# The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning

arXiv: https://arxiv.org/abs/2607.25905 · HTML: https://mlsanity.com/paper/2607.25905/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 42 references that are in the map.
- First posted on arXiv 28 Jul 2026.

## Builds on (most cited first, 10 of 10)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [Depth separation beyond radial functions](https://mlsanity.com/paper/2102.01621/) (2021)
- [Ridges, Neural Networks, and the Radon Transform](https://mlsanity.com/paper/2203.02543/) (2022)
- [Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality](https://mlsanity.com/paper/2209.01173/) (2022)
- [Minimum norm interpolation by perceptra: Explicit regularization and implicit bias](https://mlsanity.com/paper/2311.06138/) (2023)
- [Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspective](https://mlsanity.com/paper/2412.07728/) (2024)

## Abstract (arXiv)

We illustrate in several examples that even neural networks of infinite width (specifically, Barron functions) may encounter substantial obstacles when used as a model class for problems in the calculus of variations. An instance of practical relevance concerns the bending, stretching and folding of a thin elastic shell with anchored or clamped boundary conditions where elastic energy could be reduced by folding along a circular line, but the neural networks can only describe straight folds along entire lines. Conversely, we show that there is no gap between the energy that Barron functions and Lipschitz functions can achieve for a large class of integral first-order functionals.