# Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential

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## Abstract

**:**

## 1. Introduction

#### The Motivation and the Contribution of this Paper

## 2. Theoretical Background

**Definition**

**1.**

**Definition**

**2.**

#### Algorithms

Algorithm 1: Certified Deep Ritz Algorithm. |

## 3. Results

#### 3.1. Direct Approximations of the Ground State in 1D

#### 3.2. Direct Approximations of the Ground State in Higher-Dimensional Spaces

#### 3.3. Approximations of the Landscape Function in 1D

#### 3.4. Direct VPINN Approximation of the Landscape Function in 2D

#### 3.5. Encoder–Decoder Network as a Reduced-Order Model for a Family of Landscape Functions

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Sample Availability

## Abbreviations

PDE | partial differential equation |

ReLU | rectified linear unit |

FEM | finite element method |

DOF | degrees of freedom |

VPINN | Variational Physics Informed Neural Networks |

FCNN | fully convolutional neural network |

## Appendix A. Implementation Details

## Appendix B. Estimating Residuals

#### Appendix B.1. Finite Element Quadrature for 2D Problems

#### Appendix B.2. Direct Approximations for Higher Dimensional Problems

## Appendix C. Architecture of the VPINN Neural Network

**Figure A1.**VPINN architecture with k blocks, l layers in each block and m neurons in each dense layer.

**Figure A2.**FCNN encoder–decoder architecture inspired by the U-Net concept from [30].

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**Figure 1.**(

**a**) Comparison of the ground state obtained in chebfun (${\psi}_{chebfun}\left(x\right)$) and as the VPINN solution (${\psi}_{NN}\left(x\right)$) with the architecture ${\overrightarrow{N}}_{\mathtt{DenseNet}}=(n,k,l,m)=(1,4,2,10)$; (

**b**) Residual and Rayleigh quotient error estimate metrics during the training process.

**Figure 2.**The effective potential and its 6 local minima, which define localization of the first six eigenstates is shown on the right. Eigenstates ${\psi}_{i},i=0,1,...,5$ were computed in chebfun.

**Figure 3.**A surface plot of the effective potential $W=1/u$ (

**a**) and the landscape function u (

**b**). In (

**a**) we plot the boundaries of the sets $\{x\phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{3.33333pt}{0ex}}\epsilon \phantom{\rule{3.33333pt}{0ex}}u(x)\ge 1\}$ that localize the eigenstates. In (

**b**) we plot the circles of radius $1/\tilde{{\epsilon}_{i}}$, for ${\tilde{\epsilon}}_{i-1}=3{W}_{\mathrm{min},i}/2$, $i=1,2,3$, centered at the i-th lowermost local minimum ${W}_{\mathrm{min},i}$.

**Figure 4.**A benchmarking comparison of the encoder–decoder prediction of the landscape function against the FEniCS solution.

**Figure 5.**Comparing the Chebyshev series expansion with 149 terms and a VPINN solution with the architecture ${\overrightarrow{N}}_{\mathtt{DenseNet}}=(1,2,2,2)$ and 30 trainable parameters.

**Table 1.**Convergence rates for the ground state energy of the harmonic oscillator in relation to the dimension. QMC: quasi-Monte Carlo.

n | ${\mathit{\epsilon}}_{0}$ | M for the Loss Function | Adam Optimizer Epochs | M for the Smolyak Quadrature | Smolyak Relative Error % | Relative Error for QMC with $\mathit{M}={10}^{5}$ Points% |
---|---|---|---|---|---|---|

1 | 1 | 100 | 50,000 | 127 | 0.004 | 0.003 |

2 | 2 | 1000 | 20,000 | 769 | 1.416 | 1.226 |

3 | 3 | 5000 | 50,000 | 2815 | 1.110 | 1.608 |

6 | 6 | 50,000 | 80,000 | 40,193 | - | 1.40 |

9 | 9 | 50,000 | 50,000 | 242,815 | 230.366 | 5.816 |

**Table 2.**We tested the accuracy of the predictor ${\tilde{\u03f5}}_{i-1}=\left(1+{\displaystyle \frac{n}{4}}\right){W}_{min,i}$ for 16 lowermost eigenvalues. The chebfun solution was used to benchmark the error.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |

Minimum values of W | 0.747201 | 0.918677 | 0.918754 | 0.933014 | 1.028903 | 1.057663 | 1.174706 | 1.245278 |

chebfun eigenvalues | 0.979730 | 1.071839 | 1.230230 | 1.282611 | 1.301724 | 1.485232 | 1.577349 | 1.588252 |

Relative error in % | 4.6675 | 7.1379 | 6.6481 | 9.0708 | 1.1981 | 1.9850 | 6.9082 | 1.9930 |

8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |

Minimum values of W | 1.256498 | 1.273980 | 1.326926 | 1.613203 | 1.848415 | 1.868003 | 1.907063 | 1.931723 |

chebfun eigenvalues | 1.625253 | 1.758768 | 1.780166 | 2.095899 | 2.161778 | 2.265704 | 2.270798 | 2.278380 |

Relative error in % | 3.3614 | 9.4551 | 6.8257 | 3.7882 | 6.8805 | 3.05864 | 4.9776 | 5.9811 |

**Table 3.**A report on the convergence in k and m for the family of architectures ${\overrightarrow{N}}_{\mathtt{DenseNet}}=(2,k,2,m)$. We benchmark the error against the highly accurate ${P}_{3}$ FEniCS solution.

Parameters | k | m | Relative ${\mathit{L}}^{2}$ Error 100,000 Epoch | Relative ${\mathit{H}}^{1}$ Error 100,000 Epoch | Relative ${\mathit{L}}^{2}$ Error 200,000 Epoch | Relative ${\mathit{H}}^{1}$ Error 200,000 Epoch | Relative Error of the First Three Eigenvalues Respectively |
---|---|---|---|---|---|---|---|

803 | 4 | 8 | 2.5852% | 5.6216% | 2.0527% | 4.9876% | 0.1638%, 1.4479%, 1.1472% |

1203 | 4 | 10 | 2.7487% | 5.3611% | 1.2354% | 3.6960% | 0.0839%, 2.3489%, 0.6341% |

1753 | 5 | 10 | 1.9314% | 4.2386% | 1.0679% | 3.3851% | 0.5957%, 1.9264%, 0.3822% |

2403 | 6 | 10 | 1.1745% | 3.0548% | 0.7998% | 2.6994% | 0.4539%, 1.7883%, 1.5112% |

4403 | 4 | 20 | 1.9037% | 3.6929% | 0.7233% | 2.5757% | 0.3242%, 1.8831%, 1.2586% |

9603 | 4 | 30 | 1.8217% | 3.7451% | 0.6689% | 2.3609% | 0.3639%, 2.0083%, 0.9685% |

16,803 | 4 | 40 | 0.6372% | 1.9704% | 0.3920% | 1.5497% | 0.3269%, 1.8606%, 0.6983% |

26,003 | 4 | 50 | 3.6993% | 7.3510% | 0.4207% | 1.6748% | 0.3127%, 1.5756%, 0.3559% |

**Table 4.**Validation of the encoder–decoder representation of the mapping $\mathcal{L}:V\mapsto u$ on a collection of test examples. Recall that the effective potential is defined as $W=1/u$.

Average ${\mathit{L}}^{2}$ error | 1.7545% |

Maximal ${L}^{2}$ error | 2.9769%, example: 58 |

Average ${H}^{1}$ error | 9.2233% |

Maximal ${H}^{1}$ error | 12.6765%, example: 65 |

Mean relative error in $1/{W}_{min,1}$ | 0.4887% |

Maximal relative error in $1/{W}_{min,1}$ | 2.1402%, example: 70 |

The worst ten relative errors in $1/{W}_{min,1}$ (%) | 2.1402, 1.5909, 1.5560, 1.4816, 1.4151, 1.4626, 1.3441, 1.3377, 1.3181, 1.3132 |

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**MDPI and ACS Style**

Grubišić, L.; Hajba, M.; Lacmanović, D. Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential. *Entropy* **2021**, *23*, 95.
https://doi.org/10.3390/e23010095

**AMA Style**

Grubišić L, Hajba M, Lacmanović D. Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential. *Entropy*. 2021; 23(1):95.
https://doi.org/10.3390/e23010095

**Chicago/Turabian Style**

Grubišić, Luka, Marko Hajba, and Domagoj Lacmanović. 2021. "Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential" *Entropy* 23, no. 1: 95.
https://doi.org/10.3390/e23010095