# Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator

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## Abstract

**:**

## 1. Introduction

- We show how the derivative reproducing properties of kernels can be used to approximate differential operators such as the Koopman generator and the Schrödinger operator, as well as their eigenvalues and eigenfunctions from data. Additionally, we derive a kernel-based method tailored to reversible dynamics, which does not require estimating drift and diffusion terms, but only an equilibrated trajectory.
- Furthermore, we exploit the fact that, under certain conditions, the Schrödinger operator can be turned into a Kolmogorov backward operator (see, e.g., [24]), which allows for the interpretation of a quantum-mechanical system as a drift-diffusion process and, as a consequence, the application of methods developed for the analysis of stochastic differential equations or their generators.
- We demonstrate potential applications in molecular dynamics, using the example of a quadruple-well problem, and quantum mechanics, describing how to apply the proposed methods directly to the Schrödinger equation or the associated stochastic process. This will be illustrated with two well-known examples, the quantum harmonic oscillator and the hydrogen atom.

## 2. Koopman Theory and Reproducing Kernel Hilbert Spaces

#### 2.1. The Koopman Operator and Its Generator

**Remark**

**1.**

#### 2.2. Generator EDMD

#### 2.3. Second-Order Differential Operators

**Remark**

**2.**

**Lemma**

**1.**

**Corollary**

**1.**

#### 2.4. Reproducing Kernel Hilbert Spaces and Derivative Reproducing Properties

**Definition**

**1.**

- (i)
- ${\left(\right)}_{f}\mathbb{H}$ for all $f\in \mathbb{H}$ and
- (ii)
- $\mathbb{H}=\overline{span\left\{k\right(x,\xb7)\mid x\in \mathbb{X}\}}$.

**Theorem**

**1**

- (i)
- ${D}^{\alpha}k(x,\xb7)\in \mathbb{H}$ for any $x\in \mathbb{X}$ and $\alpha \in {I}_{p}$.
- (ii)
- $\left({D}^{\alpha}f\right)\left(x\right)={\left(\right)}_{{D}^{\alpha}}\mathbb{H}$ for any $x\in \mathbb{X}$, $f\in \mathbb{H}$, and $\alpha \in {I}_{p}$.

**Example**

**1.**

- For the polynomial kernel, we obtain:$${D}^{{e}_{i}}k(x,{x}^{\prime})=q\phantom{\rule{1.00006pt}{0ex}}{x}_{i}^{\prime}{(c+{x}^{\top}{x}^{\prime})}^{q-1}\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{D}^{{e}_{i}+{e}_{j}}k(x,{x}^{\prime})=q\phantom{\rule{1.00006pt}{0ex}}(q-1)\phantom{\rule{1.00006pt}{0ex}}{x}_{i}^{\prime}\phantom{\rule{1.00006pt}{0ex}}{x}_{j}^{\prime}{(c+{x}^{\top}{x}^{\prime})}^{q-2}.$$Thus, $\nabla \phantom{\rule{1.00006pt}{0ex}}k(x,{x}^{\prime})=q\phantom{\rule{1.00006pt}{0ex}}{x}^{\prime}\phantom{\rule{1.00006pt}{0ex}}{(c+{x}^{\top}{x}^{\prime})}^{q-1}$ and ${\nabla}^{2}\phantom{\rule{1.00006pt}{0ex}}k(x,{x}^{\prime})=q\phantom{\rule{1.00006pt}{0ex}}(q-1)\phantom{\rule{1.00006pt}{0ex}}{x}^{\prime}\phantom{\rule{1.00006pt}{0ex}}{x}^{\prime \top}{(c+{x}^{\top}{x}^{\prime})}^{q-2}$.
- Similarly, for the Gaussian kernel, this results in:$$\begin{array}{cc}\hfill {D}^{{e}_{i}}k(x,{x}^{\prime})& =-\frac{1}{{\varsigma}^{2}}({x}_{i}-{x}_{i}^{\prime})\phantom{\rule{1.00006pt}{0ex}}k(x,{x}^{\prime}),\hfill \\ \hfill {D}^{{e}_{i}+{e}_{j}}k(x,{x}^{\prime})& =\left(\right)open="\{"\; close>\begin{array}{c}\left(\right)open="["\; close="]">{\displaystyle \frac{1}{{\varsigma}^{4}}}{({x}_{i}-{x}_{i}^{\prime})}^{2}-{\displaystyle \frac{1}{{\varsigma}^{2}}}k(x,{x}^{\prime}),\hfill \\ i=j,\hfill \end{array}{\displaystyle \frac{1}{{\varsigma}^{4}}}({x}_{i}-{x}_{i}^{\prime})({x}_{j}-{x}_{j}^{\prime})\phantom{\rule{1.00006pt}{0ex}}k(x,{x}^{\prime}),\hfill & i\ne j,\hfill \hfill \end{array}$$$\nabla \phantom{\rule{1.00006pt}{0ex}}k(x,{x}^{\prime})=-\frac{1}{{\varsigma}^{2}}(x-{x}^{\prime})\phantom{\rule{1.00006pt}{0ex}}k(x,{x}^{\prime})$, and ${\nabla}^{2}\phantom{\rule{1.00006pt}{0ex}}k(x,{x}^{\prime})=\left(\right)open="["\; close="]">\frac{1}{{\varsigma}^{4}}(x-{x}^{\prime}){(x-{x}^{\prime})}^{\top}-\frac{1}{{\varsigma}^{2}}I$.

## 3. Kernel-Based Representation of Differential Operators

#### 3.1. Galerkin Projection of Operators

**Definition**

**2.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

#### 3.2. Empirical Estimates

**Remark**

**3.**

#### 3.3. Weak Formulation and Numerical Algorithm

**Lemma**

**4.**

- (i)
- In the general case, u solves ${G}_{2}\phantom{\rule{1.00006pt}{0ex}}u=\widehat{\lambda}\phantom{\rule{1.00006pt}{0ex}}{G}_{0}\phantom{\rule{1.00006pt}{0ex}}u$, where the entries of the matrices ${G}_{2}$ and ${G}_{0}$ are given by:$$\begin{array}{cc}\hfill {\left[{G}_{2}\right]}_{mr}& =\left(\right)open="["\; close="]">\mathrm{d}\varphi \left({x}_{m}\right)\left({x}_{r}\right),\hfill \\ \hfill {\left[{G}_{0}\right]}_{mr}& =\left(\right)open="["\; close="]">\varphi \left({x}_{m}\right)\left({x}_{r}\right).\hfill \end{array}$$
- (ii)
- Analogously, for the symmetric case, we obtain $\frac{1}{2}{\sum}_{l=1}^{d}{G}_{1}^{\left(l\right)}\phantom{\rule{1.00006pt}{0ex}}{G}_{1}^{\left(l\right)}\phantom{\rule{1.00006pt}{0ex}}u=\widehat{\lambda}\phantom{\rule{1.00006pt}{0ex}}{G}_{0}\phantom{\rule{1.00006pt}{0ex}}{G}_{0}\phantom{\rule{1.00006pt}{0ex}}u$, where we define:$${\left(\right)}_{{G}_{1}^{\left(l\right)}}mr$$and ${\sigma}_{l}\left({x}_{m}\right)$ is the ${l}^{\mathrm{th}}$ column of the matrix $\sigma \left({x}_{m}\right)$.

**Algorithm**

**1.**

- (1)
- Choose a kernel k and compute all its required derivatives, either analytically or with the aid of automatic differentiation.
- (2)
- Assemble the Gram matrices ${G}_{2}$ and ${G}_{0}$ or, if the system is symmetric, ${G}_{1}^{\left(l\right)}$, for $l=1,\cdots ,d$, and ${G}_{0}$.
- (3)
- Solve the corresponding eigenvalue problem described in Lemma 4 to obtain an eigenvector u.
- (4)
- An eigenfunction is then given by $\phi =\mathsf{\Phi}\phantom{\rule{1.00006pt}{0ex}}u$.

#### 3.4. Analysis

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

- (i)
- The operators ${C}_{00}$, ${\widehat{C}}_{00}$, ${\mathcal{T}}_{\mathbb{H}}$, and ${\widehat{\mathcal{T}}}_{\mathbb{H}}$ are Hilbert–Schmidt.
- (ii)
- Let $\delta \in (0,1]$. Assume the coefficients of the operator $\mathcal{T}$ are all globally bounded, and let ${sup}_{x\in \mathbb{X}}{D}^{\alpha}k(x,x)<\infty $ for all $|\alpha |\le 4$ ($|\alpha |\le 2$ in the symmetric case). If the data are drawn i.i.d. from the distribution μ, then there are constants ${\kappa}_{0},{\kappa}_{1}$ such that with probability at least $1-\delta $,$$\begin{array}{cccc}\hfill \parallel {\mathcal{C}}_{00}-{\widehat{C}}_{00}{\parallel}_{HS}& \le \frac{2{\kappa}_{0}\sqrt{2}}{\sqrt{M}}{log}^{1/2}\frac{2}{\delta},\hfill & \hfill \parallel {\mathcal{T}}_{\mathbb{H}}-{\widehat{\mathcal{T}}}_{\mathbb{H}}{\parallel}_{HS}& \le \frac{2{\kappa}_{1}\sqrt{2}}{\sqrt{M}}{log}^{1/2}\frac{2}{\delta},\hfill \end{array}$$where the ${\u2225\xb7\u2225}_{HS}$ is the Hilbert–Schmidt norm.

**Proof.**

**Proposition**

**1.**

**Proof.**

## 4. Applications

#### 4.1. Molecular Dynamics

**Example**

**2.**

#### 4.2. Quantum Mechanics

#### 4.2.1. Generator EDMD for the Schrödinger Equation

**Example**

**3.**

**Example**

**4.**

#### 4.2.2. SDE Formulation of the Schrödinger Equation

**Example**

**5.**

**Example**

**6.**

**Theorem**

**2.**

#### 4.3. Manifold Learning

**Example**

**7.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Lemma 1.**

**Proof**

**of**

**Lemma 2.**

**Proof**

**of**

**Lemma 4.**

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**Figure 1.**(

**a**) Quadruple-well potential. The color blue corresponds to small values and yellow to large values. (

**b**) Clustering into four metastable sets based on sparse eigenbasis approximation (SEBA). (

**c**) Eigenvalues computed using kernel generator extended dynamic mode decomposition (gEDMD) and a Markov state model. The bars indicate the estimated standard deviation.

**Figure 2.**(

**a**) Numerically computed eigenfunctions ${\psi}_{\ell}$ and associated energy levels ${E}_{\ell}$ of the quantum harmonic oscillator. The results are virtually indistinguishable from the analytical results. (

**b**) Corresponding probability densities ${p}_{\ell}$.

**Figure 3.**Numerically computed eigenfunctions of the Schrödinger equation associated with the hydrogen atom. Only points where the absolute value of the eigenfunction is larger than a given threshold are plotted. The shapes clearly resemble the well-known hydrogen atom orbitals shown next to the scatter plots. The eigenfunctions (or rotations thereof) correspond to the following quantum numbers $(\underline{n},\underline{\ell},\underline{m})$: (

**a**) $(1,0,0)$, (

**b**) $(2,1,1)$, (

**c**) $(3,2,1)$, and (

**d**) $(4,3,1)$.

**Figure 4.**Eigenfunctions of the Schrödinger equation associated with the hydrogen atom computed by applying kernel gEDMD to the corresponding Koopman generator. The quantum numbers $(\underline{n},\underline{\ell},\underline{m})$ are: (

**a**) $(3,2,0)$ and (

**b**) $(4,3,2)$.

**Figure 5.**Swiss roll colored with respect to the eigenfunctions (

**a**) ${\phi}_{0}$ and (

**b**) ${\phi}_{5}$, which parametrize the angular and vertical direction, respectively. (

**c**) Resulting two-dimensional embedding.

**Figure 6.**Relationships between the Koopman, Kolmogorov, and Schrödinger operators for a drift-diffusion process of the form $\mathrm{d}{X}_{t}=-\nabla V\left({X}_{t}\right)\phantom{\rule{1.00006pt}{0ex}}\mathrm{d}t+\sqrt{2{\beta}^{-1}}\phantom{\rule{1.00006pt}{0ex}}\mathrm{d}{B}_{t}$. Here, ${\rho}_{0}$ denotes the invariant density, i.e., ${\mathcal{L}}^{*}{\rho}_{0}=0$. In our setting, the transformation of the Schrödinger operator requires a strictly positive real-valued ground state ${\psi}_{0}$.

${X}_{t}$ | stochastic process |

$\mathbb{X}$ | state space |

$k,\varphi $ | kernel and associated feature map |

$\mathbb{H}$ | reproducing kernel Hilbert space induced by k |

${\mathcal{K}}^{t}$ | Koopman operator with lag time t |

$\mathcal{L}$ | generator of the Koopman operator |

$\mathcal{H}$ | Schrödinger operator |

$\mathcal{T}$ | general differential operator |

${\mathcal{T}}_{\mathbb{H}}$ | kernel-based differential operator |

${\mathcal{C}}_{00}$ | covariance operator |

$\widehat{\mathcal{A}}$ | empirical estimate of operator $\mathcal{A}$ |

${G}_{0},{G}_{1},{G}_{2}$ | (generalizations of) Gram matrices |

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**MDPI and ACS Style**

Klus, S.; Nüske, F.; Hamzi, B.
Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator. *Entropy* **2020**, *22*, 722.
https://doi.org/10.3390/e22070722

**AMA Style**

Klus S, Nüske F, Hamzi B.
Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator. *Entropy*. 2020; 22(7):722.
https://doi.org/10.3390/e22070722

**Chicago/Turabian Style**

Klus, Stefan, Feliks Nüske, and Boumediene Hamzi.
2020. "Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator" *Entropy* 22, no. 7: 722.
https://doi.org/10.3390/e22070722