A Methodology for Adaptive Design of Fractal Neural Architectures for Forecasting Self-Similar and Multifractal Time Series

Conventional recurrent and convolutional-recurrent neural architectures are often effective in capturing local and sequential dependencies. However, they may be insufficient for representing hierarchical scale-dependent structures that occur in time series with fractal or multifractal properties. This study addresses this limitation by modifying the FractalNet-LSTM architecture through the modification of its fractal block. Rather than treating the fractal block as a fixed component, the proposed approach introduces parametric branching, which allows the number of parallel computational paths to be adapted to the structural complexity of the analyzed time series. The study evaluates the proposed architecture on three time series datasets from different domains. Prior to model training, the datasets were analyzed using methods to assess self-similarity, long-range dependence, and multifractal properties. The forecasting performance of the modified FractalNet-LSTM was compared with LSTM, BiLSTM, CNN-LSTM, and the classical FractalNet-LSTM model across several forecasting horizons. The experimental results indicate that the proposed architecture can improve forecasting accuracy for time series with pronounced fractal and multifractal characteristics. The advantage is most evident for datasets with heterogeneous scale-dependent behavior, while the improvement is less pronounced for weakly multifractal or near-monofractal series. For example, on the Industrial Boiler dataset, the proposed model increased R2 from 0.8046 to 0.9195 for the 8-step horizon compared with the classical FractalNet-LSTM. On the Green Energy Demand dataset, R2 increased from 0.9316 to 0.9808 for the 1-step horizon and from 0.7357 to 0.8445 for the 8-step horizon. The results indicate that shorter forecasting horizons may require fewer branches, whereas longer horizons and more complex multifractal structures may benefit from deeper or more expressive fractal blocks.