Stochastic differential equations (SDEs) are used to describe a wide variety of complex stochastic dynamical systems. Learning the hidden physics within SDEs is crucial for unraveling fundamental understanding these systems' and nonlinear behavior. We propose flexible scalable framework training artificial neural networks learn constitutive that represent SDEs. The proposed physics-informed ord...

We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those belong a certain prescribed set. establish two types sufficient conditions under non\-auto\-nomous ordinary differential equations have such property. The first criterion applies semilinear equation provided that its linear part is hyperbolic and nonlinearity small in neig...