Convergence theory for Hermite approximations under adaptive coordinate transformations

1. Problem Statement

Hermite functions form an orthonormal basis for $L^2(\mathbb{R})$ and are widely used in spectral methods for unbounded domains, particularly in quantum physics (e.g., solving the Schrödinger equation). However, standard Hermite expansions often suffer from slow convergence when approximating target functions whose decay rates at infinity do not match the intrinsic Gaussian decay ($e^{-x^2/2}$) of the Hermite basis.

Recent computational physics literature has proposed using adaptive coordinate transformations, parameterized by normalizing flows (invertible neural networks), to accelerate convergence. While these methods have shown empirical success, there has been a lack of rigorous mathematical theory explaining why they work or providing error estimates for such adaptive approximations. The paper addresses this gap by establishing the first convergence theory for Hermite approximations under adaptive coordinate transformations.

2. Methodology

Theoretical Framework

The core of the methodology relies on an Equivalence Principle derived from functional analysis:

• Weighted Composition Operator: The authors define a weighted composition operator $W_g$ induced by a diffeomorphism $g: \Omega \to \mathbb{R}$:
  $$(W_g f)(x) = f(g(x)) \sqrt{|g'(x)|}$$
  This operator maps $L^2(\mathbb{R})$ to $L^2(\Omega)$ unitarily.
• Equivalence Principle (Theorem 4.1): Approximating a function $f$ in the linear span of the transformed basis $\{W_g h_n\}$ is mathematically equivalent to approximating the pullback function $u = W_{g^{-1}}f$ in the span of the standard Hermite functions $\{h_n\}$.
  $$\|f - P_{N,W}f\|_{L^2(\Omega)} = \|W_{g^{-1}}f - P_N(W_{g^{-1}}f)\|_{L^2(\mathbb{R})}$$
  This allows the authors to leverage classical convergence theorems for Hermite expansions (which depend on the smoothness and decay of the function) to derive error estimates for the adaptive case.

Construction of Optimal Maps

To achieve spectral convergence (convergence faster than any polynomial rate), the pullback function $W_{g^{-1}}f$ must be a Schwartz function (smooth with rapid decay).

• Decay Matching: The authors utilize optimal transport theory to construct a monotone transport map $g$. This map pushes the probability density associated with the squared target function ($f^2$) to the Gaussian density associated with Hermite functions.
• Theorem 4.2: For any smooth function $f$ with at most exponential decay, there exists a diffeomorphism $g$ such that the pullback $W_{g^{-1}}f$ exhibits Gaussian decay ($e^{-x^2/2}$), guaranteeing spectral convergence.

Numerical Implementation

The paper implements these concepts using two types of coordinate transformations:

1. Parametric Models: Linear scaling/shift and power-law transformations ($g(x) = x(x^2+\epsilon)^\beta$).
2. Normalizing Flows: Specifically, iResNets (invertible Residual Neural Networks), which are differentiable bijections with efficiently computable inverses.

• Optimization: The parameters of these transformations are optimized by minimizing a loss function. This loss function includes a term that penalizes the mismatch between the decay of the transformed target function and the Gaussian decay of the Hermite basis.
• Application: The method is applied to the time-independent Schrödinger equation for the Morse oscillator, optimizing the Rayleigh quotient to find eigenvalues.

3. Key Contributions

1. First Rigorous Convergence Theory: The paper provides the first error estimates for Hermite approximations using adaptive coordinate transformations, linking them directly to the regularity of the pullback function.
2. Equivalence Principle: It establishes a unitary equivalence between adaptive Hermite approximations and standard Hermite approximations of a transformed function, allowing the transfer of classical approximation theory results.
3. Existence of Spectral Convergence: It proves that for smooth functions with exponential decay, an adaptive coordinate transformation exists that aligns the function's decay with the Hermite basis, ensuring spectral convergence.
4. Mechanism of Normalizing Flows: The work theoretically validates why normalizing flows (like iResNets) improve convergence in quantum physics applications: they effectively learn a map that transforms the target eigenfunction into a function with Gaussian-like decay.

4. Results

Numerical Experiments on Interpolation

The authors tested the method on two target functions with different decay behaviors:

• Algebraic Decay ($f_1$): Standard Hermite expansion converges slowly ($O(N^{-1.67})$). Adaptive transformations improved this significantly:
  • Linear transformation: $O(N^{-2.64})$
  • iResNet: $O(N^{-4.87})$ (approx. 3x improvement in rate).
• Super-Gaussian Decay ($f_2$): Standard expansion converges sub-optimally due to decay mismatch.
  • Linear transformation improved the exponential convergence exponent.
  • Asymptotically derived power-law maps yielded super-geometric convergence ($\kappa > 1$), outperforming the neural network approach in this specific case because the neural network's Lipschitz constraints limited its ability to model the required quadratic growth ($g(x) \sim x^2$).

Application to Schrödinger Equation (Morse Oscillator)

• The method was applied to compute the lowest 23 eigenvalues of the Morse oscillator.
• Accuracy: Using adaptive coordinates significantly reduced the relative error in eigenvalues compared to standard Hermite expansions.
• Convergence Rates:
  • Linear coordinate transformation improved the convergence rate by a factor of ~4.
  • iResNet-based transformation improved the rate by a factor of ~9.
• Decay Analysis: Moment analysis confirmed that the iResNet successfully transformed the Morse eigenfunctions (which have super-Gaussian decay) into functions with Gaussian-like decay, aligning them with the Hermite basis.

5. Significance

• Theoretical Foundation: This work bridges the gap between the empirical success of normalizing flows in computational physics and rigorous numerical analysis. It explains why these "black box" neural network transformations work: they are effectively solving an optimal transport problem to match decay rates.
• Algorithmic Improvement: It provides a theoretical basis for designing better spectral methods. Instead of relying solely on trial-and-error optimization of neural networks, the paper suggests that the goal of the optimization should be explicitly aligned with matching the decay profile of the target function to the basis.
• Broad Applicability: While focused on Hermite functions and quantum physics, the equivalence principle and decay-matching strategy are applicable to other spectral methods on unbounded domains and other orthogonal polynomial bases.
• Future Directions: The paper suggests extending these adaptive coordinate techniques to multivariate settings and other functional spaces (e.g., continuous functions), potentially revolutionizing how high-dimensional PDEs are solved in physics and engineering.