Hermite近似在自适应坐标变换下的收敛理论
Convergence theory for Hermite approximations under adaptive coordinate transformations
AI总结：
本文基于自适应坐标变换与Hermite展开，首次建立误差估计理论，揭示了通过正常流参数化坐标变换可提升Hermite近似的收敛速度。
摘要：
最近的研究表明，使用归一化流（即可逆神经网络）参数化和优化坐标变换，可以显著加速谱近似的收敛。本文首次提出了使用Hermite展开与自适应坐标变换近似函数的误差估计。我们的分析建立了等价原理：在变换后的基底跨度中近似函数f等价于在Hermite函数跨度中近似f的拉回。这使我们能够利用Hermite展开的经典近似理论，推导出在变换坐标中的误差估计，该估计取决于拉回的正则性。我们提供了一个例子，展示了非线性坐标变换如何增强Hermite展开的收敛性。聚焦于沿实轴衰减的光滑函数，我们构建了一个单调传输映射，使目标函数的衰减与Hermite基函数对齐。这保证了相应Hermite展开的谱收敛速度。我们的分析为基于正常流的自适应Hermite近似收敛行为提供了理论见解，这最近在计算量子物理文献中得到了探索。
英文摘要：
Recent work has shown that parameterizing and optimizing coordinate transformations using normalizing flows, i.e., invertible neural networks, can significantly accelerate the convergence of spectral approximations. We present the first error estimates for approximating functions using Hermite expansions composed with adaptive coordinate transformations. Our analysis establishes an equivalence principle: approximating a function $f$ in the span of the transformed basis is equivalent to approximating the pullback of $f$ in the span of Hermite functions. This allows us to leverage the classical approximation theory of Hermite expansions to derive error estimates in transformed coordinates in terms of the regularity of the pullback. We present an example demonstrating how a nonlinear coordinate transformation can enhance the convergence of Hermite expansions. Focusing on smooth functions decaying along the real axis, we construct a monotone transport map that aligns the decay of the target function with the Hermite basis. This guarantees spectral convergence rates for the corresponding Hermite expansion. Our analysis provides theoretical insight into the convergence behavior of adaptive Hermite approximations based on normalizing flows, as recently explored in the computational quantum physics literature.
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