提出可微分的分数阶传播框架，实现误差可控的高效科学机器学习。
DFSC: Error-Controlled Differentiable Mittag-Leffler Propagation for Fractional Scientific Machine Learning
- 分离已知分数阶演化与数据驱动修正，神经网络仅学未解部分。
- 自适应算法确保计算误差在容差内，支持多类型算子路径和跨设备运行。
- 提供可验证的误差界，适合需精确控制误差的分数阶建模场景。
分数阶科学机器学习需要可微分、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由Mittag-Leffler传播子描述时，重复使用历史求解器或从数据中重新学习该响应是不必要的。我们提出DFSC，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修正分离，使神经模块仅学习未解析的动力学，同时联合优化分数阶参数与残差网络参数。其自适应算法通过增加特殊函数截断深度或Lanczos维数，直至连续可微评估满足指定容差。在负实数交替级数区域，DFSC还返回经认证的第一项省略项边界；在该区域外则明确标注估计为经验值。DFSC支持密集、稀疏、矩阵自由、自伴、广义及可控复数算子路径；可训练分数阶；直接求解反问题；残差神经网络组合；以及CPU/GPU执行。经认证的级数边界覆盖全部59个合格参考案例，已解误差的中位数边界/误差效度为1.246。重用预准备的批处理Lanczos基底可获得相同固定路径结果，显著降低重复查询时间：CPU上减少4.61–7.11倍，RTX 5070上减少13.07–16.22倍（不含一次性准备时间）。27例反矩阵求解保持全局满秩局部曲率，同时明确依赖模型条件。外部求解器和混合真实数据结果表明，DFSC作为误差感知的可选原语，适用于匹配分数阶结构的场景，而非通用替代分数阶求解器或神经模型。
原文摘要 · Abstract (English)
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.
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