My current research focuses on learning dynamics from data. One topic is nonparametric learning of the interaction laws in systems of interacting particles/agents. 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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Learning systems self-interacting particles/agents
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. Also, we develop a nonparametric statistical learning theory addressing identifiability, consistency, and optimal rate of convergence of the estimators. In particular, we show that despite the high dimensionality of the systems, optimal learning rates as in 1D can be achieved.
Interacting particle systems on networks: joint inference
⭐ ⭐ ⭐ Erhan Bayrakta, F.Lu, Mauro Maggioni, Ruoyu Wu, and Sichen Yang. Probabilistic cellular automata with local transition matrices: synchronization, ergodicity, and inference. arXiv2405 PDF
⭐ ⭐ ⭐ Quanjun Lang, Xiong Wang, F.Lu, and Mauro Maggioni. Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel. arXiv2402 PDF
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. J. Machine Learning Research, vol. 22, no.32, 1-67, 2021. arXiv1910
Zhongyang Li, F. Lu, Mauro Maggioni, Sui Tang and Cheng Zhang: On the identifiability of interaction functions in systems of interacting particles. 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. Found Comput Math (2021). 1-55. arXiv2007
Zhongyang Li and F. Lu. On the coercivity condition in the learning of interacting particle systems. arXiv2011 PDF
⭐ ⭐ ⭐ Xiong Wang, Inbar Serrousi, and F.Lu. Optimal minimax rate of learning interaction kernels. arXiv2311 PDF
Mean-field equations: We introduce a least-squares algorithm to learn the interaction function from data consisting the solution of the PDE. Such an 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 cases 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. SIAM Journal on Scientific Computing 44 (1), A260–A285, 2022. PDF arXiv2010 Slides
⭐ ⭐ ⭐ Quanjun Lang and F. Lu.Identifiability of interaction kernels in mean-field equations of interacting particles. arXiv2106 PDF
Sequential Monte Carlo for opinion dynamics
Zehong Zhang and F. Lu, Cluster prediction for opinion dynamics from partial observations. arXiv2007 Slides