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

2026-07-28

AI Summary

This work investigates the expressive limitations of infinitely wide neural networks—specifically Barron functions—in variational problems involving physical phenomena with intricate local geometry, such as bending and folding of elastic shells. By integrating techniques from the calculus of variations, functional analysis, and neural network approximation theory, the authors construct explicit counterexamples and perform energy-approximation comparisons. They rigorously establish, for the first time, that while Barron functions exhibit no energy gap relative to Lipschitz functions for a broad class of first-order integral functionals, they fundamentally fail to approximate energy-minimizing curved-fold configurations in certain elastic shell models, being restricted to straight-fold solutions. This reveals an intrinsic limitation of Barron spaces in scientific machine learning and demonstrates a depth separation phenomenon rooted in geometric expressivity.

Abstract

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.

Problem

• Barron functions
• Lipschitz functions
• energy gap
• depth separation
• calculus of variations

Innovation

• Barron functions
• energy gap
• depth separation
• calculus of variations
• scientific machine learning