神经随机微分方程可逼近任意随机微分方程，具备通用近似能力。
Universal approximation property of neural stochastic differential equations
- 基于带全局线性增长约束的神经网络构造神经SDE
- 在系数足够光滑时，误差可量化且趋于零
- 适用于需要精确建模随机过程的场景
我们识别出若干类神经网络，可在固定全局线性增长约束下，局部一致地逼近连续函数。对于此类神经网络，其对应的神经随机微分方程能够任意精确地逼近各类Itô扩散型随机微分方程。此外，针对系数充分光滑的随机微分方程，推导出了定量误差估计。
原文摘要 · Abstract (English)
We identify various classes of neural networks that are able to approximate continuous functions locally uniformly subject to fixed global linear growth constraints. For such neural networks the associated neural stochastic differential equations can approximate general stochastic differential equations, both of Itô diffusion type, arbitrarily well. Moreover, quantitative error estimates are derived for stochastic differential equations with sufficiently regular coefficients.
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