# Splines Parameterization of Planar Domains by Physics-Informed Neural Networks

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## Abstract

**:**

## 1. Introduction

- The discrete description of the computational domain is achieved by using PINNs.
- The continuous representation of the computational domain is then obtained by using a suitable QI operator which provides a spline parameterization, i.e., a continuous description, of the desired smoothness.

## 2. Preliminaries

## 3. The Method

`Tensorflow`[38],

`PyTorch`[39], and

`Jax`[40]. The parameter w in (4) is determined via thresholding on the proximity of the interior points to the boundaries; $w=0$ if $\mathrm{dist}({\widehat{\mathcal{M}}}_{\mathbf{\theta}}\left(\mathbf{t}\right),\partial \mathsf{\Omega})<{10}^{-4}$, otherwise $w=1$. Training is performed on an Intel(R) Core(TM) i7-9800X processor running at 3.80 GHz using 31 GB of RAM along with a GeForce GTX 1080 Ti GPU unit by using the

`Tensorflow`package. The training phase firstly is carried on for 5000 epochs with the Adam optimizer [41] with learning rate $\lambda =0.001$, and afterwards for 10,000 epochs with the L-BFGS-B algorithm [42], following the method originally proposed in [25] to have (empirical) convergence guarantees. The weights are initialized under the Xavier initialization method [43]. The adopted non-linear activation function for each layer is the hyperbolic tangent

`tanh`. Given different initial ${\mathbf{W}}^{0},{\mathbf{b}}^{0}$, it is observed that PINNs may converge to different solutions, see, e.g., [44,45]. Hence, our experiments are performed 10 times, changing the initial random seed and producing 10 approximate solutions. Since there is no guarantee of a unique solution, as a non-convex optimization problem is solved by minimizing (4), the selected ${\widehat{\mathcal{M}}}_{\mathbf{\theta}}$ which is retained corresponds to the solution achieving the smallest residual term. The PINNs code outputs the predicted evaluation of ${\widehat{\mathcal{M}}}_{\mathbf{\theta}}$ at points $({t}_{i},{t}_{j})$ inside $\widehat{\mathsf{\Omega}}$. At this stage, a quasi-interpolant splines is constructed by adopting Formula (2).

- The boundary $\partial \mathsf{\Omega}$ is split into 4 pieces ${\mathsf{\Gamma}}_{i}$, for $i=1,\dots ,4$, by performing for example knot-insertion.
- Each ${\mathsf{\Gamma}}_{i}$ is then parametrized as a Bspline curve ${g}_{i}:[0,1]\to {\mathsf{\Gamma}}_{i}$.
- PINNs are trained to minimize the loss functional in Equation (4) over a set of boundary points and over the Laplace equation.
- The trained network ${\widehat{\mathcal{M}}}_{\mathbf{\theta}}$ represents an approximation of the sought parameterization map $\mathcal{M}$.
- Uniformly spaced grid points are generated in $\widehat{\mathsf{\Omega}}$ and mapped by ${\widehat{\mathcal{M}}}_{\mathbf{\theta}}$ to $\mathsf{\Omega}$.
- A continuous spline approximation of ${\widehat{\mathcal{M}}}_{\mathbf{\theta}}$ is obtained by using a Hermite Quasi-Interpolation operator (QI).

Algorithm 1 Pseudo-code for the proposed algorithm |

## 4. Numerical Examples

#### 4.1. Circle

#### 4.2. Wedge-Shape

#### 4.3. Quarter-Annulus-Shaped Domain

#### 4.4. Hourglass-Shaped Domain

#### 4.5. Butterfly-Shaped Domain

## 5. Post-Processing Correction

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Circular domain. (

**a**) Points generated via Coons; (

**b**) Linear parameterization with Coons) (

**c**) Points generated via inpaint; (

**d**) Linear parameterization with inpaint; (

**e**) Points with PINNs; and (

**f**) PINNs-QI parameterization.

**Figure 3.**Wedge-shaped domain. (

**a**) Points generated via Coons; (

**b**) Linear parameterization with Coons; (

**c**) Points generated via inpaint; (

**d**) Linear parameterization with inpaint; (

**e**) Points with PINNs; and (

**f**) PINNs-QI parameterization.

**Figure 4.**Quarter-annulus domain. (

**a**) Points generated via Coons; (

**b**) Linear parameterization with Coons; (

**c**) Points generated via inpaint; (

**d**) Linear parameterization with inpaint; (

**e**) Points generated with PINNs; and (

**f**) PINNs-QI parameterization.

**Figure 5.**Hourglass-shaped domain. (

**a**) Points generated via Coons; (

**b**) Linear parameterization with Coons and zoom in; (

**c**) Points generated via inpaint; (

**d**) Linear parameterization with inpaint and zoom in (

**e**) Points generated via PINNs; and (

**f**) PINNs-QI parameterization and zoom in.

**Figure 6.**Determinant of the Jacobian matrix for the bicubic parameterization. (

**a**) Method: Coons-QI; (

**b**) Method: Inpaint-QI; and (

**c**) Method: PINNs-QI.

**Figure 7.**Butterfly-shaped domain. (

**a**) Points generated via Coons; (

**b**) Linear parameterization with Coons and zoom-in; (

**c**) Points generated via inpaint; (

**d**) Linear parameterization with inpaint and zoom-in; (

**e**) Points generated with PINNs; and (

**f**) PINNs-QI parameterization and zoom-in.

**Figure 8.**Determinant of the Jacobian matrix for the obtained parameterization. (

**a**) Method: Coons; (

**b**) Method: Inpaint; and (

**c**) Method: PINNs-QI.

**Figure 9.**Correction output. (

**a**) PINNS-QI parameterization after post-processing; and (

**b**) Determinant of J for the post-procecessed mapping.

**Figure 10.**Comparison before and after post-processing. (

**a**) Zoom-in where there is no injectivity; and (

**b**) Zoom-in after the post-processing correction.

**Figure 11.**Results after the post-processing correction. (

**a**) PINNs-QI parameterization after post-processing; and (

**b**) Determinant of the Jacobian matrix after post-processing.

Method | Bij | W | min(det J) | max(det J) |
---|---|---|---|---|

Coons | yes | 2.1640 | 0.3150 | 4.7044 |

Inpaint | yes | 2.1598 | 0.4141 | 3.7948 |

PINNs | yes | 2.1639 | 0.3125 | 4.3160 |

Method | Bij | W | min(det J) | max(det J) |
---|---|---|---|---|

Coons | yes | 2.0834 | 0.9927 | 1.9635 |

Inpaint | yes | 2.0812 | 0.6329 | 2.0284 |

PINNs | yes | 2.0819 | 0.9928 | 1.8355 |

Curve | KV | ${\mathit{c}}_{\mathit{x}}$ | ${\mathit{c}}_{\mathit{y}}$ |
---|---|---|---|

${g}_{1}$ | ${\Xi}_{1}$ | ${(-4,\phantom{\rule{0.166667em}{0ex}}-2.5,\phantom{\rule{0.166667em}{0ex}}-1)}^{T}$ | ${(0,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}0)}^{T}$ |

${g}_{2}$ | ${\Xi}_{2}$ | ${(-1,\phantom{\rule{0.166667em}{0ex}}-1,\phantom{\rule{0.166667em}{0ex}}-0.7,\phantom{\rule{0.166667em}{0ex}}-0.4,\phantom{\rule{0.166667em}{0ex}}0)}^{T}$ | ${(0,\phantom{\rule{0.166667em}{0ex}}0.4,\phantom{\rule{0.166667em}{0ex}}0.7,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}1)}^{T}$ |

${g}_{3}$ | ${\Xi}_{1}$ | ${(0,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}0)}^{T}$ | ${(4,\phantom{\rule{0.166667em}{0ex}}2.5,\phantom{\rule{0.166667em}{0ex}}1)}^{T}$ |

${g}_{4}$ | ${\Xi}_{2}$ | ${(-4,\phantom{\rule{0.166667em}{0ex}}-4,\phantom{\rule{0.166667em}{0ex}}-4,\phantom{\rule{0.166667em}{0ex}}-2,\phantom{\rule{0.166667em}{0ex}}0)}^{T}$ | ${(0,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}4)}^{T}$ |

Method | Bij | W | min(det J) | max(det J) |
---|---|---|---|---|

Coons | yes | 2.4242 | 5.0519 | 31.7624 |

Inpaint | yes | 2.1631 | 2.1631 | 2.5388 |

PINNs | yes | 2.2287 | 3.3262 | 31.1962 |

Curve | ${\mathit{c}}_{\mathit{x}}$ | ${\mathit{c}}_{\mathit{y}}$ |
---|---|---|

${g}_{1}$ | ${(1.5,\phantom{\rule{0.166667em}{0ex}}3.5,\phantom{\rule{0.166667em}{0ex}}5.6,\phantom{\rule{0.166667em}{0ex}}8,\phantom{\rule{0.166667em}{0ex}}10)}^{T}$ | ${(1.5,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}2.7,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}1.8)}^{T}$ |

${g}_{2}$ | ${(10,\phantom{\rule{0.166667em}{0ex}}7,\phantom{\rule{0.166667em}{0ex}}6,\phantom{\rule{0.166667em}{0ex}}7,\phantom{\rule{0.166667em}{0ex}}10)}^{T}$ | ${(1.8,\phantom{\rule{4pt}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}7,\phantom{\rule{0.166667em}{0ex}}10,\phantom{\rule{0.166667em}{0ex}}13)}^{T}$ |

${g}_{3}$ | ${(1.2,\phantom{\rule{0.166667em}{0ex}}3.5,\phantom{\rule{0.166667em}{0ex}}5.6,\phantom{\rule{0.166667em}{0ex}}8,\phantom{\rule{0.166667em}{0ex}}10)}^{T}$ | ${(13,\phantom{\rule{0.166667em}{0ex}}12,\phantom{\rule{0.166667em}{0ex}}11.7,\phantom{\rule{0.166667em}{0ex}}12.5,\phantom{\rule{0.166667em}{0ex}}13)}^{T}$ |

${g}_{4}$ | ${(1.5,\phantom{\rule{0.166667em}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}1.2)}^{T}$ | ${(1.5,\phantom{\rule{0.166667em}{0ex}}4,\phantom{\rule{0.166667em}{0ex}}7,\phantom{\rule{0.166667em}{0ex}}10,\phantom{\rule{0.166667em}{0ex}}13)}^{T}$ |

Method | Bij | W | min(det J) | max(det J) |
---|---|---|---|---|

Coons | yes | 6.7636 | 0.4713 | 161.3302 |

Inpaint | no | 8.2491 | −15.7232 | 161.3302 |

PINNs | no | 2.1853 | −1.1140 | 151.6974 |

PINNs-Post | yes | 4.0696 | 4.7709 | 339.1136 |

Curve | KV | ${\mathit{c}}_{\mathit{x}}$ | ${\mathit{c}}_{\mathit{y}}$ |
---|---|---|---|

${g}_{1}$ | ${\Xi}_{1}$ | ${(4,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}8,\phantom{\rule{0.166667em}{0ex}}9.2,\phantom{\rule{0.166667em}{0ex}}11,\phantom{\rule{0.166667em}{0ex}}14,\phantom{\rule{0.166667em}{0ex}}16)}^{T}$ | ${(1,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}7,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}1)}^{T}$ |

${g}_{2}$ | ${\Xi}_{2}$ | ${(16,\phantom{\rule{0.166667em}{0ex}}16,\phantom{\rule{0.166667em}{0ex}}13,\phantom{\rule{0.166667em}{0ex}}13,\phantom{\rule{0.166667em}{0ex}}17)}^{T}$ | ${(15,\phantom{\rule{0.166667em}{0ex}}7,\phantom{\rule{0.166667em}{0ex}}11,\phantom{\rule{0.166667em}{0ex}}15)}^{T}$ |

${g}_{3}$ | ${\Xi}_{1}$ | ${(1,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}7,\phantom{\rule{0.166667em}{0ex}}9,\phantom{\rule{0.166667em}{0ex}}11,\phantom{\rule{0.166667em}{0ex}}14,\phantom{\rule{0.166667em}{0ex}}17)}^{T}$ | ${(15,\phantom{\rule{0.166667em}{0ex}}13,\phantom{\rule{0.166667em}{0ex}}14.5,\phantom{\rule{0.166667em}{0ex}}12,\phantom{\rule{0.166667em}{0ex}}14,\phantom{\rule{0.166667em}{0ex}}13,\phantom{\rule{0.166667em}{0ex}}15)}^{T}$ |

${g}_{4}$ | ${\Xi}_{2}$ | ${(4,\phantom{\rule{0.166667em}{0ex}}3,\phantom{\rule{0.166667em}{0ex}}6,\phantom{\rule{0.166667em}{0ex}}6,\phantom{\rule{0.166667em}{0ex}}1)}^{T}$ | ${(1,\phantom{\rule{0.166667em}{0ex}}5,\phantom{\rule{0.166667em}{0ex}}7,\phantom{\rule{0.166667em}{0ex}}11,\phantom{\rule{0.166667em}{0ex}}15)}^{T}$ |

Method | Bij | W | min(detJ) | max(det J) |
---|---|---|---|---|

Coons | no | ∞ | −51.7043 | $1.0693\times {10}^{3}$ |

Inpaint | no | ∞ | −254.3629 | $1.0695\times {10}^{3}$ |

PINNs | no | 2.9064 | −114.9197 | $1.2973\times {10}^{3}$ |

PINNs-Post | yes | 2.6513 | 0.045 | $4.3519\times {10}^{3}$ |

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**MDPI and ACS Style**

Falini, A.; D’Inverno, G.A.; Sampoli, M.L.; Mazzia, F.
Splines Parameterization of Planar Domains by Physics-Informed Neural Networks. *Mathematics* **2023**, *11*, 2406.
https://doi.org/10.3390/math11102406

**AMA Style**

Falini A, D’Inverno GA, Sampoli ML, Mazzia F.
Splines Parameterization of Planar Domains by Physics-Informed Neural Networks. *Mathematics*. 2023; 11(10):2406.
https://doi.org/10.3390/math11102406

**Chicago/Turabian Style**

Falini, Antonella, Giuseppe Alessio D’Inverno, Maria Lucia Sampoli, and Francesca Mazzia.
2023. "Splines Parameterization of Planar Domains by Physics-Informed Neural Networks" *Mathematics* 11, no. 10: 2406.
https://doi.org/10.3390/math11102406