# Neural Network-Informed Lotka-Volterra Dynamics for Cryptocurrency Market Analysis
## 4 Hybrid Neural Network--ODE Framework for Parameter Estimation and Forecasting
### Mathematical Formulation
Together, these three parameters allow ARIMA models to capture both short-term autocorrelation (through \(p\) and \(q\)) and nonstationarity (through \(d\)) in time series data. The neural ODE (NN-ODE) model is a deep learning approach for modeling time series dynamics by parameterizing the system of differential equations with neural networks. Instead of specifying fixed functional forms, NN-ODE learns the underlying evolution rules directly from the data. This allows the model to flexibly capture nonlinear and complex relationships in sequential data. The network is trained to minimize the difference between observed and predicted trajectories, using methods like Runge-Kutta for integrating the neural dynamics over time. As a result, NN-ODE offers a data-driven way to forecast system evolution, especially useful when the true governing equations are unknown or highly nonlinear. The various ARIMA configurations [41], while capable of capturing some of the underlying temporal patterns, generally yield higher root mean squared error (RMSE) values compared to the DNN-RK4 method. To facilitate an objective assessment, all models, including NN-ODE, are evaluated using the RMSE metric, and results are summarized both in graphical form and in a results table. This analysis provides strong evidence that the DNN-RK4 method offers superior predictive power for modeling the evolution of cryptocurrency market shares when compared to traditional ARIMA time series approaches and the neural ODE method. To evaluate the forecasting accuracy of the proposed DNN-RK4 approach, grounded in a nonlinear dynamical system framework, we compared its performance with that of classical time series models, specifically ARIMA, as well as the NN-ODE model. We evaluated multiple ARIMA(\(p,d,q\)) configurations, including \((2,1,2)\), \((2,1,1)\), and \((1,0,1)\). For each candidate model, we fitted the parameters on the training set and computed the RMSE on the validation set. The ARIMA(\(2,1,2\)) configuration produced the lowest RMSE, and was therefore chosen for comparison against our proposed method and NN-ODE. To optimize the NN-ODE model, we conducted an extensive grid search, varying the number of neurons in each of the three hidden layers from 3 to 10, with learning rates of 0.0001 and 0.001 and training epochs of 100 and 200. The mean RMSE (MRMSE) across all assets was used as our primary evaluation metric to identify the best configuration. This systematic process led to the selection of a 5-4-10 architecture with a learning rate of 0.001 and 200 training epochs, which consistently provided the lowest MRMSE and was adoptedfor our main experiments. The comparison focuses on the three main components of the market: Bitcoin (BTC), Ethereum (ETH), and alternative cryptocurrencies (ALTs). The DNN-RK4 model, driven by parameters estimated through a neural network, exhibited superior predictive performance across all market segments. For BTC, the DNN-RK4 model achieved an RMSE of 0.06865, significantly outperforming the ARIMA model's RMSE of 0.14742 and the NN-ODE's RMSE of 0.11757, as we notice in Figure 4. In the case of ETH, we see on Figure 5 that the differential was similarly notable--0.02682 (DNN-RK4) compared to 0.03665 (ARIMA) and 0.04507 (NN-ODE). For the ALT category, the DNN-RK4 model again showed improved accuracy, reducing the error from 0.11890 (ARIMA) and 0.11305 (NN-ODE) to 0.05581 as shown in Figure 6. These results highlight the advantage of using a model based on competitive dynamics to capture nonlinear relationships inherent to the cryptocurrency ecosystem, where investor behavior, technological updates, and market sentiment evolve rapidly. The cryptocurrency market is characterized by high volatility, interdependence between major assets, and the continuous entry of new tokens. These features make traditional linear models less effective in capturing complex feedback mechanisms and competitive interactions.