首次给出神经算子在 Sobolev 范数下的误差量化，揭示模型规模与精度的幂律关系。
Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation
- 建立 Sobolev 空间中神经算子的泛函分析框架，证明逼近误差随参数量呈幂律下降。
- 在 Burgers 方程上训练 FNO，测试误差低至 10^{-7}，相对误差约 10^{-3}，同时准确预测解和导数。
- 实证发现误差与参数量呈近似幂律关系（指数 α≈1.4），验证理论对模型缩放行为的预测能力。
神经算子已成为学习无限维函数空间间映射的强大工具，但其在 Sobolev 范数下的逼近性质尚未被充分量化。而 Sobolev 范数控制函数值和导数，是描述偏微分方程适定性、稳定性与泛化性的自然度量。本文建立了一个用于算子学习的 Sobolev 空间泛函分析框架，并将其与傅里叶神经算子（FNO）在典型 PDE 上的数值行为相联系。对于定义在 $H^{s}(D)$ 到 $H^{t}(D')$ 且 $s > d/2$、输入受限于 $H^{s}(D)$ 紧子集上的连续非线性算子 $/mathcal{G}$，我们证明其可在 $H^{t}$-范数下被神经算子一致逼近，所需可训练参数量为 $\\mathcal{O}(\varepsilon^{-d/s})$，并得到显式复杂度-误差关系：$\\\|\\mathcal{G}-\\mathcal{G}_θ\\\|_{H^{t}} \lesssim C N^{-s/d}$。随后研究一维黏性 Burgers 方程解算子 $\\mathcal{G}: u_{0}\mapsto u(\cdot,1)$ 在有界 $H^{1}$-球上的表现，使用 $H^{1}$-损失训练 FNO。在模型规模扫面中，测试 $H^{1}$-误差降至 $\\mathcal{O}(10^{-7})$，相对误差为 $10^{-3}$ 量级，预测结果精确匹配未见数据的解与空间导数。对数-对数图显示 Sobolev 误差与参数数量近似服从幂律 $\\\|\\mathcal{G}-\\mathcal{G}_θ\\\|_{H^{1}} \approx C N^{-α}$，经验指数 $α\approx 1.4$；长时程训练中大模型出现优化不稳定性，定量表明 Sobolev 逼近理论能有效预测神经算子的缩放行为。
原文摘要 · Abstract (English)
Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for PDE well-posedness, stability, and generalization. We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE. First, for a continuous nonlinear operator $\mathcal{G}: H^{s}(D)\to H^{t}(D')$ with $s > d/2$ and inputs restricted to a compact subset of $H^{s}(D)$, we prove that $\mathcal{G}$ can be uniformly approximated in $H^{t}$-norm by a neural operator with $\mathcal{O}(\varepsilon^{-d/s})$ trainable parameters. This yields an explicit complexity--error relation of the form $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{t}} \lesssim C N^{-s/d}$. We then study the one-dimensional viscous Burgers solution operator $\mathcal{G}: u_{0}\mapsto u(\cdot,1)$ on a bounded $H^{1}$-ball and train FNOs with an $H^{1}$-loss. Across a sweep of model sizes, we obtain test $H^{1}$-errors down to $\mathcal{O}(10^{-7})$ and relative errors of order $10^{-3}$, with predictions accurately matching both solutions and spatial derivatives on held-out data. A log-log plot of Sobolev error versus parameter count exhibits an approximate power law $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{1}} \approx C N^{-α}$ with empirical exponent $α\approx 1.4$, and long-horizon training reveals optimization instabilities in large FNOs, providing quantitative evidence that Sobolev-space approximation theory meaningfully predicts neural-operator scaling behavior.
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