# Neural Network-Informed Lotka-Volterra Dynamics for Cryptocurrency Market Analysis
## 4 Hybrid Neural Network--ODE Framework for Parameter Estimation and Forecasting
### Mathematical Formulation
The DNN-RK4 method, supported by neural network-driven parameter learning, demonstrates strong adaptability in forecasting such nonlinear systems. It is particularly useful in capturing turning points and saturation effects, which are common in speculative and innovation-driven markets like crypto. Further supporting these observations, Table 3 now compares both RMSE and MAE values across multiple modeling strategies. To underscore the practical benefits of our DNN-RK4 hybrid model, we present a concise overview of its forecasting performance relative to ARIMA and NN-ODE. The following summary distills the key RMSE improvements for each asset class. * BTC: DNN-RK4 halves the error of ARIMA (0.0687 vs. 0.1474 RMSE). * ETH: DNN-RK4 outperforms by approx. 27% (0.0268 vs. 0.0367 RMSE). * ALTs: DNN-RK4 reduces error by approx. 53% (0.0558 vs. 0.1189 RMSE). * Aggregate: Mean RMSE drops from 0.101 (ARIMA) and 0.092 (NN-ODE) to 0.050 (DNN-RK4). While RMSE has been our main evaluation criterion throughout the study, we also report MAE to provide a more robust and comprehensive assessment of forecasting accuracy, as MAE is less sensitive to large outliers. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & **BTC** & **ETH** & **ALTs** & **BTC** & **ETH** & **ALTs** \\  & **(RMSE)** & **(RMSE)** & **(RMSE)** & **(MAE)** & **(MAE)** & **(MAE)** \\ \hline DNN-RK4 & 0.06865 & 0.02682 & 0.05581 & 0.05432 & 0.02259 & 0.04735 \\ ARIMA (2,1,2) & 0.14742 & 0.03665 & 0.11890 & 0.13906 & 0.03370 & 0.10986 \\ NN-ODE & 0.11757 & 0.04507 & 0.11305 & 0.09701 & 0.03806 & 0.10388 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Root mean squared error (RMSE) and mean absolute error (MAE) values for different methods and components: BTC, ETH, and ALTs. Figure 6: Actual vs. predicted Altcoins market-share trajectories (February 2021–May 2023). \[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}|y_{i}-\hat{y}_{i}| \tag{27}\]

The DNN-RK4 model, in particular, demonstrated excellent performance, with RMSE values of 0.06865, 0.02682, and 0.05581 and corresponding MAE values of 0.05432, 0.02259, and 0.04735 for BTC, ETH, and ALTs, respectively. Notably, the mean MAE for DNN-RK4 (0.04142) was substantially lower than for ARIMA (0.09420) and NN-ODE (0.07965). This dual-metric analysis further confirms that DNN-RK4 consistently yields the most accurate predictions for cryptocurrency market shares among the methods compared. ## 6 Discussion

In recent years, the intersection of machine learning and dynamical systems has opened new directions for financial and economic modeling, especially in challenging environments like cryptocurrencies. Our work builds on these methodological developments, drawing inspiration from state-of-the-art deep learning architectures such as LSTM networks [42] and more recent advances using attention mechanisms and transformer models [43], which have demonstrated impressive forecasting accuracy in financial applications, including the crypto sector [44]. In addition to modeling price dynamics, modern approaches increasingly incorporate macroeconomic variables--for example, central bank policy rates--to explain asset fluctuations and regime shifts [45]. By grounding our approach in this broader context, we aim to bridge endogenous competition models with the latest advances in machine learning, highlighting the relevance and potential extensions of our framework to wider financial systems. To further improve the robustness and accuracy of cryptocurrency market modeling, future research will investigate a range of advanced machine learning and hybrid approaches tailored to the unique challenges of crypto dynamics. One promising direction involves temporal convolutional networks (TCNs) [46], which can outperform recurrent models in sequence modeling tasks while offering parallel processing advantages and stable gradients over long sequences--a critical factor--in training stability when analyzing extensive crypto market histories.