Stochastic differential equations (SDEs) are widely used to model complex stochastic dynamical systems. For observational data that follows stochastic dynamical systems driven by random perturbations, modeling the data using SDEs to capture these dynamics is essential for understanding the underlying probabilistic behaviors. Recently, Neural SDEs have emerged as a powerful tool for modeling continuous-time stochastic dynamics. By naturally accommodating irregular observation intervals and leveraging the adjoint method for memory-efficient computations, Neural SDE provides a flexible framework for continuous-time modeling. This paper proposes a flexible and scalable framework for flexibly learning Neural SDE expressions with Brownian motions as functions of both observed values and time, ensuring that the correct formulations for both the drift and diffusion terms are obtained. Our data-driven Neural SDE identification framework (NSDE-ID) leverages the path signature, which is a collection of all the iterated integrals and efficiently extracts features from time-series data, and incorporates path-dependent distributions, enabling SDE estimation that extends beyond point-in-time predictions. Building upon recent advancements in Neural SDE, our approach is designed to enforce statistical consistency in the learning process, thereby enabling robust modeling of complex dynamics. We demonstrate the efficacy of NSDE-ID on three benchmark SDE parameter estimation problems and analyze its numerical performance and robustness. Overall, NSDE-ID offers a promising new direction for systematically unraveling the continuous stochastic dynamics within observational data through a flexible SDE representation.

Understanding the formation mechanism of magnetic domain patterns is important to improve the performance of magnetic materials. The magnetic domain patterns depend on the parameters of the time-dependent Ginzburg-Landau (TDGL) equation and the sweep rate of the external magnetic field. Although conventional analytical approaches can predict the patterns formed in the case of a high sweep rate, more versatile methods are required to understand the formation mechanism of complicated patterns for any temporal variation in the external field. This study proposes a method that extracts dominant modes from a posterior distribution based on the four-dimensional variational method (4DVar). The method decomposes the magnetic domain patterns into eigenvectors of the Hessian matrix of the cost function, defined as the difference between the observed and simulated magnetic domain patterns. The eigenvectors are extracted using a second-order adjoint method (SOA) and power iteration. The patterns are reconstructed by superimposing the extracted eigenvectors, and their time evolution can be obtained from the weights of the eigenvectors. Experiments demonstrate that the patterns are sufficiently reconstructed using a small number of the eigenvectors. This enables us to understand the pattern evolution as the change of the dominant eigenvectors.