  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.
