Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measure. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, a hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases. We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multi-scale gradient system, and the 3D stochastic Lorenz equation with degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.

中文翻译：

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ISALT：适用于本地Lipschitz遍历系统的基于大时间步长的基于推理的方案

对SDE的高效仿真在许多应用中至关重要，尤其是对于需要对短时动态和长时间统计进行高效仿真的遍历系统。但是，本地的Lipschitz SDE经常需要特殊处理，例如具有较小时间步长的隐式方案，以准确地模拟遍历度。我们引入了一个框架，该框架可用于构造基于推理的方案，以适应数据中的大时间步长（ISALT），从而将时间减少了几个数量级。关键是对无限维离散时间流图的近似值的统计学习。我们探索了使用数值方案（例如Euler-Maruyama，混合RK4和隐式方案）来导出信息基础函数，从而导致参数推断的问题。我们引入了一种可伸缩的算法，以最小二乘估计参数，并且证明了随着数据大小的增加，估计器的收敛性。我们在三个非全局性Lipschitz SDE上测试ISALT：1D双阱势，2D多尺度梯度系统和具有退化噪声的3D随机Lorenz方程。数值结果表明，与普通数值方案相比，ISAL可以容忍更大的时间步幅。当时间步长为中等时，在再现不变度量时达到最佳精度。数值结果表明，与普通数值方案相比，ISAL可以容忍更大的时间步幅。当时间步长为中等时，在再现不变度量时达到最佳精度。数值结果表明，与普通数值方案相比，ISAL可以容忍更大的时间步幅。当时间步长为中等时，在再现不变度量时达到最佳精度。