## 6 Metric  
We designed a metric LOE to measure the gradient problem, where \(loss_{t}\) denotes the loss value at moment \(t\), epoch denotes the current training round, and total_epoch denotes the total number of training rounds. \[\text{LOE}=\left|\frac{loss_{t}-loss_{t-1}}{loss_{t-1}+\alpha}\right|\cdot e^{ \frac{\rho\text{ epoch}}{\text{total\_epoch}}} \tag{18}\]  
As training proceeded, we noticed that the negative impact of the gradient problem on model training gradually increased in the later stages of training. In order to quantify this, we designed a correction mechanism containing an exponential term for correcting the severity of the gradient problem. In the index we designed, the parameter LOE represents the degree of impact of the gradient problem on training, with larger values indicating a more significant impact of gradient explosion. Specifically, we first calculated the proportion of the increase in loss at each time node relative to the previous time node; this proportion requires an absolute value operation because both gradient vanishing and gradient explosion can occur. We then multiplied this proportion by an e-exponential correction term, where the value of the exponent is related to the number of training rounds and model parameters. By introducing such a correction mechanism, we can more accurately visualize the extent to which the model is experiencing gradient problems and can effectively improve the model based on these indicators. The performance comparison of the models using WGN and without WGN method is shown in Tables 3-6 and the performance comparison of using the WGN method with gradient clipping and the GRU model is shown in Tables A1 and A2. Their experimental configurations are all the same as those described in the previous experimental section. Figure 10: Plots of the ECG5000 dataset in UCR with and without normalized training effects under different models. Figure 9: Plots of the Computers dataset in UCR with and without normalized training effects under different models. In the language models, the perplexity of the models utilizing WGN is typically lower than that of the models not employing WGN. Moreover, both the perplexity and the LOE of the models utilizing WGN demonstrated a decline as the number of layers increased. The perplexity reached a minimum in the two-layer LSTM model with WGN, and the LOE was found to be average. In the context of the image dataset classification, the highest accuracy was observed when the WGN method was employed with a network comprising two layers, utilizing either the RNN or the LSTM model. Furthermore, the incorporation of the WGN method into the RNN model resulted in a significant reduction in the LOE. In contrast, the introduction of WGN into the LSTM model led to a notable enhancement in accuracy, although this was accompanied by an increase in LOE. For the prediction task of the ETTm1 dataset, we observed that the performance of recurrent neural networks or LSTM models with different numbers of layers combined with the WGN method varied under different prediction length conditions. Firstly, at a prediction length of 96, the two-layer RNN-WGN model exhibited the lowest mean absolute error, while the single-layer RNN-WGN model demonstrated the lowest LOE. Conversely, at a prediction length of 24, the one-layer LSTM-WGN model exhibits the most favorable outcomes, while the one-layer RNN-WGN model demonstrates the lowest LOE. This indicates that the utilization