Levenberg-Marquardt Method for Training Takagi-Sugeno Fuzzy Systems with Implementation Insights

The Levenberg-Marquardt method is a training technique used for optimizing Takagi-Sugeno fuzzy systems. This algorithm effectively optimizes neural network weights and biases to enhance system accuracy through iterative parameter updates. By combining gradient descent and Gauss-Newton methods, it provides an efficient solution for nonlinear least squares problems, making it particularly suitable for complex fuzzy system training. In implementation, the method calculates the Jacobian matrix of the error function with respect to the fuzzy system parameters (antecedent and consequent parameters). The core update equation follows: parameters_new = parameters_old - (J^T * J + λ * I)^(-1) * J^T * error, where λ serves as a damping factor that adaptively switches between gradient descent (large λ) and Gauss-Newton (small λ) behaviors. This adaptive mechanism ensures stable convergence while maintaining fast optimization speed. The method supports both supervised and unsupervised learning scenarios. For supervised learning, the implementation typically involves minimizing the mean squared error between fuzzy system outputs and target values through batch parameter updates. The training process iteratively adjusts membership function parameters and rule consequents, contributing to the development of fuzzy systems with high generalization capabilities. Key implementation considerations include proper initialization of damping factors, handling of matrix singularity conditions, and convergence criteria based on error tolerance thresholds.