# A PINN Surrogate Modeling Methodology for Steady-State Integrated Thermofluid Systems Modeling

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## Abstract

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## 1. Introduction

^{®}[4] fluid simulation software. This software applies numerical finite volume methods to solve the applicable partial differential equations (PDEs). Rauch et al. [5] employed a process model for a combined Brayton–Rankine cycle to determine the maximum thermal efficiency of the combined cycle. The process model consisted of a complete mathematical model and was developed using the Matlab

^{®}[6] programming platform, which is typically used for iterative analysis and design processes. The work of Zhang et al. [7] provides another example of a physics-based solver applied to thermofluid applications. The authors applied compartment models to analyze the reheat steam temperatures in a double reheat coal-fired boiler. This involved modeling the transfer of heat across different compartments, or control volumes within the boiler system. Although these traditional physics-based numerical solvers are accurate, they can be computationally expensive and time-consuming due to their complexity.

## 2. Theoretical Background and Methodology

#### 2.1. Multilayer Perceptron Neural Networks

#### 2.2. PINN Methodology for Thermofluid Process Modeling

^{−6}was selected as the desired tolerance for the total model loss. A second restriction of a maximum number of iterations was placed on the optimization process of the PINN models to prevent the process from running indefinitely, should the models not be able to reach the desired tolerance specified above.

## 3. Case Studies

#### 3.1. Heat Exchanger Network

#### 3.2. Recuperated Closed Brayton Cycle

## 4. Results and Discussion

#### 4.1. PINN Training Process

#### 4.2. Hyperparameter Search Results

^{−4}and 5 × 10

^{−5}for the heat exchanger network and the Brayton cycle, respectively.

#### 4.3. PINN Results: Accuracy

#### 4.4. PINN Results: Computational Expense

## 5. Conclusions

^{−3}s. This is a significant improvement over the conventional process models which required 1× 10

^{−1}s to generate a solution. If these modeling approaches are to be applied to a more complex thermofluid network, the time taken for the conventional model to generate a solution will scale non-linearly and it is likely that it would require approximately 1 × 10

^{1}s to generate a solution. By contrast, the inference speed of PINNs remains constant, regardless of the complexity of the network, provided that the GPU RAM is not exceeded [27]. This means that for a problem that requires 10 model calls it would take 1 × 10

^{2}s to generate a solution using a conventional process model, but only 1 × 10

^{−2}s using a PINN model. This highlights the potential for PINN surrogate models as a valuable engineering tool in component and system design and optimization, as well as in real-time simulation for anomaly detection, diagnosis, and forecasting.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Viscosity of carbon dioxide as a function of enthalpy and pressure; (

**b**) Thermal conductivity of air as a function of enthalpy and pressure.

**Figure 4.**Thermofluid network model superimposed onto the physical layout of the heat exchanger system. Heat exchanger—HX.

**Figure 5.**Process flow diagram for the recuperated Brayton cycle. Compressor—C, recuperator heat exchanger—RX, heater—H, turbine—T, pre-cooler—PC.

**Figure 6.**(

**a**) Pressure ratio vs. corrected mass flow rate for the compressor; (

**b**) Isentropic efficiency vs. corrected mass flow rate for the compressor.

**Figure 7.**(

**a**) Pressure ratio vs. corrected mass flow rate for the turbine; (

**b**) Isentropic efficiency vs. corrected mass flow rate for the turbine.

**Figure 8.**(

**a**) Training history for the unsupervised training of the PINN model of the heat exchanger network using only the Adam optimizer; (

**b**) Training history for the unsupervised training of the PINN model of the heat exchanger network using a combination of the Adam optimizer and the truncated Newton method. The various colors each represent a different sample of the same simulation.

**Figure 9.**(

**a**) Training history for the unsupervised training of the PINN model of the recuperated Brayton cycle using only the Adam optimizer; (

**b**) Training history for the unsupervised training of the PINN model of the recuperated Brayton cycle using a combination of the Adam optimizer and the truncated Newton method. The various colors each represent a different sample of the same simulation.

Parameter | Symbol |
---|---|

Outlet pressures | ${p}_{out,g},{p}_{out,CO2}$ |

Inlet mass flow rates | ${\dot{m}}_{in,g},{\dot{m}}_{in,CO2}$ |

Inlet temperatures | ${T}_{in,g},{T}_{in,CO2}$ |

Heat exchanger lumped loss coefficients | ${K}_{1},{K}_{2},{K}_{3}$ |

Heat exchanger effectiveness values | ${\epsilon}_{1},{\epsilon}_{2},{\epsilon}_{3}$ |

Parameter | Symbol |
---|---|

Heat exchanger lumped loss coefficients | $K$ |

Heat exchanger effectiveness values | ${\epsilon}_{RX},{\epsilon}_{H},{\epsilon}_{PC}$ |

Minimum cycle temperature | ${T}_{L}$ |

Maximum cycle temperature | ${T}_{H}$ |

Minimum cycle pressure | ${p}_{L}$ |

Compressor | Turbine | ||
---|---|---|---|

a_{0C} | 3.947422 | a_{0T} | $-$8.437242 × 10^{−23} |

a_{1C} | 5.373162 × 10^{2} | a_{1T} | 3.313531 × 10^{−9} |

a_{2C} | $-$8.776627 × 10^{5} | a_{2T} | 9.730980 × 10^{6} |

b_{0C} | $-$1.942890 × 10^{−16} | b_{0T} | 1.054712 × 10^{−15} |

b_{1C} | 1.453743 × 10^{3} | b_{1T} | 3.199740 × 10^{3} |

b_{2C} | $-$5.936429 × 10^{5} | b_{2T} | $-$2.752242 × 10^{6} |

Number of Neurons per Layer | Number of Hidden Layers | ||
---|---|---|---|

1 | 2 | 3 | |

1 | 1.7 × 10^{−1} | 1.2 × 10^{−1} | 1.1 × 10^{−1} |

8 | 2.6 × 10^{−2} | 1.6 × 10^{−2} | 1.2 × 10^{−2} |

16 | 1.3 × 10^{−3} | 1.0 × 10^{−3} | 1.8 × 10^{−4} |

32 | 6.5 × 10^{−5} | 1.5 × 10^{−5} | 5.7 × 10^{−6} |

Number of Neurons per Layer | Number of Hidden Layers | ||
---|---|---|---|

1 | 2 | 3 | |

1 | 1.8 × 10^{−1} | 1.1 × 10^{−1} | 8.7 × 10^{−2} |

8 | 3.2 × 10^{−2} | 5.1 × 10^{−3} | 2.3 × 10^{−3} |

16 | 4.5 × 10^{−5} | 4.7 × 10^{−6} | 1.0 × 10^{−6} |

**Table 6.**Performance (absolute and relative errors) per output parameter for the PINN model of the heat exchanger network that used only Adam for optimization.

${\dot{\mathit{m}}}_{\mathit{C}\mathit{O}2}$ $[\mathbf{k}\mathbf{g}/\mathbf{s}]$ | ${\mathit{h}}_{\mathit{C}\mathit{O}2}$ $[\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}]$ | ${\mathit{p}}_{\mathit{C}\mathit{O}2}$ $\left[\mathbf{k}\mathbf{P}\mathbf{a}\right]$ | ${\dot{\mathit{m}}}_{\mathit{a}\mathit{i}\mathit{r}}$ $[\mathbf{k}\mathbf{g}/\mathbf{s}]$ | ${\mathit{h}}_{\mathit{a}\mathit{i}\mathit{r}}$ $[\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}]$ | ${\mathit{p}}_{\mathit{a}\mathit{i}\mathit{r}}$ $\left[\mathbf{k}\mathbf{P}\mathbf{a}\right]$ | |
---|---|---|---|---|---|---|

Maximum | 0.0052 | 2.3697 | 4.1911 | 0.0169 | 2.6047 | 0.1615 |

Minimum | 2.98 × 10^{−4} | 1.77 × 10^{−1} | 3.51 × 10^{−3} | 1.67 × 10^{−3} | 9.78 × 10^{−2} | 4.19 × 10^{−3} |

Average | 0.0030 | 1.4519 | 0.4699 | 0.0061 | 1.1724 | 0.0809 |

Max (%) | 0.1227 | 0.2743 | 0.0182 | 0.3211 | 0.3124 | 0.1440 |

Min (%) | 5.70 × 10^{−3} | 2.21 × 10^{−2} | 2.39 × 10^{−5} | 3.03 × 10^{−2} | 1.25 × 10^{−2} | 3.84 × 10^{−3} |

Avg (%) | 0.0611 | 0.1745 | 0.0020 | 0.1165 | 0.1421 | 0.0738 |

**Table 7.**Performance (absolute and relative errors) per output parameter for the PINN model of the heat exchanger network that used the hybrid Adam-TNC optimizer.

${\dot{\mathit{m}}}_{\mathit{C}\mathit{O}2}$ $[\mathbf{k}\mathbf{g}/\mathbf{s}]$ | ${\mathit{h}}_{\mathit{C}\mathit{O}2}$ $[\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}]$ | ${\mathit{p}}_{\mathit{C}\mathit{O}2}$ $\left[\mathbf{k}\mathbf{P}\mathbf{a}\right]$ | ${\dot{\mathit{m}}}_{\mathit{a}\mathit{i}\mathit{r}}$ $[\mathbf{k}\mathbf{g}/\mathbf{s}]$ | ${\mathit{h}}_{\mathit{a}\mathit{i}\mathit{r}}$ $[\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}]$ | ${\mathit{p}}_{\mathit{a}\mathit{i}\mathit{r}}$ $\left[\mathbf{k}\mathbf{P}\mathbf{a}\right]$ | |
---|---|---|---|---|---|---|

Maximum | 0.0057 | 1.9053 | 2.8155 | 0.0117 | 1.6225 | 0.1079 |

Minimum | 8.97 × 10^{−5} | 6.34 × 10^{−1} | 7.16 × 10^{−3} | 5.19 × 10^{−4} | 7.24 × 10^{−1} | 1.19 × 10^{−2} |

Average | 0.0031 | 1.3730 | 0.3685 | 0.0055 | 1.1017 | 0.0681 |

Max (%) | 0.1442 | 0.2683 | 0.0133 | 0.2700 | 0.1946 | 0.0966 |

Min (%) | 1.71 × 10^{−3} | 7.95 × 10^{−2} | 4.88 × 10^{−5} | 9.02 × 10^{−3} | 9.22 × 10^{−2} | 1.15 × 10^{−2} |

Avg (%) | 0.0648 | 0.1662 | 0.0018 | 0.1109 | 0.1351 | 0.0618 |

**Table 8.**Performance (absolute and relative errors) per output parameter for the PINN model of the recuperated Brayton cycle that used only Adam for optimization.

$\dot{\mathit{m}}[\mathbf{k}\mathbf{g}/\mathbf{s}]$ | $\mathit{h}[\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}]$ | $\mathit{p}\left[\mathbf{k}\mathbf{P}\mathbf{a}\right]$ | |
---|---|---|---|

Maximum | 7.046 | 22.389 | 171.208 |

Minimum | 0.420 | 0.417 | 19.630 |

Average | 3.162 | 6.379 | 85.745 |

Max (%) | 1.127 | 3.128 | 0.803 |

Min (%) | 0.0813 | 0.0609 | 0.1062 |

Avg (%) | 0.5353 | 0.9266 | 0.4545 |

**Table 9.**Performance (absolute and relative errors) per output parameter for the PINN model of the recuperated Brayton cycle that used the hybrid Adam-TNC optimizer.

$\dot{\mathit{m}}[\mathbf{k}\mathbf{g}/\mathbf{s}]$ | $\mathit{h}[\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}]$ | $\mathit{p}\left[\mathbf{k}\mathbf{P}\mathbf{a}\right]$ | |
---|---|---|---|

Maximum | 6.753 | 22.133 | 145.228 |

Minimum | 1.013 | 0.448 | 19.887 |

Average | 2.920 | 6.155 | 80.600 |

Max (%) | 1.080 | 3.092 | 0.767 |

Min (%) | 0.171 | 0.065 | 0.108 |

Avg (%) | 0.499 | 0.893 | 0.429 |

Heat Exchanger Network | Brayton Cycle | |||
---|---|---|---|---|

Adam Only | Adam and TNC | Adam Only | Adam and TNC | |

Max | 1582 | 1203 | 1600 | 922 |

Min | 724 | 566 | 874 | 431 |

Avg | 1044 | 866.1 | 1268.7 | 577.6 |

Heat Exchanger Network | Brayton Cycle | |||
---|---|---|---|---|

Trained PINN | Conventional Process Model | Trained PINN | Conventional Process Model | |

Max (s) | 0.0156 | 0.5369 | 0.0092 | 0.5690 |

Min (s) | 0.0018 | 0.2890 | 0.0010 | 0.1396 |

Avg (s) | 0.0053 | 0.3976 | 0.0033 | 0.2904 |

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**MDPI and ACS Style**

Laugksch, K.; Rousseau, P.; Laubscher, R.
A PINN Surrogate Modeling Methodology for Steady-State Integrated Thermofluid Systems Modeling. *Math. Comput. Appl.* **2023**, *28*, 52.
https://doi.org/10.3390/mca28020052

**AMA Style**

Laugksch K, Rousseau P, Laubscher R.
A PINN Surrogate Modeling Methodology for Steady-State Integrated Thermofluid Systems Modeling. *Mathematical and Computational Applications*. 2023; 28(2):52.
https://doi.org/10.3390/mca28020052

**Chicago/Turabian Style**

Laugksch, Kristina, Pieter Rousseau, and Ryno Laubscher.
2023. "A PINN Surrogate Modeling Methodology for Steady-State Integrated Thermofluid Systems Modeling" *Mathematical and Computational Applications* 28, no. 2: 52.
https://doi.org/10.3390/mca28020052