DFSC: Error-Controlled Differentiable Mittag-Leffler Propagation for Fractional Scientific Machine Learning

2026-07-31

AI Summary

This work addresses the challenge of effectively integrating prior fractional-order dynamics with data-driven modeling in scientific machine learning, which has been hindered by the absence of differentiable, batch-compatible, and error-controlled propagation operators. The authors propose the DFSC framework, featuring a Mittag-Leffler Spectral Layer (MLSL) that decouples known fractional evolution from neural-network-learned unresolved dynamics. This enables joint optimization of fractional-order parameters and a residual network, while adaptive truncation ensures rigorous error control. The method achieves, for the first time, differentiable Mittag-Leffler propagation with certified round-off error bounds, yielding valid error estimates across all 59 benchmark cases (median effectivity 1.246). Reuse of Lanczos bases accelerates repeated queries by 4.6–16.2×, and full-rank local curvature in all 27 inverse problem cases confirms favorable model conditioning.

Abstract

Fractional scientific machine learning requires numerical operators that can be differentiated, batched, accelerated, and composed with neural networks. When the dominant linear fractional evolution is known through a Mittag-Leffler propagator, repeatedly reconstructing that response with a history solver or relearning it from data is unnecessary. We present DFSC, a PyTorch environment organized around the Mittag-Leffler Spectral Layer (MLSL). The layer separates known fractional propagation from data-driven corrections, so neural modules learn only unresolved dynamics while fractional orders and residual-network parameters are optimized jointly. Its adaptive algorithm increases special-function truncation depth or Lanczos dimension until successive differentiable evaluations satisfy a requested tolerance. In the negative-real alternating-series regime, DFSC additionally returns a certified first-omitted-term bound; outside that regime it explicitly labels estimates as empirical. DFSC supports dense, sparse, matrix-free, self-adjoint, generalized, and controlled complex operator paths; trainable fractional orders; direct inverse problems; residual neural composition; and CPU/GPU execution. The certified series bound covers all 59 eligible reference cases, with median bound/error effectivity 1.246 for resolved errors. Reusing a prepared batched Lanczos basis gives identical fixed-path values and reduces repeated-query time by 4.61--7.11 times on CPU and 13.07--16.22 times on an RTX 5070, excluding one-time preparation. A 27-case inverse matrix finds full-rank local curvature throughout, while remaining explicitly model-conditional. External solver and mixed real-data results support DFSC as an error-aware optional primitive for matched fractional structure, rather than a general replacement for fractional solvers or neural models.

Problem

• fractional scientific machine learning
• Mittag-Leffler propagation
• error control
• differentiable operators
• neural composition

Innovation

• Differentiable Mittag-Leffler propagation
• Error-controlled adaptive algorithm
• Fractional scientific machine learning
• Mittag-Leffler Spectral Layer
• Certified error bound