PSI - Issue 78

Pooria Mesbahi et al. / Procedia Structural Integrity 78 (2026) 1839–1846

1842

2.2. Transfer learning (TL) using Bayesian network (BN)

The BN approach is particularly advantageous in transfer learning, as it can operate e ff ectively with small datasets and supports backward inference Nguyen et al. (2025). Therefore, in the proposed transfer learning framework, a BN is employed to transfer information from a monitored bridge to a non-monitored one. The objective of this transfer learning is to estimate the probability that a model parameter of Bridge2 lies within a specific range, given that the corresponding parameter of Bridge1 is known to fall within a certain range (as obtained from Bayesian FEMU). The proposed BN is illustrated in Fig. 2. The design of the BN begins with the definition of a comparative variable,

( Δ ) W 0.1 S 0.2 B 0.7

2 1

( Δ ) W 0.1 S 0.3 B 0.6

Difference in environmental condition Difference in loading condition

Δ

Δ Δ Δ

( Δ ) W 0.1 S 0.2 B 0.7

Difference in external factors

Difference in internal factors

( Δ | Δ ,Δ ) S

Δ

Δ

Difference in all factors

W Δ W S B W S B W S B W 0.6 0.5 0.2 0.5 0.1 0.2 0.2 0.2 0.1 S 0.3 0.3 0.6 0.3 0.8 0.3 0.6 0.3 0.3 B 0.1 0.2 0.2 0.2 0.1 0.5 0.2 0.5 0.6 B

i -th model parameter of Bridge1 i -th model parameter of Bridge2

( Δ | Δ ,Δ )

Δ B Δ W S B W S B W S B W 0.6 0.5 0.2 0.5 0.1 0.2 0.2 0.2 0.1 S 0.3 0.3 0.6 0.3 0.8 0.3 0.6 0.3 0.3 B 0.1 0.2 0.2 0.2 0.1 0.5 0.2 0.5 0.6 W S

( ,1 ) L1 0.2 L2 0.2 L3 0.2 L4 0.2 L5 0.2 Bayesian FEMU for Bridge1

,2 ,1

( ,2 | ,1 , Δ )

Δ B ,1 L1 L2 L3 L4 L5 L1 L2 L3 L4 L5 L1 L2 L3 L4 L5 L0 0.6 0.3 0.2 0.15 0.12 0.077 0.067 0.06 0.056 0.053 0.19 0.121 0.086 0.066 0.052 L1 0.083 0.3 0.2 0.15 0.12 0.6 0.074 0.067 0.062 0.058 0.21 0.133 0.095 0.072 0.057 L2 0.076 0.096 0.2 0.15 0.12 0.077 0.6 0.073 0.067 0.064 0.12 0.146 0.104 0.079 0.063 L3 0.069 0.087 0.115 0.15 0.12 0.071 0.074 0.6 0.074 0.071 0.12 0.15 0.115 0.087 0.069 L4 0.063 0.079 0.104 0.146 0.12 0.064 0.067 0.073 0.6 0.077 0.12 0.15 0.2 0.096 0.076 L5 0.057 0.072 0.095 0.133 0.21 0.058 0.062 0.067 0.074 0.6 0.12 0.15 0.2 0.3 0.083 L6 0.052 0.066 0.086 0.121 0.19 0.053 0.056 0.06 0.067 0.077 0.12 0.15 0.2 0.3 0.6 Fig. 2: Proposed BN to transfer learning from observed structure to non-observed one W S

denoted as ∆ AF , which shows how the e ff ect of the general condition on the selected model parameter di ff ers between the two assets. ∆ AF is conditioned on two random variables, namely the di ff erence in external factors ( ∆ EF ) and the di ff erence in internal factors ( ∆ IF ). Furthermore, ∆ EF is modeled as dependent on two parent nodes: ∆ EC , representing the di ff erence in environmental conditions, and ∆ LC , representing the di ff erence in loading conditions. Conditional probability tables (CPTs) for all nodes (i.e., random variables) are constructed based on empirical evidence, statistical data, or engineering judgment and show probabilistic dependency between nodes. The BN considers information on θ i , 1 (i-th model parameter of Bridge1), including its prior distribution and Bayesian FEMU-updated evidence, together with expert judgment or data on condition di ff erences between Bridge1 and Bridge2. The prior of θ i , 1 follows a uniform distribution derived from design values and field tests, while Bayesian FEMU results are discretized into five levels ( L 1 – L 5 ) and treated as virtual evidence; in contrast, θ i , 2 (i-th model pa rameter of bridge 2) is represented with seven levels for higher specificity. Comparative evidence is introduced through random variables ∆ EC = EC 1 − EC 2 , ∆ LC = LC 1 − LC 2 , ∆ EF = EF 1 − EF 2 , ∆ IF = IF 1 − IF 2 , ∆ AF = AF 1 − AF 2 , where EC i , LC i , EF i , IF i , AF i denote environmental, load-related, external, internal, and general conditions, respec tively. These positive real-valued quantities, obtained from inspections, field tests, or expert judgment, take larger values when conditions are better; thus, ∆ > 0 ( W ) indicates Asset 2 is worse, ∆ = 0 ( S ) reflects similar conditions, and ∆ < 0 ( B ) means Asset 2 is better. The CPTs shown in Fig. 2 are examples and should be adapted case by