多尺度扩散中漂移识别的连续时间随机梯度下降
Stochastic gradient descent in continuous time for drift identification in multiscale diffusions
AI总结：
本文针对多尺度过阻尼朗之万方程，提出用连续时间随机梯度下降学习均质化漂移，通过核滤波解决模型误设定，数值验证精度并理论上证明渐近无偏性。
摘要：
我们考虑多尺度过阻尼朗之万随机微分方程的设定，并研究从多尺度系统的连续时间观测中学习均质化动力学的漂移函数的问题。我们将漂移项分解为基函数的截断级数，并采用连续时间随机梯度下降来推断展开系数。由于多尺度数据与均质化模型之间的不兼容性，仅凭估计器无法重建精确的漂移。因此，我们提出通过适当的核过滤原始轨迹，并将过滤后的数据纳入估计器的随机微分方程中，这确实解决了误设定问题。多个数值实验突出了我们方法的准确性。此外，我们在一个简化框架中从理论上证明了，在无限数据极限以及描述最快尺度的多尺度参数消失时，我们的估计器具有渐近无偏性。
英文摘要：
We consider the setting of multiscale overdamped Langevin stochastic differential equations, and study the problem of learning the drift function of the homogenized dynamics from continuous-time observations of the multiscale system. We decompose the drift term in a truncated series of basis functions, and employ the stochastic gradient descent in continuous time to infer the coefficients of the expansion. Due to the incompatibility between the multiscale data and the homogenized model, the estimator alone is not able to reconstruct the exact drift. We therefore propose to filter the original trajectory through appropriate kernels and include filtered data in the stochastic differential equation for the estimator, which indeed solves the misspecification issue. Several numerical experiments highlight the accuracy of our approach. Moreover, we show theoretically in a simplified framework the asymptotic unbiasedness of our estimator in the limit of infinite data and when the multiscale parameter describing the fastest scale vanishes.
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