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.


  1. Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality


  1. Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activation

(with Josiah Park) 2022 ArXiv

  1. Stochastic gradient descent with noise of machine learning type. Part II: Continuous time analysis

2021 ArXiv

  1. Stochastic gradient descent with noise of machine learning type. Part I: Discrete time analysis

2021 ArXiv

  1. A priori estimates for classification problems using neural networks

(with Weinan E) 2020 ArXiv

  1. On the Convergence of Gradient Descent Training for Two-layer ReLU-networks in the Mean Field Regime

2020 ArXiv

Published and accepted articles

  1. Keeping it together: a phase field version of path-connectedness and its implementation

(with P. Dondl), 2018 Journal ArXiv

  1. Representation Formulas and Pointwise Properties for Barron Functions

(with Weinan E) 2020 Journal ArXiv

  1. Connected Coulomb Columns: Analysis and Numerics

(with P. Dondl, M. Novaga and S. Wolff-Vorbeck), 2018 Journal ArXiv

  1. On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers

(with Weinan E) 2020 Conference ArXiv

  1. Some observations on partial differential equations in Barron and multi-layer spaces

(with Weinan E) 2020 Conference ArXiv

  1. Towards 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 ArXiv

  1. On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics

(with Weinan E) 2020 Journal ArXiv

  1. The motion of curved dislocations in three dimensions: Simplified linearized elasticity

(with I. Fonseca and J. Ginster), 2020 Journal ArXiv

  1. Can shallow neural networks beat the curse of dimensionality? A mean field training perspective

(with Weinan E), 2020 Journal ArXiv

  1. Kolmogorov 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 ArXiv

  1. Confined elasticae and the buckling of cylindrical shells

Adv Calc Var (2020) Journal ArXiv

  1. Approximation 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 ArXiv

  1. The 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 ArXiv

  1. Phase Field Models for Thin Elastic Structures with Topological Constraint

(with P. Dondl and A. Lemenant), Arch Rational Mech Anal (2017) 223 Journal ArXiv

  1. Uniform Regularity and Convergence of Phase Fields for Willmore's Energy

(with P. Dondl), Calc. Var. PDE (2017) 56 Journal ArXiv

  1. On the Boundary Regularity of Phase-Fields for Willmore's Energy

(with P. Dondl), Proc A Royal Soc of Edinburgh (2017) 149:4 Journal ArXiv

  1. Helfrich's Energy and Constrained Minimisation

Comm in Math Sci (2017) 15:8 Journal ArXiv

  1. On 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.