# Neural Network-Informed Lotka-Volterra Dynamics for Cryptocurrency Market Analysis
## 4 Hybrid Neural Network--ODE Framework for Parameter Estimation and Forecasting
### Mathematical Formulation
Prior studies in similar settings [38] have found that even a modest number of bootstrap replications (e.g., 5-10) can still provide a reasonable indication of variability and uncertainty, particularly when computational costs are high and data are limited. By applying this resampling technique, we obtained a realistic and statistically grounded estimate of the network's prediction accuracy under data variability. During training, a dynamic plot was used to monitor convergence and track loss reduction in real time. Once training was complete, the model was used to predict the time-dependent variables x(t), y(t), and z(t) across the full time horizon. The performance of the proposed method is highly influenced by both the choice of activation function and the strategy used for initializing weights and biases. The tanh function was chosen over sigmoid and ReLU because it centers outputs around zero, aiding in faster convergence and improving gradient propagation. Unlike sigmoid, tanh offers stronger gradients and mitigates the vanishing gradient issue, while also avoiding the \"dead neuron\" problem often associated with ReLU. Initial weights were drawn from a distribution of small values to keep the activation functions within their linear regions during early training, which promotes more stable learning dynamics. Similarly, initializing biases with small values prevents early saturation and preserves symmetry breaking across neurons. These combined practices contribute to improved stability, faster convergence, and higher accuracy, making the proposed framework well suited to estimating parameters in complex nonlinear systems such as those modeling cryptocurrency market dynamics. To address model sensitivity, we performed an extensive grid search varying the number of neurons in each hidden layer from 2 to 5, and from 10 to 15, as well as different learning rates and epochs. The most promising performance was consistently observed when the architecture ranged between 5 to 10 neurons per hidden layer, which we adopted as the focus of our main experiments. Table 1 presents the RMSE results across this comprehensive grid of neural network architectures and hyperparameter settings. The results show that, while some variation in RMSE exists across architectures, the overall pattern remains consistent, with the 10-6-9 architecture providing the lowest mean RMSE. This analysis demonstrates that our findings are robust to reasonable changes in network structure and training configuration. By presenting both the numerical results and our interpretation, we provide evidence that the main conclusions of the study do not depend sensitively on a single neural network architecture or hyperparameter setting. From Algorithm 2, we see that the DNN searched over all \(6^{3}=216\) possible three-hidden-layer DNN architectures (from 5 to 10 neurons per layer), for each learning rate and epoch combination. ```
1:Input: Crypto market share data; Search over hidden layer sizes \(h_{1},h_{2},h_{3}=5\ldots 10\); Learning rates \(\{0.00005,0.0001\}\); Epochs \(\{100,200\}\)
2:for each combination of \((h_{1},h_{2},h_{3})\), learning rate, and epoch count do
3:for each bootstrap replicate \(b=1\) to 5 do
4: Train the FFNN on a moving-block bootstrap sample of training data
5: Evaluate out-of-bag (OOB) validation cost for this replicate
6: Save weights of the replicate with lowest OOB cost
7:endfor
8: Estimate ODE parameters via least-squares using NN output; simulate with RK4; compute RMSE for BTC, ETH, ALTs
9: Store architecture configuration and RMSE
10:endfor
11: Sort all architectures by mean RMSE (RK4 simulation); select the top 10
```

**Algorithm 2** Selection of best neural network architecture. From Table 1, we select the best architecture based on the mean RMSE.