## Mathematical Foundations of Learning Interaction Kernels and Latent Functions in Operators

Time: 2022-06-22
Published By: He Liu

**Speaker(s): ** Fei LU (Johns Hopkins University)

**Time: **August 2 - August 11, 2022

**Venue: ** Online

This mini-course presents preliminary developments on the mathematical foundations of learning latent functions in operators from data. Examples of such latent functions include interaction kernels in particle systems and kernels in nonlocal operators. The function is latent because the data depends on it non-locally. The operator can be either linear or nonlinear, but it depends linearly on the latent function.

We develop a learning theory covering identifiability, the convergence of the estimator, and regularization. It provides a unified framework to treat both random and non-random data. When the data are random samples, the theory extends the classical learning theory by including identifiability and a coercivity condition for convergence. But when the data are non-random, it provides a statistical learning formulation for the deterministic inverse problem. Various open questions will be discussed.

Lecture 1. Introduction and a review of classical learning theory

Lecture 2. Learning interaction kernels in interacting particle systems

Lecture 3. Learning interaction kernels in mean-field equations

Lecture 4. Learning latent functions in operators and regularization

References:

- [CS02] Cucker, F. and Smale, S., 2002. On the mathematical foundations of learning. Bulletin of the American mathematical society, 39(1), pp.1-49.

- [LMT21] Lu, F., Maggioni, M. and Tang, S., 2021. Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories. Foundations of Computational Mathematics, pp.1-55.

- [LLMTZ21] Li, Z., Lu, F., Maggioni, M., Tang, S. and Zhang, C., 2021. On the identifiability of interaction functions in systems of interacting particles. Stochastic Processes and their Applications, 132, pp.135-163.

- [LL22] Lang, Q. and Lu, F., 2022. Learning interaction kernels in mean-field equations of first-order systems of interacting particles. SIAM Journal on Scientific Computing, 44(1), pp.A260-A285.

- [LL21] Lang, Q. and Lu, F., 2021. Identifiability of interaction kernels in mean-field equations of interacting particles. arXiv preprint arXiv:2106.05565.

- [LLA22] Lu, F., Lang, Q. and An, Q., 2022. Data adaptive RKHS Tikhonov regularization for learning kernels in operators. arXiv preprint arXiv:2203.03791.

Time：

**2022/08/02,04,09,11 09:00-10:30am (Beijing Time)**

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