PSI - Issue 79

Henrik Petersson et al. / Procedia Structural Integrity 79 (2026) 298–305

300

is very poor at extrapolating outside the training data. To overcome this, physics-informed neural networks can be utilized to guide the network into specific physical principals via mathematical models. In this study, the data used is a fatigue data set taken from JIS databook, code-series: 173-080, material: JIS S35C . The fatigue limit for this material is set to 248 MPa, an upper stress limit is set to 398 MPa (this actual value is probably higher but here this value is only used as an illustrative value). The stress and fatigue data used as both input and output are log-transformed and subsequently normalized within the range of 0.1 to 0.9, X ′ = 0 . 8 · X − X min X max − X min + 0 . 1 (3) where X is the log-transformed value from the fatigue data set, X ′ is the normalized value of X , X max and X min are the maximum and minimum value of X respectively. The hyperparameters was selected through a preliminary tuning process to identify the configuration of the hyperparameters. The neural network architect is based on a simple neural network with one input layer, i.e. stress amplitude, one output layer, i.e. cyclic life, and one hidden layer. The number of neurons in the hidden layer is set to 12 and a tanh activation function is used. Xavier Glorot (2010) was used for initialization (initial weights of the NN) and the Adam optimizer Kingma (2017) was used for training the NN in PyTorch Paszke (2019) The loss function L True , typically used in neural networks, is the mean squared error (MSE), which is also used in this work and defined as L True = 1 m m i = 1 N f Predicted i − N f True i 2 (4) In PINNs a physical loss function L Physical is setup and contains underlying equations / residuals that shall be min imised. This loss function does not depend on the existing data, which enables extrapolation outside the training data. The total loss function for the whole network is the sum of the true and the physical loss functions, described in the following equation, where all loss contributions are summed to a total loss value, L Total = L True + L Physical (5) where the physical loss function is decomposed into three parts L Physical = L 1 + L 2 + L 3 (6) In this work, the behaviour of the regression model is controlled by feeding the training with physical principles modelled mathematically. By setting up physically relevant equations and incorporate them into the physical loss function, the model will be biased into complying with these equations. The desired behaviour within the data range should follow a linear behaviour in the log 10 − log 10 diagram. Hence, based on this the following equation can be used log 10 N f = C 1 · log 10 ( σ ) + C 2 (7) where C 1 and C 2 are model parameters in the loss function. This equation should be active in the range where data points exists. The loss function is setup in the following way

n

j = 1

log 10 N f

− C 1 · log 10 σ j + C 2 2

1 n

Predicted j

, σ j ∈ [ σ min ,σ max ]

(8)

L 1 =

where σ is a vector containing stress amplitudes over an interval that covers the experimental data points.

In the high stress part of the regression, close to the upper stress limit, a loss function is set to span over an interval of N f less than the lowest fatigue life of the experimental data points. Thus, the loss function will act in an interval [0 , min( N f )]. In order to capture the upper stress limit σ limit , the following expression is employed

= log

1 − log

1

1 C 3

σ limit

σ limit

Upper k

+ N f 0 , σ k ∈ [ σ 0 ,σ limit ]

(9)

N f

σ k −

σ 0 −