Transfer learning via interpolating structures

T. A. Dardeno; A. J. Hughes; L. A. Bull; R. S. Mills; N. Dervilis; K. Worden

cs.LGMar 23, 2026

Transfer learning via interpolating structures

Authors: T. A. Dardeno, A. J. Hughes, L. A. Bull, R. S. Mills, N. Dervilis, K. Worden

Organizations: Dynamics Research Group, School of Mechanical, Aerospace and Civil Engineering, University of Sheffield, Sheffield S1 3JD, UK · School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8SQ, Scotland

Abstract

Despite recent advances in population-based structural health monitoring (PBSHM), knowledge transfer between highly-disparate structures (i.e., heterogeneous populations) remains a challenge. The current work proposes that heterogeneous transfer may be accomplished via intermediate structures that bridge the gap in information between the structures of interest. A key aspect of the technique is the idea that by varying parameters such as material properties and geometry, one structure can be continuously morphed into another. The approach is demonstrated via a case study involving the parameterisation of (and transfer between) simulated heterogeneous bridge designs (Case 1). Transfer between simplified physical representations of a 'bridge' and 'aeroplane' is then demonstrated in Case 2, via a chain of finite-element models. The facetious question 'When is a bridge not an aeroplane?' has been previously asked in the context of predicting positive transfer based on structural similarity. While the obvious answer to this question is 'Always,' the results presented in the current paper show that, in some cases, positive transfer can indeed be achieved between highly-disparate systems.

Figures & tables

Figure 1: Schematic of geometrical model of PBSHM in terms of a fibre bundle.

Figure 2: A geodesic (red curve) on a sphere, with tangent spaces (planes) shown at successive points along the path. The blue arrows show the tangent directions evolving continuously along the geodesic, illustrating the geodesic flow.

Figure 3: Geodesic flow kernel, adapted from [ 4 ] .

Component	L (m)	W (m)	H (m)	E (Pa)	$\rho$ (kg/m 3 )	$k$ (N/m)
Deck	32	7	0.7	$2\times 10^{11}$	7850	$1\times 10^{12}$
Supports	3	1	5	$1.9\times 10^{10}$	2392	$1\times 10^{12}$

Table 1: Model Properties for Case 1.

Panel of Figure 4 — Figure 4: Two-span source (a) and three-span target bridges (b) in the damaged state.

Panel of Figure 5 — Figure 5: Schematic of model generation process. The first support remained fixed while the second support moved to the right and materialised as $\alpha$ increased.

Figure 6: Cosine alignment along the transfer path for Case 1. The alignment between the current source and target is shown in black with $\mathbf{\times}$ markers, with values close to 1 indicating that adjacent structures in the chain are consistently similar. The alignment between the current source and the original source is shown in black with $\bullet$ markers, with decreasing values showing progressively less similarity as the chain moves farther away from the original source.

Figure 7: Heterogeneous transfer approach via intermediate structures.

$K$	Accuracy (%)	Best chain	Accuracy (%)	Best chain
	Linear		Geodesic Flow Kernel
0	$44.44\pm 0.00$	(no chain)	$46.76\pm 9.51$	(no chain)
1	$44.70\pm 4.08$	[1 10 18]	$67.46\pm 25.24$	[1 11 18]
2	$57.16\pm 22.11$	[1 5 13 18]	$94.75\pm 15.83$	[1 9 13 18]
3	$74.03\pm 26.96$	[1 4 12 13 18]	$98.82\pm 7.57$	[1 4 12 13 18]
4	$93.59\pm 17.85$	[1 8 11 13 14 18]	$99.82\pm 3.11$	[1 8 11 13 14 18]

Table 2: Best chains per number of intermediates

K

for linear and GFK classification approaches. Values are mean

\pm

standard deviation over 1000 noise realisations.

Panel of Figure 8 — Figure 8: Experimental setup for bridge (a) and aeroplane (b) structures.

Panel of Figure 9 — Figure 9: Damage to the deck (D1) (a) and (b) column (D2).

Panel of Figure 10 — Figure 10: Lower bending modes for the bridge and aeroplane. Bridge modes at approximately (a) 100 Hz, (b) 116 Hz, and (c) 150 Hz. Aeroplane repeated root modes at approximately 117 Hz (d), (e).

Figure 11: Representative input spectra, coherence, and FRFs for healthy and damaged states for the bridge configuration.

Panel of Figure 12 — Figure 12: Sensor layout for all analyses. Accelerometers are shown in red, except for the drive point location which is shown in blue. All units are in mm, and measured from the right-hand edge of the deck/wing.

Figure 13: FRFs from a variability study on the bridge, where four different decks, and four different columns, were removed and reinstalled using consistent torque for the bolts.

Structure	Component	L (m)	W (m)	H (m)	E (Pa)	$\rho$ (kg/m 3 )
Bridge	Deck	1.00	0.10	0.010	$6.8\times 10^{10}$	2830
	Supports	0.0254	0.0254	0.220	$\downarrow$	$\downarrow$
	Foot pads	0.0762	0.0762	0.010	$\downarrow$	$\downarrow$
Aeroplane	Wings	1.00	0.10	0.010	$6.8\times 10^{10}$	2830
	Supports	0.0254	0.0254	0.220	$\downarrow$	$\downarrow$
	Foot pads	0.0762	0.0762	0.010

Table 3: Geometry and material properties for the bridge and aeroplane.

Panel of Figure 14 — Figure 14: FRFs at drive-point location for experimental bridge, subset of FEM intermediates, and experimental aeroplane. The blue, green, and red curves correspond to the healthy, D1, and D2 classes, respectively.

Panel of Figure 15 — Figure 15: Subset of intermediate geometries (1, 85, 168, 251, 293, 376, 459, and 500). Images are read top-to-bottom in the left column, then top-to-bottom in the right column.

Panel of Figure 16 — Figure 16: Cosine alignment along the transfer path at each sensor location for Case 2. The alignment between the current source and target is shown in black with $\mathbf{\times}$ markers, with values close to 1 indicating that adjacent structures in the chain are consistently similar. The alignment between the current source and the original source is shown in black with $\bullet$ markers, with decreasing values showing progressively less similarity as the chain moves farther away from the original source.

Sensor	Linear Kernel (%)	Geodesic Flow Kernel (%)
1	$28.57\pm 0.00$	$60.64\pm 11.90$
2	$64.29\pm 0.00$	$64.28\pm 0.05$
3	$64.29\pm 0.00$	$64.29\pm 0.00$
5	$64.29\pm 0.00$	$67.17\pm 13.74$
6	$63.92\pm 9.41$	$63.59\pm 12.51$
7	$64.29\pm 0.00$	$61.26\pm 11.89$

Table 4: Transfer accuracy at the final hop without intermediates (

K=0

) for Linear and GFK approaches in Case 2. Values are mean

\pm

standard deviation over 400 noise realisations.

Sensor	Linear Kernel (%)	Geodesic Flow Kernel (%)
1	$35.71\pm 0.00$	$35.71\pm 0.00$
2	$35.71\pm 0.00$	$100.00\pm 0.00$
3	$100.00\pm 0.00$	$100.00\pm 0.00$
5	$100.00\pm 0.00$	$100.00\pm 0.00$
6	$64.29\pm 0.00$	$100.00\pm 0.00$
7	$64.29\pm 0.00$	$100.00\pm 0.00$

Table 5: Transfer accuracy at the final hop for Linear and GFK approaches in Case 2 using

K=4

intermediates: 71, 201, 351, and 500. Values are mean

\pm

standard deviation over 25 noise realisations.

Sensor	Linear Kernel (%)	Geodesic Flow Kernel (%)
1	$35.71\pm 0.00$	$35.69\pm 0.38$
2	$35.71\pm 0.00$	$52.27\pm 17.82$
3	$35.71\pm 0.00$	$36.45\pm 4.53$
5	$35.71\pm 0.00$	$35.71\pm 0.00$
6	$35.71\pm 0.00$	$40.86\pm 10.99$
7	$35.71\pm 0.00$	$48.20\pm 14.19$

Table 6: Transfer accuracy at the final hop for Linear and GFK approaches in Case 2 using

K=5

intermediates: 71, 201, 251, 351, and 500. Values are mean

\pm

standard deviation over 350 noise realisations.

Sensor	Linear Kernel (%)	Geodesic Flow Kernel (%)
1	$64.29\pm 0.00$	$87.94\pm 16.91$
2	$35.71\pm 0.00$	$38.37\pm 4.33$
3	$100.00\pm 0.00$	$100.00\pm 0.00$
5	$100.00\pm 0.00$	$100.00\pm 0.00$
6	$100.00\pm 0.00$	$100.00\pm 0.00$
7	$57.37\pm 13.86$	$100.00\pm 0.00$

Table 7: Transfer accuracy at the final hop for Linear and GFK approaches in Case 2 using

K=9

intermediates: 26, 31, 91, 101, 181, 219, 291, 401, 500. Values are mean

\pm

standard deviation over 385 noise realisations.

Sensor	Linear Kernel (%)	Geodesic Flow Kernel (%)
1	$100.00\pm 0.00$	$100.00\pm 0.00$
2	$91.71\pm 0.99$	$100.00\pm 0.00$
3	$100.00\pm 0.00$	$100.00\pm 0.00$
5	$100.00\pm 0.00$	$100.00\pm 0.00$
6	$100.00\pm 0.00$	$100.00\pm 0.00$
7	$100.00\pm 0.00$	$100.00\pm 0.00$

Table 8: Transfer accuracy at the final hop for Linear and GFK approaches in Case 2 using all intermediates (

K=500

). Values are mean

\pm

standard deviation over 25 noise realisations.

Panel of Figure 17 — Figure 17: A subset of the confusion matrices for Case 2 transfer configurations. Top row: Sensor 1, (a) GFK with $K=0$ (no intermediates) and (b) $K=9$ . Middle row: Sensor 2, (c) GFK with $K=0$ and (d) $K=5$ . Bottom: Sensor 2, (e) Linear with the full chain ( $K=500$ ). Matrices are averaged over the corresponding noise realisations reported in Tables 4 to 8 .

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Domain Adaptation Transfer Learning

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This paper presents a methodology-centered transfer learning framework for fragility adaptation under domain shift, class imbalance, and scarce target labels while preserving engineering interpretability and supporting decision-making under uncertainty. Four transfer learning strategies (instance-based, parameter-based, hierarchical Bayesian, and multi-source) are demonstrated through three complementary case studies: (i) instance-based transfer learning via importance weighting, demonstrated on coastal bridge fragility using Hurricane Katrina observations; (ii) parameter-based transfer learning together with hierarchical Bayesian transfer learning, enabling partial pooling across strata and posterior uncertainty quantification, demonstrated on residential building fragility using Hurricane Ian observations; and (iii) multi-source transfer learning that fuses multiple analytical fragility models with learned source weights and regularized target-domain adaptation, demonstrated on seismic bridge fragility using observations from the 2001 Nisqually earthquake. Across these case studies, direct transfer of source models (i.e. using existing state-of-the-art models) fails under domain shift and severe class imbalance, while targeted adaptation substantially improves failure detection and predictive stability in low-data regimes. These findings highlight the need for systematic guidance on diagnostics, strategy selection, and uncertainty reporting when developing and adapting fragility models.

Abstract

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