科学机器学习中的巴伦 - 利普希茨能量差距与深度分离现象
The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning
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中文总结 AI 辅助
研究变分法问题中无限宽度神经网络（巴伦函数）的障碍，如弹性壳折叠问题，同时表明巴伦函数与利普希茨函数在一类积分一阶泛函能量上无差距。
中文摘要 AI 辅助
我们通过几个例子说明，即使是无限宽度的神经网络（具体为巴伦函数）在用作变分法问题的模型类时也可能遇到重大障碍。一个实际相关的例子是具有固定或夹紧边界条件的薄弹性壳的弯曲、拉伸和折叠，其中沿圆形线折叠可降低弹性能量，但神经网络只能描述沿整条线的直线折叠。相反，我们表明对于一大类积分一阶泛函，巴伦函数和利普希茨函数能够实现的能量之间没有差距。
英文摘要
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.
