Seminars in Numerical Analysis

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Seminar series in Numerical Analysis

Our seminar series is starting again, this year we're back on campus!

The Numerical Analysis seminars are a series of lectures to showcase and foster state-of-the-art research in the fields of computational sciences and engineering, mathematical modelling, and applied mathematics. These lunch lectures take place on a monthly basis, the dates are announced below. If you want to sign up for the monthly invitation, please sign up for our newsletter.

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03 mei 2024 12:30 t/m 13:15

[NA] Jakob Zech: Nonparametric Distribution Learning via Neural ODEs

In this talk, we explore approximation properties and statistical aspects of Neural Ordinary Differential Equations (Neural ODEs). Neural ODEs are a recently established technique in computational statistics and machine learning, that can be used to characterize complex distributions. Specifically, given a fixed set of independent and identically distributed samples from a target distribution, the goal is either to estimate the target density or to generate new samples. We first investigate the regularity properties of the velocity fields used to push forward a reference distribution to the target. This analysis allows us to deduce approximation rates achievable through neural network representations. We then derive a concentration inequality for the maximum likelihood estimator of general ODE-parametrized transport maps. By merging these findings, we are able to determine convergence rates in terms of both the network size and the number of required samples from the target distribution. Our discussion will particularly focus on target distributions within the class of positive \(C^k\) densities on the \(d\)-dimensional unit cube \([0,1]^d\).

31 mei 2024 12:30 t/m 13:15

[NA] Hyea Hyun Kim: Partitioned neural network approximation to partial differential equations and its training performance enhancement utilizing domain decomposition algorithms

With the success of deep learning technologies in many scientific and engineering applications, neural network approximation methods have emerged as an active research area in numerical partial differential equations. However, the new approximation methods still need further validations on their accuracy, stability, and efficiency so as to be used as alternatives to classical approximation methods. In this talk, we first introduce partitioned neural network approximation to partial differential equations, where neural network functions localized in each small subdomains are employed as a solution surrogate in order to reduce the approximation and optimization errors in the standard single large neural network approximation. The parameters in each local neural network function are then optimized to minimize the corresponding cost function to the model problem. To enhance the parameter training efficiency further, iterative algorithms for the partitioned neural network function can be developed by utilizing classical domain decomposition algorithms and their convergence theory. We finally present promising features in this new approach as a way of enhancing the neural network solution accuracy, stability, and efficiency with some supporting numerical results.

Organizers

D. (Deepesh) Toshniwal

A. (Alexander) Heinlein

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The accompanying interview series of the Numerical Analysis seminars.

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