DFSC Enables Error-Controlled Fractional Scientific Machine Learning

Fractional scientific machine learning, which involves models with fractional derivatives, requires specialized numerical operators that are differentiable, batchable, and compatible with neural networks. When the dominant linear fractional evolution is already understood through a Mittag-Leffler propagator, repeatedly solving or relearning this response is inefficient. This research presents DFSC, a PyTorch-based environment built around the Mittag-Leffler Spectral Layer (MLSL). DFSC's MLSL intelligently separates the known fractional propagation from any data-driven corrections, allowing neural modules to focus solely on learning unresolved dynamics. This architecture enables the joint optimization of fractional orders and residual-network parameters. A key feature is its adaptive algorithm, which adjusts truncation depth or Lanczos dimension until differentiable evaluations meet a specified tolerance. For specific regimes, DFSC even provides a certified bound on the first omitted term, ensuring high accuracy. The system supports various operator paths, trainable fractional orders, inverse problems, and CPU/GPU execution, demonstrating significant speedups and robust performance in inverse matrix problems.