ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems
This work addresses the problem of computational inefficiency in simulating ergodic SDEs for researchers and practitioners in fields like physics and engineering, offering a novel method that is incremental in its adaptation of existing numerical schemes.
The paper tackles the challenge of efficiently simulating stochastic differential equations (SDEs) with locally Lipschitz properties, which typically require small time-steps, by introducing an inference-based framework (ISALT) that learns flow maps from data, achieving a reduction in simulation time by several orders of magnitude and optimal accuracy in reproducing invariant measures with medium-large time-steps.
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.