An optimization-driven physics-informed neural network for variable-order fractal–fractional differential equations: application to the bloch model

In machine learning, interest in neural fractional differential equations has surged due to their capabilities in modeling memory-dependent system dynamics. While most existing approaches concentrate on constant-order fractional derivatives, employing variable-order fractional operators provides a more adaptable and descriptive means of capturing intricate memory effects. Also, fractal–fractional derivatives extend classical fractional calculus by combining memory effects with fractal geometry, yielding more accurate models of complex real-world phenomena. In this study, we introduce the Physics-informed Neural Variable-Order Fractal–Fractional Differential Equation (PiNVoFFDE) network, a novel architecture that integrates variable-order fractal–fractional derivative in the Caputo sense with a trainable neural network guided by prior physical knowledge for solving variable-order fractal–fractional differential equations. By allowing the derivative order to adjust dynamically based on time, we obtain a more general model that captures richer update dynamics and delivers greater modeling flexibility. The Adams–Bashforth–Moulton predictor–corrector scheme is employed to provide the numerical solution to the PiNVoFFDE model, utilizing the recently introduced average-and-subtraction-based optimizer (ASBO) for model training. Using our proposed framework, we obtain the numerical solution of the system of fractal–fractional Bloch equations, which are a fundamental system of differential equations widely used in physics, chemistry, magnetic resonance imaging (MRI), and nuclear magnetic resonance (NMR) to describe how magnetization evolves under external magnetic fields and internal relaxation processes. The results reveal that PiNVoFFDE consistently outperforms constant-order fractional models and exceeds alternative approaches, demonstrating its superior adaptability and overall performance.

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