BP Algorithm for Function Fitting Using Multi-Layer Feedforward Networks

Backpropagation (BP) neural networks represent a classic multi-layer feedforward architecture commonly employed for regression tasks such as function fitting. To accomplish the specified objective, we can follow this structured approach:

Network Architecture Design The input layer size corresponds to the function's input dimension – for univariate function fitting, only one input node is required. Hidden layer neurons are typically determined experimentally, starting with a moderate quantity (e.g., 5-10 nodes) for iterative optimization. The output layer matches the function's output dimensionality, requiring just one node for single-output function approximation. Code implementation involves defining these layers using matrix dimensions, where weight matrices connect adjacent layers.

Data Preparation Partition datasets into training and testing subsets. The training set adjusts network weights through backpropagation, while the testing set evaluates generalization performance. Normalize input data to accelerate convergence and enhance training stability – commonly implemented using z-score standardization or min-max scaling algorithms.

Training Process Employ backpropagation for weight updates. Each iteration comprises: forward propagation to compute outputs, loss function calculation (e.g., Mean Squared Error), error backpropagation using chain rule differentiation, and weight adjustment via gradient descent. Learning rate selection is critical – excessive rates cause oscillation while insufficient values slow convergence. Momentum terms can be incorporated to optimize training by smoothing weight updates across iterations. Code implementation typically involves nested loops for epochs and data samples, with matrix operations for efficient gradient computation.

Performance Validation Post-training, evaluate network performance using test data by comparing predicted versus actual outputs. Plot fitted curves to identify underfitting or overfitting phenomena. Optimize models by tuning hidden layer neurons, learning rates, and regularization parameters. Key metrics include R-squared values and residual analysis, implemented through validation scripts that visualize prediction accuracy.

With proper network architecture design and training strategy configuration, BP neural networks effectively approximate complex nonlinear functional relationships through iterative error minimization.