# Multi-Objective Robust Optimization Reconstruction Algorithm for Electrical Capacitance Tomography
## 5 Multimodal Neural Network
### The Proposed Multimodal Neural Network  
Multimodal machine learning is a technique that integrates data from different sources and formats. Its advantage lies in the ability to comprehensively utilize various information sources, thereby improving the accuracy and robustness of models. This approach can overcome the limitations of single-modal data by integrating multiple types of data such as images, text, and audio, thereby enhancing feature representation capabilities [69, 70].  
We propose the MNN to enhance the inference accuracy of the MLPIs by jointly processing image and capacitance data. Its architecture includes input and output layers and multiple fully connected hidden layers (Figure 1). Unlike conventional networks, the MNN uniquely accepts dual-modal inputs (image and capacitance data), enabling cross-modal information fusion to surpass single-source limitations and boost accuracy.  
Training the MNN represents a pivotal step, necessitating the solution to the following optimization problem:  
\[\min_{\mathbf{\Theta}}\Bigl{\{}\sum_{i=1}^{Sam}\Bigl{|}\Bigl{|}\psi_{MDNN}(\mathbf{c}_ {i};\mathbf{\Theta})-\mathbf{\theta}_{i}\Bigr{|}\Bigr{|}_{2}^{2}\Bigr{\}} \tag{25}\]  
where \(\mathbf{\Theta}\) denotes model parameters; \(\{(\mathbf{c}_{i},\mathbf{\theta}_{i})\}_{i=1}^{Sam}\) defines training sample pairs; and \(\psi_{MDNN}(\cdot)\) stands for a forward propagation operator, and its form is determined by the network architecture.  
The training workflow is summarized in Algorithm 3. The training of the MNN is in an offline mode, which means that all complex computations and model parameter optimization are conducted in the training phase, thus minimizing the computational burden in the inference phase. After sufficient training, the MNN is able to achieve high efficiency and speed in the inference process. Through the combination of offline training and efficient inference, the MNN shows strong potential for practical applications, not only in terms of computation time but also in terms of high accuracy and stability, providing a feasible solution to complex imaging problems.  
```
1. Input: Training samples \(\{(\mathbf{c}_{i},\mathbf{\theta}_{i})\}_{i=1}^{Sam}\).
2. Output: \(\mathbf{\Theta}\).
3. Determine the network structure.
4. Initialize model parameters.
5. Solve (25) by the stochastic gradient descent method.
```  
**Algorithm 3**MNN Training.  
This new MNN offers the following advantages:  
(1) The MNN can integrate data from various modalities (e.g., image and capacitance data), effectively utilizing information from different sources. This fusion compensates for the limitations of single-modality information, enhances model performance, and reduces the potential information loss associated with using only one modality.  
Figure 1: Multimodal neural network.  
(2) Through its multi-layer structure, the MNN extracts features that capture complex data patterns and relationships. Unlike traditional methods that require manual feature design, the MNN can automatically learn useful features from raw multimodal data, reducing dependence on manual expertise.  
(3) By integrating multimodal data, the MNN improves model performance. The complementary advantages of different modalities help reduce errors and biases associated with single modalities, providing more stable and reliable predictions. Moreover, the ability to leverage redundant information from multimodal data allows the MNN to maintain strong performance even in the presence of noise and incomplete data.