# Neural Network-Informed Lotka-Volterra Dynamics for Cryptocurrency Market Analysis
## 4 Hybrid Neural Network--ODE Framework for Parameter Estimation and Forecasting
### Mathematical Formulation
The obtained coefficients are integrated into the Lotka-Volterra system (12), which is solved over the original time span using the classical fourth-order Runge-Kutta method [31]. The simulated trajectories are then compared to the observed data, and RMSE is computed to quantify fit quality. \[\frac{dy}{dt}=f(t,y),\quad y(t_{0})=y_{0} \tag{15}\]

The Runge-Kutta method approximates the solution to (15) through an iterative process defined as follows:

\[y_{n+1} = y_{n}+h\sum_{i=1}^{s}b_{i}k_{i}, \tag{16}\] \[k_{i} = f\Bigg{(}t_{n}+c_{i}h,y_{n}+h\sum_{j=1}^{i-1}a_{ij}k_{j}\Bigg{)},\quad\text{for}\,i=1,2,\ldots,s.\]

where

\(h\) represents the time step, \(s\) denotes the number of stages in the method, \(k_{i}\) are the intermediate stage values, and \(a_{ij},b_{i},c_{i}\) are coefficients specifying the particular Runge-Kutta scheme. From the general Runge-Kutta framework (16), various schemes of different algebraic orders can be derived. In this study, we utilize a fourth-order Runge-Kutta method with four stages [31]. Given an initial condition, \(y(t_{0})=y_{0}\), the solution at the subsequent time step \(t_{n+1}=t_{n}+h\) is calculated as follows:\[k_{1} = f\big{(}t_{n},y_{n}\big{)},\] \[k_{2} = f\big{(}t_{n}+c_{2}h,\ y_{n}+\big{(}a_{21}k_{1}\big{)}h\big{)},\] \[k_{3} = f\big{(}t_{n}+c_{3}h,\ y_{n}+\big{(}a_{31}k_{1}+a_{32}k_{2}\big{)} h\big{)}, \tag{17}\] \[k_{4} = f\big{(}t_{n}+c_{4}h,\ y_{n}+\big{(}a_{41}k_{1}+a_{42}k_{2}+a_{4 3}k_{3}\big{)}h\big{)},\] \[y_{n+1} = y_{n}\ +\ h\big{(}b_{1}k_{1}+b_{2}k_{2}+b_{3}k_{3}+b_{4}k_{4}\big{)}. \tag{18}\]

For the classical RK4 method, the coefficients are as follows:

\[c_{2}=\tfrac{1}{2},\quad c_{3}=\tfrac{1}{2},\quad c_{4}=1,\] \[\frac{dy}{dt}=f(t,y),\quad y(t_{0})=y_{0} \tag{15}\]0 \[\frac{dy}{dt}=f(t,y),\quad y(t_{0})=y_{0} \tag{15}\]1

This iterative procedure is applied at each time step, generating an approximate solution to the differential equation described by (15). ### Model Evaluation

The neural network was trained using the Adam optimizer, and epochs with an initial learning rate were selected after experiments with various values. To identify the optimal architecture, three different configurations of hidden layers were evaluated. In addition, multiple activation functions were tested, but the hyperbolic tangent (tanh) proved most effective, delivering the most stable and accurate results. For performance evaluation, a moving block bootstrap validation approach was employed. This method resamples blocks of the original time series to generate multiple training and testing datasets, preserving temporal dependencies within each block. Unlike traditional cross-validation or walk-forward methods, bootstrap validation is particularly useful when dealing with limited data or non-stationary time series [37]. It allows for robust statistical inference and a more comprehensive understanding of the model's generalization performance across different subsets of the data. We chose a block length of 10 time steps to span roughly one calendar month of share data, under the assumption that correlations persist over that window. We ran five bootstrap replicates, which we found sufficient to stabilize the out-of-bag (OOB) cost minima in preliminary tests. For each replicate, the OOB indices consist of all training points not selected for any block; the OOB cost is then computed on those held-out points to estimate generalization error. The replicate producing the lowest OOB cost is retained for the final neural-network weights. While we acknowledge that a larger number of bootstrap iterations is ideal for quantifying model uncertainty, the limited size of our dataset (93 observations) imposes practical constraints on both the statistical value and computational feasibility of increasing the bootstrap count. With a small dataset, extensive bootstrapping can lead to high overlap and reduced independence among bootstrap samples, thereby providing diminishing returns in terms of additional information.