# Neural Network-Informed Lotka-Volterra Dynamics for Cryptocurrency Market Analysis
## 4 Hybrid Neural Network--ODE Framework for Parameter Estimation and Forecasting
### Mathematical Formulation
RMSE evaluates the accuracy of predictions by measuring the average error between observed and predicted value, based on a validation procedure:

\[\frac{dy}{dt}=f(t,y),\quad y(t_{0})=y_{0} \tag{15}\]2

Then, we extract the coefficients for all three and solve the system using the fourth-order Runge-Kutta method. Based on the comparison of the mean RMSE, we choose the best one, which we then include in the graphs. ## 5 Numerical Experiment

### Case Study Description

This study examines the historical market penetration and future prospects of the three crypto competitors. Monthly market share data for Bitcoin (BTC), Ethereum (ETH), and altcoins (ALTS) were sourced from CoinMarketCap historical snapshots [39], covering August 2015 to May 2023, as shown in Figure 2, a period of 93 months (almost 8 years). The market share of alternative cryptocurrencies (ALTs) is derived by subtracting the shares of Bitcoin (BTC) and Ethereum (ETH) from 1, ensuring that the total market share across all segments sums to unity. The initial step in assessing the performance of the proposed framework involves estimating the parameters of the Lotka-Volterra system defined in Equation (2). While traditional approaches typically rely on predefined assumptions derived from the data, this study adopts a data-driven strategy. Specifically, a deep neural network is employed \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline \(h_{1}\) & \(h_{2}\) & \(h_{3}\) & BTC & ETH & ALTs & MRMSE & LR & Epochs \\ \hline
10 & 6 & 9 & 0.056355 & 0.097497 & 0.096556 & 0.083469 & 0.0001 & 200 \\
10 & 6 & 9 & 0.063268 & 0.092336 & 0.097758 & 0.084454 & 0.0001 & 100 \\
8 & 10 & 10 & 0.068767 & 0.097675 & 0.101360 & 0.089267 & 0.0001 & 200 \\
7 & 7 & 6 & 0.062612 & 0.100960 & 0.102480 & 0.088684 & 0.0001 & 200 \\
7 & 10 & 10 & 0.071430 & 0.096462 & 0.102860 & 0.090251 & 0.0001 & 200 \\
6 & 5 & 10 & 0.063437 & 0.101870 & 0.102940 & 0.089416 & 0.0001 & 200 \\
9 & 10 & 5 & 0.071015 & 0.099051 & 0.103100 & 0.091055 & 0.00005 & 200 \\
9 & 9 & 10 & 0.068600 & 0.099730 & 0.104540 & 0.090957 & 0.0001 & 200 \\
10 & 8 & 10 & 0.058409 & 0.114040 & 0.105150 & 0.092533 & 0.0001 & 200 \\
9 & 10 & 5 & 0.072399 & 0.100790 & 0.105270 & 0.092820 & 0.0001 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Per-asset RMSEs (BTC, ETH, and ALTs) and mean RMSE (MRMSE) for various NN architectures and hyperparameters. Figure 2: Evolution of cryptocurrency market from August 2015 to May 2023.

to infer the system parameters by effectively \"training\" the model based on time-series data of cryptocurrency market shares. This neural network-based estimation replaces heuristic techniques and enables the model to capture the underlying dynamics through optimization and learning from observed behavior. The results of the application of DNN for the case studied provided the following values for the corresponding parameters:

\[\frac{dx}{dt} = x\big{(}0.5358-0.4521\,x+0.7681\,y-1.3624\,z\big{)},\] \[\frac{dy}{dt} = y\big{(}1.4664-1.2830\,x-1.6997\,y-1.7748\,z\big{)}, \tag{20}\] \[\frac{dz}{dt} = z\big{(}0.1941-0.1602\,x-0.5119\,y-0.1015\,z\big{)}.\]

The estimated parameters, derived from neural network-assisted system identification, provide a quantitative picture of the mutual influences and self-regulatory mechanisms among Bitcoin (BTC, \(x\)), Ethereum (ETH, \(y\)), and Altcoins (ALTs, \(z\)) in the cryptocurrency market. The associated uncertainty quantification, as reported on Table 2, indicates that while some interactions are well identified, others remain uncertain, and thus, all findings must be interpreted in this context. Intrinsic growth rates--\(p_{1}=0.5358\) for BTC, \(p_{5}=1.4664\) for ETH, and \(p_{9}=0.1941\) for ALTs--suggest that BTC and ETH may experience positive baseline growth in isolation, while ALTs are closer to neutral. The confidence interval for BTC is relatively narrow (low uncertainty), while those for ETH and ALTs are much wider, reflecting high or medium uncertainty and indicating that considerable variability is possible.