# Mathematics of Machine Learning

The One World Seminar Series on the Mathematics of Machine Learning is an online platform for research seminars, workshops and seasonal schools in theoretical machine learning. The focus of the series lies on theoretical advances in machine learning and deep learning as a complement to the one world seminars on probability, on Information, Signals and Data (MINDS), on methods for arbitrary data sources (MADS), and on imaging and inverse problems (IMAGINE).

The series was started during the Covid-19 epidemic in 2020 to bring together researchers from all over the world for presentations and discussions in a virtual environment. It follows in the footsteps of other community projects under the One World Umbrella which originated around the same time.

We welcome suggestions for speakers concerning new and exciting developments and are committed to providing a platform also for junior researchers. We recognize the advantages that online seminars provide in terms of flexibility, and we are experimenting with different formats. Any feedback on different events is welcome.

## Next Event

Wed Mar 3
12 noon ET

Structure preservation and convergence in scientific machine learning

Physics-informed techniques have emerged as a means of incorporating prior knowledge into machine learning. These techniques generally function by minimizing a hybrid loss, regularizing a traditional $\ell_2$ error with a PDE residual. While remarkably effective, these approaches suffer two major shortcomings. Firstly, such neural network (NN) solutions of PDEs generally fail to converge with increasing architecture size. Despite recent work establishing NNs may approximate at least as well as hp-finite element spaces, in practice when training with gradient methods O(1) optimization errors prevent realizing consistency. Secondly, the regularized losses introduce physics via a penalized residual, and it is well known from classical numerical analysis that the approximation space must be designed in tandem with the residual to ensure converge to a given PDE.

We conjecture that the same tools used to design convergent and structure-preserving properties in forward simulation may be used to design scientific ML architectures with similar guarantees. In this talk, we present two current works which address each of these issues. First, we introduce partition of unity networks (POUnets) to develop convergent approximation with deep networks. It has been shown that traditional feed forward networks may approximate by emulating partitions of unity (POU), and then emulating monomials on each partition, ultimately yielding a localized polynomial approximation and associated hp-convergence. Rather than emulating these components, POUnets function by directly incorporating both the POU and polynomials into the architecture. The resulting approximation breaks the curse of dimensionality and admits a fast least-squares optimization strategy. Predictions are competitive with high-order finite element spaces, and provide superior approximation for problems with reduced regularity.

Secondly, we introduce a data-driven exterior calculus (DDEC) which may be used to endow scientific ML architectures with the structure-preserving properties of mimetic PDE discretization. Traditional mimetic methods function by exploiting the exterior calculus structures offered by a mesh to construct discrete operators that exactly mimic the topological properties of continuum operators. We show how graphs may be used as a surrogate for the topology offered by graphs, and present new network architectures which allows "physics-informed" machine learning which exactly preserves conservation, guarantees extraction of well-posed problems, and allows handling of the non-trivial null-spaces occurring in fields such as electromagnetics.

If time permits, we will additionally share some current results applying these tools in challenging data-driven modeling effort at Sandia, related to data-driven shock hydrodynamics in metals and discovery of surrogates for semiconductors in radiation environments.

## Mailing List

Sign up here to join our mailing list and receive announcements. If your browser automatically signs you into a google account, it may be easiest to join on a university account by going through an incognito window. With other concerns, please reach out to one of the organizers.

## Format

Seminars are held online on Zoom. The presentations are recorded and video is made available on our youtube channel. A list of past seminars can be found here. All seminars, unless otherwise stated, are held on Wednesdays at 12 noon ET. The invitation will be shared on this site before the talk and distributed via email.

## Board

Simon Shaolei Du (University of Washington)

Surbhi Goel (Microsoft Research NY)

Song Mei (UC Berkeley)

Matthew Thorpe (University of Manchester)

Franca Hoffmann (University of Bonn)

Chao Ma (Stanford University)

Philipp Petersen (University of Vienna)

Stephan Wojtowytsch (Princeton University)