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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Learning kernels in operators
Kernels are efficient in representing nonlocal dependencies and are widely used to design operators between function spaces. Thus, learning kernels in operators from data is of broad interest, presenting a new topic at the intersection of statistical learning and inverse problems.
Related topics: ill-posed inverse problems, deconvolution, regularization, functional linear regression, minimax rate of convergence.
Deep Learning approach: attention mechanism for multi-systems
Yue Yu, Ning Liu, Fei Lu, Tian Gao, Siavash Jafarzadeh, Stewart Silling. Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery. arXiv2408
Nonparametric regression with DA-RKHS regularization
F. Lu, Q.Lang and Q. An. Data adaptive RKHS Tikhonov regularization for learning kernels in operators. MATLAB code
This project focuses on a vector estimator that views the kernel as a vector on the grid points (equivalent to using piecewise constant basis functions on grid points). In this setting, there is no additional regularization from the basis functions. We illustrate its performance in examples including integral operators, nonlinear operators, and nonlocal operators with discrete synthetic data. Numerical results show that DARTR leads to an accurate estimator robust to both numerical errors due to discrete data and noise in data, and the estimator converges at a consistent rate as the data mesh refines under different levels of noises, outperforming two improved baseline regularizers using $l^2$ and $L^2$ norms.
F. Lu, Q.An and Y. Yu Nonparametric learning of kernels in nonlocal operators.
This project focuses on kernels in nonlocal operators. It uses B-spline basis functions. We systematically study the method with synthetic data, showing the convergence of estimators. Furthermore, the method successfully learns a homogenized model for stress wave propagation in a heterogeneous solid, revealing the unknown governing laws from real-world data at the microscale.
Our DARTR outperforms baseline methods in robustness, generalizability, and accuracy in synthetic and real-world datasets.
Haibo Li and Fei Lu. Automatic reproducing kernel and regularization for learning convolution kernels. arXiv2507
Luxuan Yang, Fei Lu, Ting Gao, Wei Wei, and Jinqiao Duan. Learning Lévy density via adaptive RKHS regression with bi-level optimization arXiv2512
Data adaptive RKHS priors for Bayesian inverse problems
Neil K. Chada, Quanjun Lang, F.Lu, and Xiong Wang. A data-adaptive prior for Bayesian learning of kernels in operators. arXiv2212 PDF