Operator learning has emerged as a new paradigm for the data-driven approximation of nonlinear operators. Despite its empirical success, the theoretical underpinnings governing the conditions for efficient operator learning remain incomplete. The present work develops theory to study the data complexity of operator learning, complementing existing research on the parametric complexity. We investigate the fundamental question: How many input/output samples are needed in operator learning to achieve a desired accuracy $\epsilon$? This question is addressed from the point of view of $n$-widths, and this work makes two key contributions. The first contribution is to derive lower bounds on $n$-widths for general classes of Lipschitz and Fr\'echet differentiable operators. These bounds rigorously demonstrate a ``curse of data-complexity'', revealing that learning on such general classes requires a sample size exponential in the inverse of the desired accuracy $\epsilon$. The second contribution of this work is to show that ``parametric efficiency'' implies ``data efficiency''; using the Fourier neural operator (FNO) as a case study, we show rigorously that on a narrower class of operators, efficiently approximated by FNO in terms of the number of tunable parameters, efficient operator learning is attainable in data complexity as well. Specifically, we show that if only an algebraically increasing number of tunable parameters is needed to reach a desired approximation accuracy, then an algebraically bounded number of data samples is also sufficient to achieve the same accuracy.

翻译：算子学习已成为数据驱动非线性算子逼近的新范式。尽管其经验上取得了成功，但支撑高效算子学习条件的理论基础仍不完善。本研究发展了算子学习数据复杂度的理论，补充了现有关于参数复杂度的研究。我们探讨了一个基本问题：在算子学习中需要多少输入/输出样本才能达到期望精度 $\epsilon$？该问题从 $n$-宽度的角度进行研究，本工作做出了两个关键贡献。第一个贡献是推导了 Lipschitz 算子和 Fréchet 可微算子一般类别的 $n$-宽度下界。这些下界严格证明了“数据复杂度灾难”，揭示了在此类一般算子上进行学习所需的样本量随期望精度 $\epsilon$ 的倒数呈指数增长。本工作的第二个贡献是证明了“参数效率”蕴含“数据效率”；以傅里叶神经算子（FNO）为案例研究，我们严格证明了在一个更窄的算子类别上，若 FNO 在可调参数数量方面能高效逼近，则算子学习在数据复杂度上同样可实现高效。具体而言，我们证明若仅需要可调参数数量代数增长即可达到期望逼近精度，则代数有界的数据样本量也足以实现相同精度。