# Publications

My articles on deep learning are arranged by topic and connection in this mindmap. The brief describtions may miss some important details. For the full titles, links and co-authors, please see the bibliography below.

## Preprints

A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces

(with Jonathan Siegel) 2023 ArXivAchieving acceleration despite very noisy gradients

(with Kanan Gupta and Jonathan Siegel) 2023 ArXivOptimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality

2022 ArXivQualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activation

(with Josiah Park) 2022 ArXivA priori estimates for classification problems using neural networks

(with Weinan E) 2020 ArXivOn the Convergence of Gradient Descent Training for Two-layer ReLU-networks in the Mean Field Regime

2020 ArXiv

## Published and accepted articles

Minimum norm interpolation by perceptra: Explicit regularization and implicit bias

(with Jiyoung Park and Ian Pelakh) 2023 ArXiv ConferenceGroup Equivariant Fourier Neural Operators for Partial Differential Equations

(with Jacob Helwig, Xuan Zhang, Cong Fu, Jerry Kurtin and Shuiwang Ji) 2023 ArXiv ConferenceStochastic gradient descent with noise of machine learning type. Part I: Discrete time analysis

2021 Journal ArXivStochastic gradient descent with noise of machine learning type. Part II: Continuous time analysis

2021 Journal ArXivKeeping it together: a phase field version of path-connectedness and its implementation

(with P. Dondl), 2018 Journal ArXivRepresentation Formulas and Pointwise Properties for Barron Functions

(with Weinan E) 2020 Journal ArXivConnected Coulomb Columns: Analysis and Numerics

(with P. Dondl, M. Novaga and S. Wolff-Vorbeck), 2018 Journal ArXivOn the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers

(with Weinan E) 2020 Conference ArXivSome observations on partial differential equations in Barron and multi-layer spaces

(with Weinan E) 2020 Conference ArXivTowards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't

(with Weinan E, Chao Ma and Lei Wu) 2020 Journal ArXivOn the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics

(with Weinan E) 2020 Journal ArXivThe motion of curved dislocations in three dimensions: Simplified linearized elasticity

(with I. Fonseca and J. Ginster), 2020 Journal ArXivCan shallow neural networks beat the curse of dimensionality? A mean field training perspective

(with Weinan E), 2020 Journal ArXivKolmogorov Width Decay and Poor Approximators in Machine Learning: Shallow Neural Networks, Random Feature Models and Neural Tangent Kernels

(with Weinan E), Res Math Sci, 2020 Journal ArXivConfined elasticae and the buckling of cylindrical shells

Adv Calc Var (2020) Journal ArXivApproximation of the relaxed perimeter functional under a connectedness constraint by phase-fields

(with P. Dondl, M. Novaga and B. Wirth), SIAM Journal on Mathematical Analysis (2019) 51:5 Journal ArXivThe Effect of Forest Dislocations on the Evolution of a Phase-Field Model for Crystal Dislocations

(with P. Dondl and M. Kurzke), Arch Rational Mech Anal (2018) 232 Journal ArXivPhase Field Models for Thin Elastic Structures with Topological Constraint

(with P. Dondl and A. Lemenant), Arch Rational Mech Anal (2017) 223 Journal ArXivUniform Regularity and Convergence of Phase Fields for Willmore's Energy

(with P. Dondl), Calc. Var. PDE (2017) 56 Journal ArXivOn the Boundary Regularity of Phase-Fields for Willmore's Energy

(with P. Dondl), Proc A Royal Soc of Edinburgh (2017) 149:4 Journal ArXivHelfrich's Energy and Constrained Minimisation

Comm in Math Sci (2017) 15:8 Journal ArXivOn the Alexandrov Topology of Sub-Lorentzian Manifolds

(with I. Markina), Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Series (2014) Book ArXiv

You can also find my publications on Orcid, Google Scholar and ArXiv or in my CV. Some publications are also available on cvgmt or DeepAI respectively.