An Adaptive BFO-based Framework for Neural Network Function Approximation and Optimization for Sustainable Systems

In view of the multi-variable, multi-objective and complex nonlinear dynamic characteristics existing in sustainable systems, this study proposes an Adaptive Bacterial Foraging Optimization-Neural Network (Adaptive BFO-NN) joint optimization framework. This framework is used for function approximation and performance optimization in a breakthrough way. Based on the traditional BFO, the framework introduces an adaptive step size and direction update mechanism, a local search enhancement strategy and multi-objective joint optimization method. First, the framework uses the covariance matrix to initialize the position of the bacterial population to improve the population diversity. Subsequently, by combining the energy function with the adaptive inertia weight adjustment of the neighborhood information, the framework achieves a balance between global exploration and local development. As an optimization, local fine-tuning operators are regularly introduced during the iteration process to improve the convergence accuracy. The neural network part uses a time-delay neural network (TDNN) to model the input-output mapping. And the weight update is driven by a multi-objective fitness function. This fusion comprehensively considers prediction errors, resource utilization constraints and sustainability indicators. The comprehensive performance of this method was verified across three types of test tasks. Experimental results show that in the synthetic time-delay system test, the RMSE was $0.0126 \pm 0.0037$, an average reduction of approximately $15.53 \%$ compared to the Adam algorithm. In the sustainable energy simulation with high noise and multi-dimensional inputs, the RMSE was $0.4208 \pm 0.0591$, an average reduction of $22.06 \%$ compared to Adam. In the short-term prediction of chaotic sequences, the MSE was $0.0852 \pm 0.0181$, a $22.28 \%$ reduction compared to Adam.