My current research focuses on learning dynamics from data. One topic is nonparametric learning of the interaction laws in systems of interacting particles/agents, and another is data-driven model reduction for complex systems in computation such as fluid dynamics and molecular dynamics simulation. I view dynamical systems as a description of stochastic processes and take an inference approach to learn the dynamics from data, so I am also interested in closely related topics such as data assimilation, sequential Monte Carlo methods, deterministic and stochastic dynamical systems and PDEs, ergodicity theory, and learning theory.
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Data assimilation and inference
Data assimilation originates from numerical weather prediction, where the aim is to combine dynamical models with partial/noisy observation data to estimate states of the models. In general, it refers to statistical inference of the states and parameters of dynamical models. In practice, common methods for state estimation are filters (Kalman filters, ensemble Kalman filters, particle filters), which are empirical approximations of the high-dimensional Bayesian posterior in the framework of sequential Monte Carlo. A major challenge in DA is joint parameter-state estimation, in particular, when the inverse problem of parameter estimation is ill-posed. Such ill-posed problems arise when parameters reside on an unknown low-dimensional manifold due physical constraints. A new data-adaptive regularization method
DARTR is the state-of-the-art for linear inverse problems of learning hidden functions.
DA with stochastic reduced models.
F. Lu, X. Tu and A. J. Chorin. Accounting for model error from unresolved scales in ensemble Kalman filters by stochastic parametrization. Mon. Wea. Rev., 145(2017), no. 9, 3709--3723. Slides
DA with joint parameter-state estimation.
F. Lu, N. Weitzel and A. Monahan. Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data. Nonlin. Processes Geophys., 26, 227- 250, 2019. Poster Slides MATLAB code
Shock (extreme event) prediction by reduced model.
Nan Chen, Honghu Liu and F. Lu. Shock trace prediction by reduced models for a viscous stochastic Burgers equation arXiv2112 PDF