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报告题目：Learning stochastic nonlinear dynamics via maximising transition likelihood

报告人：Henrik Sykora 博士 （布达佩斯科技与经济大学）

时间：2025年10月27日（周一）15:00

地点：明故宫校区础18号楼705会议室

主办单位：航空航天结构力学及控制全国重点实验室、智能装备动力学中心、振动工程研究所、青青草国产免费观看、校科协、国际合作处

报告内容摘要：

In recent years, the application of statistical and machine learning methods has gained considerable ground in the measurement-based identification of nonlinear dynamical systems that describe phenomena in various disciplines in science. Finding the dynamical system generating the measured time-series data has many benefits over directly fitting a statistical model, such as regression with a linear combination of features or neural networks. Directly applying such methods results in models that can reproduce the measured data, however, they usually do not extrapolate well to out-of-sample scenarios. An additional issue is that the parameters of these models often lack physical meaning, and the models are not limited by physical principles.

To remedy this limitation, a novel trend, physics-based machine learning, has arisen, where first-principles models are directly integrated into statistical models. This includes techniques such as fitting a neural ordinary differential equations, where the state space is specified by a neural network, or the sparse identification of nonlinear dynamics (SINDy) where the dynamics of the system is assumed as the linear combination of nonlinear candidate functions. Most works focus on identifying deterministic dynamics, however, stochastic effects can influence the behaviour of dynamical systems, both qualitatively and quantitatively. However, capturing the stochastic effects inherent in many dynamical systems is challenging, especially when the data acquisition is subjected to measurement noise. Classical statistical models that aims such systems include autoregressive, moving average, integrated processes, and their combinations. However, these models are restricted to capture linear behaviour, lack physical meaning, and have short forecasting horizons, as they converge to a constant mean.

This talk aims to introduce a method to extend the library of data-driven methods to identify dynamical systems with nonlinear and inherently stochastic behaviour from measured time series data corrupted by measurement noise. The method is based on maximising the likelihood of the transitions between the data points. That is, with this method we aim to find the stochastic nonlinear dynamical system that generates the measured time series data with the highest probability. An advantage of this formulation is its flexibility, therefore it can be generalised for various scenarios, classes of complex dynamical systems (e.g. for systems with delays) and different noise models. After providing a detailed description of the method, we demonstrate its capabilities and behaviour on synthetic datasets generated by stochastic nonlinear oscillators and investigate the statistical properties of the estimators through numerical experiments. We discuss challenges and strategies for estimating model and observation noise parameters, such as the interaction between observation noise intensity and numerical stability, which reveals the properties of the estimators and trade-offs between observation time step and accuracy. Furthermore, we demonstrate that the flexible formulation we present allows even the use of neural networks (a highly nonlinear structure with model parameters embedded in multiple layers of nonlinearities) to describe both the drift and diffusion dynamics, enabling the identification of stochastic dynamics without knowledge of the underlying phenomena driving the system. Finally, we review the current challenges, open questions, and directions for generalising the presented estimation method to different types of dynamical systems.