## 2 Materials and Methods
### Neural Network Design and Implementation  
Our analytical framework primarily leverages ANNs, specifically Nonlinear Autoregressive Networks with Exogenous Inputs (NARX) [16], to derive metabolic costs from biomechanical time series data. Our choice to utilize ANNs, particularly NARX, over traditional statistical methods in this investigation is driven by their superior capability in modeling complex nonlinear dynamics often inherent in time series datasets, such as those encountered in biomechanical data analysis [15]. In ANNs, 'hidden layers' are intermediary layers between the input and output, crucial for learning complex data patterns. These layers do not interact directly with the input or output but process and transform the information, allowing the ANN to handle intricate problems. 'Delays' in the context of ANNs, mainly when dealing with time series data, are essentially a way to incorporate historical information into the model's current prediction. The concept revolves around using data from previous time steps as input for predicting future outcomes.  
In our investigation, we operationalized two distinct NARX models tailored to the two types of biomechanical data: the GRF and joint moments. This bifurcation was strategically aimed at harnessing the specific strengths of NARX networks in handling the diverse complexities inherent in each data type [16]. Each NARX network was configured with an input layer tailored to accommodate the stride-averaged time series data, hidden layers whose size was optimized based on preliminary trials to balance the trade-off between model complexity and predictive accuracy, and an output layer designed to estimate the metabolic cost. The networks underwent rigorous training, validation, and testing processes, utilizing a subset of the compiled dataset, with performance metrics, such as the mean squared error (MSE) and correlation coefficients (R-values), serving as key evaluative criteria [24].  
We developed two distinct ANN models using time series data from the GRF and joint moments across a dataset of 270 trials (Figure 1). We selected a 70% (188 trials) training, 15% validation (41 trials), and 15% testing (41 unseen trials) split for each model to ensure a robust learning foundation while maintaining enough data for meaningful validation and testing. The NARX model was trained using the Levenberg-Marquardt backpropagation algorithm. The training process utilized a learning rate of 0.001, a batch size of 15, and 1000 epochs.  
Figure 1: The architecture of the neural network model. This schematic represents the ANN models for predicting metabolic cost from biomechanical data. The upper network illustrates the model using hip, knee, and ankle moments as inputs to predict metabolic cost. The lower network demonstrates the model utilizing vertical (\(\text{GRF}_{\text{ver}}\)) and anterior–posterior (\(\text{GRF}_{\text{AP}}\)) ground reaction forces as inputs to predict metabolic cost. Both models highlight the data flow from the input through the hidden layers to the output, depicting the ANN’s ability to process complex biomechanical inputs to predict energetic expenditure.