then output feature mapping is carried out through a fully connected layer and layer normalization. During this process, residual network structures are also used twice to retain feature information during the two layer normalizations.
[0033] Specifically, during the model training process, first, through and respectively, the potential representations of gene expression and gene mutation data pairs are obtained and . Then, is input into the cross-modal feature alignment network to obtain the aligned features . Then, feature alignment is performed on and . During model training, in order to better align cross-modal features, mean squared error (MSE) is used as the alignment loss function . represents the number of samples, and the cosine similarity loss function is used to strengthen the similarity between features, obtaining the loss function used for training the cross-modal feature alignment network: ; ; .
[0034] (4) Joint fine-tuning is performed on the unconditional guided latent diffusion model (LDM) and the cross-modal feature alignment network , where the output of the cross-modal feature alignment network is used as the guiding condition.
[0035] Specifically, as shown in Figure 5 , the joint fine-tuning of the unconditional guided latent diffusion model (LDM) and the cross-modal feature alignment network is carried out (freezing the parameters of the encoder , the encoder and the decoder ). The output of the cross-modal feature alignment network is used as the guiding condition, and the loss function used for the joint fine-tuning of the model is : ; Among them, represents the true noise, Denotes the noise predicted by the denoising neural network at the t-th step under the guidance condition c. The conditional vector c is the gene expression data input to the encoder to obtain the latent representation , and then through the cross-modal feature alignment network is calculated.
[0036] Specifically, the feature fusion method of the guidance condition c and the denoising U-Net network adopts the cross-attention mechanism, where the conditional guidance c is used as K and V, and the output feature of each layer of the U-Net network is used as Q.
[0037] (5) When generating the corresponding gene mutation data using the gene expression data, first, the preprocessed gene expression data uses the pre-trained gene expression encoder to output the latent representation , and then inputs it into the jointly fine-tuned cross-modal feature alignment network to obtain the conditional vector c, and then runs the conditional generation of the jointly fine-tuned latent diffusion model LDM for initial noise and iterative denoising to obtain the latent representation of the gene mutation , and finally uses the pre-trained gene mutation decoder to decode and generate the corresponding gene mutation data.
[0038] Specifically, the initial noise is .
[0039] Corresponding to the foregoing embodiments of the genomics cross-modal conditional generation method based on the latent diffusion model, the present invention also provides an embodiment of a genomics cross-modal conditional generation system based on the latent diffusion model.
[0040] See Figure 7 , an embodiment of a genomics cross-modal conditional generation system provided by an embodiment of the present invention includes one or more processors for implementing a genomics cross-modal conditional generation method in the foregoing embodiments.
[0041] The processor may be a Central Processing Unit (CPU), or other general-purpose processors, Digital Signal Processors (DSPs), Application Specific Integrated Circuits (ASICs), Field-Programmable Gate Arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor may be a microprocessor or any conventional processor, etc.
[0042] Embodiments of the genomics cross-modal conditional generation system based on the latent diffusion model of the present invention can be applied to any device with data processing capabilities, and the any device with data processing capabilities can be a device or apparatus such as a computer. This embodiment can be implemented by software, or by hardware or a combination of software and hardware. Taking software implementation as an example, as a logically meaningful device, it is formed by the processor of any device with data processing capabilities reading the corresponding computer program instructions in the non-volatile memory into the memory for operation. At the hardware level, as Figure 7 shown, it is a hardware structure diagram of any device with data processing capabilities where the genomics cross-modal conditional generation system based on the latent diffusion model of the present invention is located. In addition to Figure 7 the shown processor, memory, network interface, and non-volatile memory, any device with data processing capabilities where the device in the embodiment is located usually also includes other hardware according to the actual functions of the any device with data processing capabilities, which will not be elaborated here.
[0043] For the specific implementation process of the functions and roles of each unit in this embodiment, please refer to the implementation process of the corresponding steps in the above method, which will not be elaborated here.
[0044] For this embodiment, since it basically corresponds to the method embodiment, the relevant parts can be referred to the partial description of the method embodiment. The device embodiments described above are only illustrative. The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed to multiple network units. Some or all of the modules can be selected according to actual needs to achieve the purpose of the solution of the present invention. Those of ordinary skill in the art can understand and implement it without creative efforts.
[0045] An embodiment of the present invention further provides a computer-readable storage medium, on which a program is stored. When the program is executed by a processor, it implements a genomics cross-modal conditional generation method based on a latent diffusion model in the above embodiment.
[0046] The computer-readable storage medium may be an internal storage unit of any device with data processing capabilities described in any of the foregoing embodiments, such as a hard disk or memory. The computer-readable storage medium may also be any device with data processing capabilities, such as a plug-in hard disk, a Smart Media Card (SMC), an SD card, a Flash Card, etc. equipped on the device. Further, the computer-readable storage medium may also include both an internal storage unit of any device with data processing capabilities and an external storage device. The computer-readable storage medium is used to store the computer program and other programs and data required by any device with data processing capabilities, and may also be used to temporarily store data that has been output or will be output.
[0047] The above embodiments are only used to illustrate the design concept and features of the present invention, and their purpose is to enable those skilled in the art to understand the content of the present invention and implement it accordingly. The protection scope of the present invention is not limited to the above embodiments. Therefore, all equivalent changes or modifications made according to the principles and design concepts disclosed by the present invention are within the protection scope of the present invention.
Claims
1. A method for generating cross-modal conditions for genomics based on a latent diffusion model, characterized in that: The following steps are involved: Constructing variational autoencoders for gene expression data and gene mutation data respectively for training, and obtaining a first encoder and a first decoder based on the gene expression data and a second encoder and a second decoder based on the gene mutation data; constructing a latent feature space according to the second encoder, freezing the parameters of the second encoder and the second decoder, and training an unconditionally guided latent diffusion model; A cross-modal feature alignment network is constructed based on the attention mechanism, the parameters of the first encoder and the second encoder are frozen, and the gene expression data is input into the cross-modal feature alignment network based on the potential representation obtained by the first encoder, and then the gene mutation data is aligned with the potential representation obtained by the second encoder to train the cross-modal feature alignment network; The unconditionally guided latent diffusion model and the cross-modal feature alignment network are jointly fine-tuned, and the output of the cross-modal feature alignment network is used as the guiding condition of the latent diffusion model; When using gene expression data to generate corresponding gene mutation data, the gene expression data is passed through the trained first encoder and the cross-modal feature alignment network in turn, and then input into the latent diffusion model for initialization noise and iterative denoising, and then decoded by the second decoder to generate gene mutation data.
2. The method for generating cross-modal conditions for genomics based on a potential diffusion model according to claim 1, characterized in that: The gene expression data were calculated using TPM and logarithmically transformed; the gene mutation data were taken as 0 or 1, 0 indicating no mutation and 1 indicating mutation.
3. The method for generating cross-modal conditions for genomics based on a potential diffusion model according to claim 1, characterized in that: The variational autoencoder includes an encoder and a decoder, the encoder includes multiple fully connected layers with gradually reduced dimensions and LeakyReLU activation function layers and dropout layers between layers, and the decoder includes multiple fully connected layers with gradually increased dimensions and LeakyReLU activation function layers between layers; when the variational autoencoder is trained, the reconstruction loss function and the KL divergence loss function are added as the loss function, and the reconstruction loss function uses the mean square error.
4. The method for generating cross-modal conditions for genomics based on a potential diffusion model according to claim 1, characterized in that: The potential diffusion model includes gradually adding noise to the potential representation of the gene mutation data obtained based on the second encoder until it is converted into pure noise, and gradually denoising the pure noise using a denoising neural network to obtain a clean potential representation; the denoising neural network uses a U-Net structure.
5. The method for generating cross-modal conditions for genomics based on a potential diffusion model according to claim 4, characterized in that: When the latent diffusion model is trained, the loss function is the square of the L2 norm of the difference between the real noise added at each step and the predicted noise of the denoising neural network.
6. The method for generating cross-modal conditions for genomics based on a potential diffusion model according to claim 1, characterized in that: In the cross-modal feature alignment network constructed based on the attention mechanism, the input features are mapped to Q, K, and V respectively through linear layers, and multi-level features are captured through a multi-head attention mechanism, and are output after passing through the first normalization layer, the fully connected layer, and the second normalization layer in sequence; when training the cross-modal feature alignment network, the addition of the alignment loss function and the cosine similarity loss function is used as the loss function, and the alignment loss function uses the mean square error.
7. The method for generating cross-modal conditions for genomics based on a potential diffusion model according to claim 4, characterized in that: During the joint fine-tuning, the guided condition and the denoising neural network use a cross-attention mechanism to perform feature fusion; the loss function used in the joint fine-tuning is the square of the L2 norm of the difference between the real noise under the guided condition and the predicted noise of the denoising neural network.
8. The method for generating cross-modal conditions for genomics based on a potential diffusion model according to claim 4, characterized in that: The initialization noise is Gaussian noise, and the iterative denoising uses a trained denoising neural network to gradually denoise the Gaussian noise.
9. A genomics cross-modal condition generation system based on a latent diffusion model, characterized in that: The method comprises one or more processors for implementing the genomics cross-modal condition generation method based on a potential diffusion model as described in any one of claims 1 to 8.
10. A computer-readable storage medium having a program stored thereon, characterized in that: When the program is executed by a processor, it is used to implement the genomics cross-modal condition generation method based on a potential diffusion model described in any one of claims 1 to 8.
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