# Modeling of 3D Blood Flows with Physics-Informed Neural Networks: Comparison of Network Architectures

^{*}

## Abstract

**:**

## 1. Introduction

#### Physics-Informed Neural Networks

## 2. Methods

#### 2.1. Fluid Model and Geometries

#### 2.2. Network Architectures

#### 2.2.1. Fully Connected Neural Network (FCNN)

#### 2.2.2. Fully Connected Neural Network with Adaptive Activation (FCNNaa)

#### 2.2.3. Fully Connected Neural Network with Skip Connections (FCNNskip)

#### 2.2.4. Fourier Network (FN)

#### 2.2.5. Modified Fourier Network (modFN)

#### 2.2.6. Multiplicative Filter Network (MFN)

#### 2.2.7. Deep Galerkin Method (DGM)

#### 2.3. Network Details

#### 2.4. Computational Fluid Dynamics (CFD)

#### 2.5. Error Analysis

## 3. Results

#### 3.1. Comparison of Accuracy

#### 3.2. Comparison of Training and Inference Runtimes

## 4. Discussion

#### Limitations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PINN | Physics-informed neural network |

PDE | Partial differential equation |

DL | Deep learning |

FCNN | Fully connected neural network |

FCNNaa | Fully connected neural network with adaptive activations |

FCNNskip | Fully connected neural network with skip connections |

FN | Fourier network |

modFN | Modified Fourier network |

MFN | Multiplicative filter network |

DGM | Deep Galerkin Method |

CNN | Convolutional neural network |

CFD | Computational fluid dynamics |

WSS | Wall shear stress |

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**Figure 1.**(

**a**) The parabolic velocity profile at the inlet of aneurysm #1. The CFD mesh including the two boundary layers is superimposed. (

**b**) The simulation geometries included two idealized structures (cylinder, bifurcation) and two aneurysms that were segmented from brain DSA images. The inlets are indicated by red arrows. (

**c**) The randomly sampled points on the surface of aneurysm #1 used to evaluate the no-slip boundary condition during PINN training.

**Figure 2.**Schematic of the DGM architecture. The detailed operations within the ℓ-th DGM layer are highlighted. $\sigma $ is the activation function, ⊙ is the Hadamard (element-wise) multiplication. $\theta $ comprises the network’s trainable parameters, i.e., the various $\mathbf{V},\mathbf{W}$ and $\mathbf{b}$ terms.

**Figure 3.**Pressure and absolute velocity distributions across a slice through the geometries for the different network architectures: the cylinder and the bifurcation ($xz$-plane through $y=0$, respectively) and both aneurysms. The absolute errors compared to the CFD reference are also shown.

**Figure 4.**Temporal evolution of the mean inlet pressure during the training process of the different network architectures. The dashed line indicates the reference mean inlet pressure obtained from the CFD simulation.

**Table 1.**Boundary conditions used in PINN training. The number of randomly sampled points to evaluate the conditions in the network loss functions is also provided.

Boundary Condition | #Points | |
---|---|---|

Inlet | Parabolic inlet velocity | 2000 |

Outlet | Zero-pressure: $p=0$ | 1000 |

Lateral surface | No-slip: $(u,v,w)=0$ | 2000 |

Interior | Navier-Stokes residual | 3000 |

Continuity plane | Mass flow continuity | 8000 |

**Table 2.**Mean absolute errors (MAEs) of the learned solutions for the different geometries and network architectures compared to the CFD references. Bold numbers indicate the best results.

Cylinder | Bifurcation | |||||
---|---|---|---|---|---|---|

p [Pa] | abs vel [cm/s] | WSS [Pa] | p [Pa] | abs vel [cm/s] | WSS [Pa] | |

FCNN | 0.640 ± 0.422 | 0.153 ± 0.121 | 0.085 ± 0.050 | 0.813 ± 0.338 | 0.210 ± 0.094 | 0.169 ± 0.215 |

FCNNaa | 0.642 ± 0.428 | 0.153 ± 0.122 | 0.086 ± 0.050 | 0.826 ± 0.350 | 0.208 ± 0.091 | 0.167 ± 0.213 |

FCNNskip | 0.642 ± 0.419 | 0.153 ± 0.121 | 0.085 ± 0.050 | 0.785 ± 0.329 | 0.203 ± 0.091 | 0.167 ± 0.218 |

FN | 0.634 ± 0.425 | 0.154 ± 0.121 | 0.086 ± 0.050 | 0.945 ± 0.415 | 0.214 ± 0.094 | 0.171 ± 0.212 |

modFN | 0.718 ± 0.425 | 0.116 ± 0.109 | 0.071 ± 0.061 | 0.898 ± 0.395 | 0.213 ± 0.091 | 0.173 ± 0.207 |

MFN | 0.663 ± 0.421 | 0.154 ± 0.122 | 0.084 ± 0.053 | 0.628 ± 0.375 | 0.198 ± 0.138 | 0.157 ± 0.110 |

DGM | 0.290 ± 0.257 | 0.107 ± 0.105 | 0.065 ± 0.054 | 0.136 ± 0.164 | 0.118 ± 0.072 | 0.145 ± 0.220 |

Aneurysm $\#1$ | Aneurysm $\#2$ | |||||

p [Pa] | abs vel [cm/s] | WSS [Pa] | p [Pa] | abs vel [cm/s] | WSS [Pa] | |

FCNN | 0.924 ± 0.622 | 0.384 ± 0.323 | 0.607 ± 1.416 | 2.055 ± 0.672 | 1.125 ± 1.203 | 0.419 ± 0.458 |

FCNNaa | 0.968 ± 0.625 | 0.386 ± 0.323 | 0.610 ± 1.426 | 2.056 ± 0.673 | 1.114 ± 1.201 | 0.417 ± 0.461 |

FCNNskip | 0.868 ± 0.606 | 0.364 ± 0.311 | 0.611 ± 1.428 | 1.996 ± 0.659 | 1.123 ± 1.136 | 0.412 ± 0.433 |

FN | 1.051 ± 0.731 | 0.512 ± 0.451 | 0.614 ± 1.412 | 2.348 ± 0.733 | 1.214 ± 1.617 | 0.491 ± 0.655 |

modFN | 0.931 ± 0.647 | 0.408 ± 0.345 | 0.611 ± 1.424 | 2.182 ± 0.685 | 1.079 ± 1.263 | 0.441 ± 0.520 |

MFN | 0.732 ± 0.729 | 0.369 ± 0.340 | 0.600 ± 1.370 | 2.203 ± 0.703 | 1.140 ± 1.391 | 0.446 ± 0.549 |

DGM | 1.752 ± 1.229 | 0.363 ± 0.319 | 0.583 ± 1.380 | 5.489 ± 1.081 | 0.557 ± 0.768 | 0.205 ± 0.338 |

FCNN | FCNNaa | FCNNskip | FN | modFN | MFN | DGM | |
---|---|---|---|---|---|---|---|

Runtime/FCNN | 100% | 115% | 102% | 101% | 169% | 114% | 363% |

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**MDPI and ACS Style**

Moser, P.; Fenz, W.; Thumfart, S.; Ganitzer, I.; Giretzlehner, M.
Modeling of 3D Blood Flows with Physics-Informed Neural Networks: Comparison of Network Architectures. *Fluids* **2023**, *8*, 46.
https://doi.org/10.3390/fluids8020046

**AMA Style**

Moser P, Fenz W, Thumfart S, Ganitzer I, Giretzlehner M.
Modeling of 3D Blood Flows with Physics-Informed Neural Networks: Comparison of Network Architectures. *Fluids*. 2023; 8(2):46.
https://doi.org/10.3390/fluids8020046

**Chicago/Turabian Style**

Moser, Philipp, Wolfgang Fenz, Stefan Thumfart, Isabell Ganitzer, and Michael Giretzlehner.
2023. "Modeling of 3D Blood Flows with Physics-Informed Neural Networks: Comparison of Network Architectures" *Fluids* 8, no. 2: 46.
https://doi.org/10.3390/fluids8020046