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

1. A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces
   (with Jonathan Siegel) 2023 ArXiv

2. Achieving acceleration despite very noisy gradients
   (with Kanan Gupta and Jonathan Siegel) 2023 ArXiv

3. Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality
   2022 ArXiv

4. Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activation
   (with Josiah Park) 2022 ArXiv

5. A priori estimates for classification problems using neural networks
   (with Weinan E) 2020 ArXiv

6. On the Convergence of Gradient Descent Training for Two-layer ReLU-networks in the Mean Field Regime
   2020 ArXiv

Published and accepted articles

1. Minimum norm interpolation by perceptra: Explicit regularization and implicit bias
   (with Jiyoung Park and Ian Pelakh) 2023 ArXiv Conference

2. Group Equivariant Fourier Neural Operators for Partial Differential Equations
   (with Jacob Helwig, Xuan Zhang, Cong Fu, Jerry Kurtin and Shuiwang Ji) 2023 ArXiv Conference

3. Stochastic gradient descent with noise of machine learning type. Part I: Discrete time analysis
   2021 Journal ArXiv

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

5. Keeping it together: a phase field version of path-connectedness and its implementation
   (with P. Dondl), 2018 Journal ArXiv

6. Representation Formulas and Pointwise Properties for Barron Functions
   (with Weinan E) 2020 Journal ArXiv

7. Connected Coulomb Columns: Analysis and Numerics
   (with P. Dondl, M. Novaga and S. Wolff-Vorbeck), 2018 Journal ArXiv

8. On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers
   (with Weinan E) 2020 Conference ArXiv

9. Some observations on partial differential equations in Barron and multi-layer spaces
   (with Weinan E) 2020 Conference ArXiv

10. 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

11. On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics
    (with Weinan E) 2020 Journal ArXiv

12. The motion of curved dislocations in three dimensions: Simplified linearized elasticity
    (with I. Fonseca and J. Ginster), 2020 Journal ArXiv

13. Can shallow neural networks beat the curse of dimensionality? A mean field training perspective
    (with Weinan E), 2020 Journal ArXiv

14. 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

15. Confined elasticae and the buckling of cylindrical shells
    Adv Calc Var (2020) Journal ArXiv

16. 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

17. 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

18. Phase Field Models for Thin Elastic Structures with Topological Constraint
    (with P. Dondl and A. Lemenant), Arch Rational Mech Anal (2017) 223 Journal ArXiv

19. Uniform Regularity and Convergence of Phase Fields for Willmore's Energy
    (with P. Dondl), Calc. Var. PDE (2017) 56 Journal ArXiv 

20. 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

21. Helfrich's Energy and Constrained Minimisation
    Comm in Math Sci (2017) 15:8 Journal ArXiv 

22. 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.