# Physics-Informed Surrogate Modelling in Fire Safety Engineering: A Systematic Review
## 3 Implementing Physics-Informed Surrogate Model
### Injecting Physics into Models
#### 3.4.2 Model Selection (Architecture-Constrained Technique--ACT)  
As mentioned in Section 1.2, the surrogate model is a mathematical tool which relates features to the outputs. This algorithm can be as simple as linear regression or as complex as a network of complex functions. Selecting an appropriate model or algorithm is a crucial step in developing a PISM and should be done taking into account the identified sub-models and constraints (see Section 3.3). Where distinct series models have been identified, implementing a series architecture as part of the surrogate model setup is recommended, as it greatly supports interpretability and quality control. The same applies to other types of identified sub-models and constraints. The choice of model algorithm (e.g., regression, neural network, \(\ldots\)) depends on the system's complexity, available data, and the necessary balance between interpretability and predictive accuracy. In fire safety engineering, interpretability, transparency, and feasibility are highly valued. White-box models, such as regression-based approaches, offer explicit mathematical formulations, making them preferable when the underlying physics is relatively well understood and can be represented with simpler functional relationships. In the following, different model algorithms are discussed to help readers choose the ones which are more suited to their application, with examples. Regression-Based Models  
When using a regression-based approach, different functional forms can be employed based on the nature of the relationship between inputs and outputs. Exponential functions are well-suited for modelling growth or decay processes, while harmonic functions capture periodic phenomena, such as vibrational responses in mechanical systems. Choosing a function aligned with known physical behaviour ensures more realistic and physically consistent predictions [101]. It is also possible to inherently enforce physical constraints through the choice of regression functions, such as the sigmoid function, which is particularly useful when outputs must remain within a bounded range (see Equation (1)). In applications like temperature prediction, where physical laws define permissible output limits, the basic sigmoid function--naturally restricting values between 0 and 1--can be applied to constrain predictions within realistic temperature ranges [143]. Equation (1). Sigmoid function  
\[0<f(h(x))=\frac{1}{1+e^{-h(x)}}<1 \tag{1}\]  
with  
\(x\): the feature vector  
\(h(. )\): the regression function  
Neural Networks-Based Models  
Different neural network-based models have been combined with physical considerations, as highlighted in the literature review in Section 2.1. Physics-informed neural networks (PINNs) are well-suited for problems governed by known PDEs, integrating PDE residuals into the loss function to enforce physical constraints. Using standard feed-forward architectures, they minimise both data mismatch and PDE violations, making them effective in heat transfer, fluid dynamics, and structural mechanics. Convolutional neural networks (CNNs) excel in processing image-based data, including thermal and satellite imagery for fire detection. They enhance feature extraction and pattern recognition [79]. Example applications are in Section 2.1.2. Graph neural networks (GNNs) are ideal for fire safety problems involving graph-structured data, such as fire spread modelling in buildings orwildfire propagation across spatial grids [82]. GNNs capture spatial dependencies through graph convolution operations that respect physical principles. Irrespective of the type of neural network, the architecture of the network significantly impacts its performance and ability to capture physical behaviour. Several key factors must be considered when designing a neural network for a physics-informed surrogate model, the most important of which are (i) sub-models (i.e., number of networks); (ii) number of neurons and hidden layers; and (iii) activation functions.