用分数阶布朗运动建模神经网络，提升长时记忆与鲁棒性。
Fractional Stochastic Neural Networks
- 引入分数阶布朗运动驱动的残差动态网络
- 在噪声回归中实现不确定性量化，生成效果优于传统方法
- 适合处理长时依赖与结构化扰动问题的研究者
本文提出一种由分数阶布朗运动驱动的残差动力学分数阶随机神经网络。通过构建离散随机最大值原理，推导出对应的伴随递归关系。对于确定性网络参数，证明了投影样本路径随机梯度下降的均方收敛性。数值实验包括闭式收敛测试、带不确定性的噪声回归、长记忆时间序列生成以及结构化扰动下的图像分类。结果表明，在长记忆恢复和鲁棒性方面，分数阶驱动模型优于布朗运动和确定性基线。
原文摘要 · Abstract (English)
In this paper, we develop a fractional stochastic neural network with residual dynamics driven by fractional Brownian motion. By introducing a discrete stochastic maximum principle for the network, we construct the corresponding adjoint recursion. For deterministic network parameters, we prove mean square convergence of projected samplewise stochastic gradient descent. Numerical experiments include a closed form convergence test, noisy regression with uncertainty quantification, long memory time series generation and image classification under structured perturbations. The results identify settings in which fractional drivers improve long memory recovery or robustness relative to Brownian and deterministic baselines.
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