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.

## Learning self-interacting particle systems

** Nonparametric inference of interaction kernel.** Self-interacting systems of particles/agents arise in many areas of science, such as particle systems in physics, flocking and swarming models in biology, and opinion dynamics in social science. A natural question is to learn the laws of interaction between the particles/agents from data consisting of trajectories. In the case of distance-based interaction laws, we present efficient regression algorithms to estimate the interaction kernels, and we develop a nonparametric statistical learning theory addressing identifiability, consistency and optimal rate of convergence of the estimators. Especially, we show that despite the high-dimensionality of the systems, optimal learning rates as in 1D can be achieved.

** Systems of particles (multi-agent systems):**
F. Lu, M Zhong, S Tang and M Maggioni. Nonparametric inference of interaction laws in systems of agents from trajectory data. Proc. Natl. Acad. Sci. USA. 116 (29) 14424--14433. 2019 Journal PDF (SI) Slides
F. Lu, M. Maggioni and S. Tang: Learning interaction kernels in heterogeneous systems of agents from multiple trajectories. arXiv1910
Zhongyang Li, F. Lu, Mauro Maggioni, Sui Tang and Cheng Zhang: On the identifiability of interaction functions in systems of interacting particles. to appear on Stoch.Process.Their Appl. arXiv1912 PDF
F. Lu, M. Maggioni and S. Tang. Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories. arXiv2007
** Mean field equations: ** We introduce a least-squares algorithm to learn the interaction function from data consisting the solution of the PDE. Such a efficient algorithm is possible by introducing a probabilistic error functional (cost function). We show that the estimator converges at an optimal rate, which is the order of the numerical integrator used to approximate the expectations in the cost function. This algorithm can be generalized to high-dimensional case using Monte Carlo approximation, and the optimal rate is 1/2. This result bridges the deterministic inverse problem with statistical learning.

⭐ ⭐ ⭐ Quanjun Lang and F. Lu. Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles. 2020 PDF arXiv2010
** Sequential Monte Carlo for particle systems: **
Zehong Zhang and F. Lu, Cluster prediction for opinion dynamics from partial observations. arXiv2007