## 2 Materials and Methods
### Layer-Wise Locally Adaptive Activation Function  
Since the optimization parameters depend on the derivatives of the activation function, the activation function \(\sigma(x)\) is actually a basic function in neural networks, has an important role in the training process, and [PERSON] et al. [43] proposed that the activation function \(\sigma(x)\) in PINNs is usually used as the following nonlinear functions.  
Hyperbolic tangent function:  
\[\sigma(x)=\tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \tag{23}\]  
Sine function:  
\[\sigma(x)=\sin(x) \tag{24}\]  
Logistic sigmoid function:  
\[\sigma(x)=\frac{1}{1-e^{-x}} \tag{25}\]  
Figure 2: Schematic diagram of the PINN saving 1D consolidation equation.  
The choice of the activation function \(\sigma(x)\) depends on the problem to be solved. [PERSON] et al. [35] proposed a PINN with a global adaptive function (GAAF-PINN) based on the global adaptive activation function (GAAF) and introduced an additional scalable parameter \(na\) in the network, where \(n\) is the scaling factor and the hyperparameter \(a\) is the slope of the activation function. The results proved that introducing scalable parameter \(na\) in the activation function not only accelerates the convergence of the network, but also obtains better prediction accuracy. Subsequently, [PERSON] et al. [9] proposed two locally adaptive activation functions: layer-wise locally adaptive activation functions (L-LAAF) and neuron-wise locally adaptive activation functions (N-LAAF). The locally adaptive activation function is implemented by introducing scalable parameters in each network layer or each neuron \(na\). [PERSON] et al. demonstrated that the local adaptive activation function outperforms the global adaptive activation function in terms of training speed and accuracy. In addition, compared to the fixed activation function, although the locally adaptive activation function adds extra parameters, nevertheless the computational cost of both is comparable as the number of network layers and the number of neurons per layer increase. The code program for solving 1D consolidation problems with the PINN with layer-wise locally adaptive activation functions (L-LAAF-PINN) is shown in Algorithm 1. It is worth noting that the method is applied to the nonlinear layer of the neural network, thus can be flexibly and quickly applied to complex consolidation problems like 2d and 3d consolidation problems. For instance, the transfer of the method can be achieved by adjusting the input and output dimensions of the neural network and the construction of the loss function accordingly.  
The PINNs in this paper all use a hyperbolic tangent function that introduces a scalable hyper-parameter \(a\) and sets initial \(a=0.01\). Figure 3 shows the image of the hyperbolic tangent function with different activation slopes. As the activation function slope \(a\) changes, the expression of hyperbolic tangent function is  
\[\sigma(x)=\frac{e^{ax}-e^{-ax}}{e^{ax}+e^{-ax}} \tag{26}\]  
```
Initialize network parameters \(\lambda\) and dataset \(\mathcal{D}\)  Initialize adaptive activation slope \(a\) for\(t=0\)to\(max\_train\_epochs\)do  Compute predicted excess pore water pressure \(\widehat{u}\)  Compute the PDE residual \(\hat{f}\) using automatic differentiation  Compute loss function \(\mathcal{L}(\mathcal{D};\lambda)=w_{train}\mathcal{L}(\mathcal{D}_{train}; \lambda)+w_{f}\mathcal{L}\Big{(}\mathcal{D}_{f};\lambda\Big{)}\)  Compute the gradient of loss function \(\
abla_{\lambda}\mathcal{L}(\mathcal{D};\lambda)\)  Update the network parameters \(\lambda(t)\rightarrow\lambda(t+1)\)  Update adaptive activation slope \(a(t)\to a(t+1)\) endfor
```  
**Algorithm 1** L-LAAF-PINN