Fractional Optimizers for LSTM Networks in Financial Time Series Forecasting

Mustapha Ez-zaiym, Yassine Senhaji, Meriem Rachid, Karim El Moutaouakil () and Vasile Palade ()
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Mustapha Ez-zaiym: Mathematics and Data Science Laboratory, Taza Multidisciplinary Faculty, Sidi Mohamed Ben Abdellah University, Fez 30000, Morocco
Yassine Senhaji: Mathematics and Data Science Laboratory, Taza Multidisciplinary Faculty, Sidi Mohamed Ben Abdellah University, Fez 30000, Morocco
Meriem Rachid: Faculty of Law, Economics and Social Sciences—LRMEF Fez, Sidi Mohamed Ben Abdellah University, Fez 30060, Morocco
Karim El Moutaouakil: Mathematics and Data Science Laboratory, Taza Multidisciplinary Faculty, Sidi Mohamed Ben Abdellah University, Fez 30000, Morocco
Vasile Palade: Centre for Computational Science and Mathematical Modelling, Coventry University, Priory Road, Coventry CV1 5FB, UK

Mathematics, 2025, vol. 13, issue 13, 1-44

Abstract: This study investigates the theoretical foundations and practical advantages of fractional-order optimization in computational machine learning, with a particular focus on stock price forecasting using long short-term memory (LSTM) networks. We extend several widely used optimization algorithms—including Adam, RMSprop, SGD, Adadelta, FTRL, Adamax, and Adagrad—by incorporating fractional derivatives into their update rules. This novel approach leverages the memory-retentive properties of fractional calculus to improve convergence behavior and model efficiency. Our experimental analysis evaluates the performance of fractional-order optimizers on LSTM networks tasked with forecasting stock prices for major companies such as AAPL, MSFT, GOOGL, AMZN, META, NVDA, JPM, V, and UNH. Considering four metrics (Sharpe ratio, directional accuracy, cumulative return, and MSE), the results show that fractional orders can significantly enhance prediction accuracy for moderately volatile stocks, especially among lower-cap assets. However, for highly volatile stocks, performance tends to degrade with higher fractional orders, leading to erratic and inconsistent forecasts. In addition, fractional optimizers with short-memory truncation offer a favorable trade-off between computational efficiency and modeling accuracy in medium-frequency financial applications. Their enhanced capacity to capture long-range dependencies and robust performance in noisy environments further justify their adoption in such contexts. These results suggest that fractional-order optimization holds significant promise for improving financial forecasting models—provided that the fractional parameters are carefully tuned to balance memory effects with system stability.

Keywords: fractional derivative; long short-term memory; financial forecasting (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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