PSI - Issue 79

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

301

where C 3 is a model parameter, σ k is the stress amplitude, σ limit is the upper stress limit, σ 0 and N f 0 are start values where the loss function is active. σ 0 is typically selected as σ max . Here, a fictive value of σ limit = 398 MPa is used to describe the upper stress limit. This equation is derived from a sigmoid function, where a sigmoid function is linear at its ”middle” and goes towards a value, which can be used for capturing the desired behaviour of the stress limit. The physical loss function for the upper stress part is then defined as the following

p  k = 1 

Upper k 

1 p

2

Predicted k

− N f

(10)

N f

L 2 =

To capture the behaviour at low stress, i.e. close to the fatigue limit N f limit , the following expression is used

= 

1 σ l − σ limit 

C 4

Lower l

N f

+ C 5 , σ l ∈ [ σ limit ,σ 0 ]

(11)

where C 4 and C 5 are model parameter, σ limit is the lower stress limit, σ 0 and N f 0 the start values where the loss function is active. This equation goes towards an infinitive value when the stress approaches σ limit , which is the desired behaviour for capturing the fatigue limit. According to the material data from JIS databook, code-series: 173 080, material: JIS S35C , a value of σ limit = 248 MPa is used as σ limit to describe the lower stress limit. σ 0 is typically selected as σ 0 = σ min The physical loss function for the low stress expression is then defined as

q  l = 1 

Lower l 

1 q

2

Predicted l

N f

− N f

(12)

L 3 =

The training sequence was terminated when the loss of the last 500 epochs had a positive slope, i.e. the trend of the loss in the training was no longer achieving better results.

2.2. Error Estimation

A variable safety factor can be achieved by performing an uncertainty quantification, by using the PINN regression model in combination with an error estimation. The error is estimated using the Gaussian distribution as described in Equation 2, where the mean function will be the PINN regression model and the variance Ω used is defined as Ω ( x i ) =  j = 1 K h ( x i , x j ) · r 2 i  j = 1 K h ( x i , x j ) (13) with r 2 i as the squared residual and K h ( x i , x j ) as the Gaussian kernel function where x i and x j are input vectors,       x i − x j       is the Euclidean distance and reflects the distance between x i data points and x j experimental data points. h i is the adaptive bandwidth parameter, which controls the spread of the bandwidth, the adaptive bandwidth is based on the distance to its k- th nearest neighbour. An additional uncertainty factor can be multiplied to the variance, which purpose is to increase the probability interval with respect to the distance from known data. It is based on the distance to the k- th nearest experimental data neighbour, the distance is then multiplied by a scaling factor and applied to the variance, i.e. the variance becomes Ω · f scal when f scal = C 6 · exp( y ) (15) where y is the distance to the k- th nearest experimental data and C 6 is a parameter for constraining the increase in predictive uncertainty. K h ( x i , x j ) = 1 h i √ 2 π · exp    −       x i − x j       2 2 h 2 i    (14)