通过自适应坐标变换加速赫尔米特展开收敛，理论证明其有效性。
Convergence theory for Hermite approximations under adaptive coordinate transformations
- 用可逆神经网络构造自适应坐标变换，提升赫尔米特展开精度。
- 证明变换后函数的逼近误差取决于拉回函数的光滑性，实现谱收敛。
- 适用于量子物理中需快速收敛的平滑衰减函数逼近场景。
近期研究显示，利用归一化流（即可逆神经网络）参数化并优化坐标变换，可显著加速谱逼近的收敛。本文首次给出了基于自适应坐标变换的赫尔米特展开的误差估计。分析建立等价原理：在变换基底下逼近函数 $f$ 等价于在赫尔米特函数下逼近 $f$ 的拉回函数。由此可借助经典赫尔米特逼近理论，在变换坐标系中以拉回函数的正则性为依据推导误差界。文中通过一个例子展示非线性坐标变换如何提升赫尔米特展开的收敛速度。针对实轴上平滑且衰减的函数，构造单调传输映射，使目标函数的衰减与赫尔米特基底对齐，从而保证对应赫尔米特展开的谱收敛率。该分析为最近在计算量子物理文献中探索的基于归一化流的自适应赫尔米特逼近提供了理论支撑。
原文摘要 · Abstract (English)
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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