发表机构
School of Mechanical Engineering, Hangzhou Dianzi University; School of Computer Science, Hangzhou Dianzi University; School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University; University of California, Santa Cruz（杭州电子科技大学机械工程学院; 杭州电子科技大学计算机学院; 西交利物浦大学数学与物理学院; 加州大学圣克鲁兹分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出围绕Mittag-Leffler谱层构建的PyTorch环境DFSC，用于分数阶科学机器学习，可实现误差可控的可微分数阶传播，能降低计算时间并适配多种算子路径，为分数阶结构提供误差感知的可选原语。
AI 中文摘要
分数阶科学机器学习需要可微、可批处理、可加速且能与神经网络组合的数值算子。当主导线性分数阶演化可通过Mittag-Leffler传播子确定时，无需用历史求解器重复重构该响应或从数据中重新学习。我们提出DFSC，这是一个围绕Mittag-Leffler谱层（MLSL）构建的PyTorch环境。该层将已知的分数阶传播与数据驱动校正分离，使神经模块仅学习未解析的动力学，同时联合优化分数阶阶次与残差网络参数。其自适应算法会增加特殊函数截断深度或Lanczos维度，直到连续可微评估满足要求的容差。在负实交替级数区间，DFSC还返回经认证的首项遗漏项界；在该区间外，则明确将估计值标记为经验值。DFSC支持稠密、稀疏、无矩阵、自伴、广义及受控复算子路径、可训练分数阶阶次、直接逆问题、残差神经网络组合，以及CPU/GPU执行。该经认证的级数界覆盖全部59个符合条件的参考案例，针对已解析误差的中位数界/误差效能为1.246。重复使用已准备的批处理Lanczos基可得到相同的固定路径值，且在CPU上减少4.61-7.11倍的重复查询时间，在RTX 5070上减少13.07-16.22倍（不含一次性准备时间）。一个含27个案例的逆矩阵全程呈现满秩局部曲率，同时保持明确的模型条件性。外部求解器与混合真实数据结果表明，DFSC是适配分数阶结构的误差感知可选原语，而非分数阶求解器或神经模型的通用替代品。
英文摘要
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.
Comments20 pages, 8 figures. Code and reproducibility materials: https://github.com/hzhooning-art/DFSC and https://doi.org/10.5281/zenodo.21588834
