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.

翻译：近期研究表明，利用归一化流（即可逆神经网络）对坐标变换进行参数化与优化，可显著加速谱逼近的收敛速度。本文首次给出了对自适应坐标变换下埃尔米特展开逼近函数的误差估计。我们建立了一个等价原理：在变换基张成空间中对函数$f$进行逼近，等价于在埃尔米特函数张成空间中对$f$的回拉函数进行逼近。这使我们能够利用经典埃尔米特展开逼近理论，根据回拉函数的正则性推导变换坐标下的误差估计。我们通过实例展示非线性坐标变换如何增强埃尔米特展开的收敛性。针对沿实轴衰减的光滑函数，我们构造了一个单调输运映射，使目标函数的衰减特性与埃尔米特基相匹配，从而保证相应埃尔米特展开的谱收敛速率。该分析为近期计算量子物理文献中基于归一化流的自适应埃尔米特逼近方法的收敛行为提供了理论洞见。