Reproduce “Neural Fractional Differential Equations” on OpenResearch

About this paper

C. Coelho, M. Fernanda P. Costa, L.L. Ferrás

Neural Fractional Differential Equations (Neural FDEs) are introduced as a deep neural network framework to model complex systems with memory-dependent and non-local behaviors by learning both the dynamics and the fractional derivative order from observational data. The framework demonstrates enhanced performance over Neural ODEs in certain synthetic datasets, achieving MSE_avg values at least two orders of magnitude lower for the Relaxation Oscillation process, and shows improved convergence speed across synthetic and real-world tasks, though with a higher computational cost.

Key insights

• Integrating fractional calculus with neural networks enables the data-driven discovery of complex fractional dynamics, including the fractional order, for systems exhibiting memory-dependent behaviors.
• The computational complexity of the Neural FDE solver scales quadratically (O(N^2)) with the number of time steps (N), which is significantly higher than the linear scaling (O(N)) of Neural ODE solvers.
• The universal approximation capability of the `f_θ` network allows it to adapt to various fractional orders, potentially leading to learned α values that do not always precisely match the ground truth, as multiple α values can yield similar data fits.

Summary by alphaXiv.