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arxiv: 2505.04018 · v1 · submitted 2025-05-06 · 💻 cs.CE · eess.SP

Modal Decomposition and Identification for a Population of Structures Using Physics-Informed Graph Neural Networks and Transformers

Xudong Jian , Kiran Bacsa , Gregory Duth\'e , Eleni Chatzi This is my paper

Pith reviewed 2026-05-22 17:05 UTC · model grok-4.3

classification 💻 cs.CE eess.SP
keywords modal identificationstructural health monitoringphysics-informed neural networksgraph neural networkstransformersunsupervised learningmodal decompositionstructural dynamics
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The pith

Physics-informed transformers and graph networks decompose structural vibrations into modes and recover their properties without any labeled data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents a deep learning framework that pairs a transformer with a graph neural network to break multi-degree-of-freedom measurements into separate modal responses and then map those responses to mode shapes. The entire system is trained by a loss function that directly encodes the mathematical rules of modal independence and orthogonality, so no ground-truth labels or pre-known mode shapes are required. Because the loss is derived from physics rather than data examples, the same trained model can process responses from structures that differ in layout or experience different loads. Validation on both simulated populations and laboratory tests shows that frequencies, damping ratios, and shapes are recovered accurately from sparse sensor readings. If the approach holds, it removes the usual barrier of needing extensive labeled datasets for each new structural configuration.

Core claim

By embedding modal decomposition theory into the loss function of a transformer-GNN architecture, the model separates measured dynamic responses into single-degree-of-freedom modal components and simultaneously identifies the associated natural frequencies, damping ratios, and mode shapes for an entire population of structures, all in a purely unsupervised manner that does not rely on labeled data or explicit mode-shape supervision.

What carries the argument

A physics-informed loss function that enforces the independence and orthogonality of structural modes to supervise the transformer decomposition, combined with a graph neural network that uses structural connectivity to predict the corresponding mode-shape vectors.

If this is right

  • Modal properties can be extracted from sparse measurements without collecting or using any supervised training labels.
  • The same model continues to work when the underlying structure changes configuration or experiences different external loads.
  • Comparative tests show higher accuracy than established modal identification methods on the reported simulation and experiment cases.
  • Population-level monitoring becomes feasible because the framework does not need to be retrained from scratch for each new but similar structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be paired with streaming sensor data to update modal estimates continuously as a structure ages or is modified.
  • Extending the same independence constraint to nonlinear or time-varying systems might allow modal tracking under changing environmental conditions.
  • Because training requires only the physics loss, the framework might be deployed on fleets of structures with minimal additional data collection effort.

Load-bearing premise

Structural vibration modes remain sufficiently independent across different configurations that a decomposition loss alone can reliably train the network without labeled examples or direct mode-shape targets.

What would settle it

If the frequencies and damping ratios recovered by the model on laboratory test data deviate by more than a few percent from the values obtained by conventional modal analysis on the identical datasets, the unsupervised physics supervision would be shown to be insufficient.

Figures

Figures reproduced from arXiv: 2505.04018 by Eleni Chatzi, Gregory Duth\'e, Kiran Bacsa, Xudong Jian.

Figure 1
Figure 1. Figure 1: Architecture of the proposed model, in which the input and output are highlighted in red, hidden features in yellow, and deep learning blocks in blue. As can be seen in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An example of the graph dataset used in this study. A truss structure can be naturally represented by an attributed graph, where node acceleration and mode shapes are node features, and modal responses are graph (global) features. Then, we assign the dynamic measurements at each node 𝑣, which is denoted as 𝑋̃ 𝑣 (𝑡), as the initial node feature vector ℎ (0) 𝑣 that is fed to GraphSAGE. A GraphSAGE model usua… view at source ↗
Figure 3
Figure 3. Figure 3: Framework of using the GNN-based model for population-based structural modal identification varying load, material, and environmental conditions [34], while a heterogeneous population includes structures with similar yet distinct topologies along with differences in these conditions [25]. For each structure in the training set, the Feature Propagation algorithm is first applied to reconstruct complete dyna… view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of the simulated dataset: (a) Geometric configuration designed to approximate a population of simply-supported truss; (b) Representative truss samples from the dataset, displaying only nodes and elements. Based on the generated geometric configurations and boundary conditions (simply-supported type) shown in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Probability histograms of the modal properties for the first six modes of the 100 trusses: (a) Natural frequencies; (b) Damping ratios. six evenly selected DOFs from two trusses within the simulated truss population. As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Filtered vibration acceleration signals for six evenly selected DOFs from two trusses in the simulated truss population, along with their power spectral density (PSD): (a) Truss No. 3, (b) Truss No. 93. 3.2. Implementation Details Our deep learning model is trained and implemented using PyTorch 2.0.1 [38], CUDA 11.7, and DGL 1.1.1 [39]. The Adam optimizer [40] is used for training with default settings, in… view at source ↗
Figure 7
Figure 7. Figure 7: Training and validation loss curves from the training process [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Modal decomposition results for two representative trusses. Each row corresponds to a decomposed mode. The first column displays the time-domain dynamic responses of the decomposed modes. The second column presents the PSD of the dynamic modal responses, with the true mode frequencies, obtained via eigenvalue analysis, indicated by red dashed lines. The third column visualizes the corresponding mode shapes… view at source ↗
Figure 9
Figure 9. Figure 9: True and identified mode shapes of a representative truss, in which structural nodes with measurements are highlighted as solid dots. corresponding mode shapes. However, removing the GNN eliminates the model’s ability to handle graph-structured data with varying node counts 𝑁, as both the standard Transformer and MLP require a fixed input size. As a consequence, we have to constrain all trusses to only 𝑃 s… view at source ↗
Figure 10
Figure 10. Figure 10: Modal identification results of the originally proposed approach and its variations: (a) Mean MAC values of the identified mode shapes, (b) Mean absolute percentage errors in identified frequencies, and (c) Mean absolute percentage errors in identified damping ratios. performance of the proposed approach compare to classical modal identification methods when applied to real-world datasets? 4.1. Dataset De… view at source ↗
Figure 11
Figure 11. Figure 11: Scale model of a cable-stayed bridge: (a) Photo; (b) Diagram of the finite element model (unit: mm). The six deployed accelerometers are labeled from A1 to A6, with arrows indicating the sensing direction; (c) Frequencies and mode shapes of the first three vertical modes obtained from the eigenvalue analysis of the finite element model. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Filtered vibration acceleration signals from six accelerometers in two representative tests, along with their power spectral density (PSD): (a) Test 1, pull-and-release, (b) Test 7, vehicle-excited. X. Jian et al.: Preprint submitted to Elsevier Page 16 of 21 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: illustrates the loss curves from the training process, where data from eight tests were used for training and the remaining two tests for validation. Both training and validation losses exhibit a sharp decline initially and eventually converge, demonstrating the effectiveness of the training process [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Modal decomposition results for two representative tests. Each row corresponds to a decomposed mode. The first column displays the time-domain dynamic responses of the decomposed modes. The second column presents the PSD of the dynamic modal responses, with the reference frequencies, obtained via eigenvalue analysis of FEMs, indicated by red dashed lines. The third column visualizes the corresponding mode… view at source ↗
Figure 15
Figure 15. Figure 15: Modal identification results for tests 1-8, presented using box plots and scatter plots: (a) MAC values of the identified mode shapes, (b) Identified natural frequencies, and (c) Identified damping ratios. Nonetheless, , our approach requires more time for modal decomposition and identification due to the computational demands of deep learning training. Specifically, the total time needed for modal identi… view at source ↗
read the original abstract

Modal identification is crucial for structural health monitoring and structural control, providing critical insights into structural dynamics and performance. This study presents a novel deep learning framework that integrates graph neural networks (GNNs), transformers, and a physics-informed loss function to achieve modal decomposition and identification across a population of structures. The transformer module decomposes multi-degrees-of-freedom (MDOF) structural dynamic measurements into single-degree-of-freedom (SDOF) modal responses, facilitating the identification of natural frequencies and damping ratios. Concurrently, the GNN captures the structural configurations and identifies mode shapes corresponding to the decomposed SDOF modal responses. The proposed model is trained in a purely physics-informed and unsupervised manner, leveraging modal decomposition theory and the independence of structural modes to guide learning without the need for labeled data. Validation through numerical simulations and laboratory experiments demonstrates its effectiveness in accurately decomposing dynamic responses and identifying modal properties from sparse structural dynamic measurements, regardless of variations in external loads or structural configurations. Comparative analyses against established modal identification techniques and model variations further underscore its superior performance, positioning it as a favorable approach for population-based structural health monitoring.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper presents a framework that combines a transformer to decompose MDOF structural dynamic responses into SDOF modal components (for identifying natural frequencies and damping ratios) with a GNN to recover corresponding mode shapes from structural graph representations. The model is trained in a purely unsupervised, physics-informed manner by embedding modal decomposition theory, mode independence, and orthogonality into the loss function, without requiring labeled data. Effectiveness is claimed via numerical simulations and laboratory experiments on varying loads and structural configurations, with comparisons to established modal identification methods.

Significance. If the unsupervised physics-informed supervision proves robust and unique, the work would be significant for scalable, population-based structural health monitoring, as it addresses the scarcity of labeled modal data and enables identification from sparse measurements across diverse structures. The integration of GNNs for configuration-aware mode shape recovery with transformer-based decomposition is a novel direction that could reduce reliance on traditional experimental modal analysis techniques.

major comments (3)
  1. [Abstract] Abstract: The central claim that the model achieves modal decomposition 'in a purely physics-informed and unsupervised manner' by leveraging 'modal decomposition theory and the independence of structural modes' to guide learning is load-bearing, but the loss function's ability to uniquely enforce correct decompositions (rather than other minima satisfying the same constraints) is not demonstrated, particularly under sparse sensors and varying configurations.
  2. [Validation] Validation through numerical simulations and laboratory experiments: While comparisons to established techniques are mentioned, the absence of detailed quantitative metrics, error bars, data exclusion criteria, or ablation studies on loss terms leaves the support for superior performance and robustness across structural variations limited and hard to verify.
  3. [Methods] Methods (physics-informed loss): The assumption that embedding modal independence and SDOF response characteristics into the loss reliably supervises the transformer and GNN without labeled data or explicit mode shape supervision risks circularity, as the loss may primarily reinforce the same modal assumptions the model is intended to discover rather than providing external grounding.
minor comments (2)
  1. [Abstract] The abstract could more explicitly state the form of the physics-informed loss terms (e.g., orthogonality penalties or SDOF oscillator residuals) to improve immediate clarity for readers.
  2. [Introduction] Notation for MDOF/SDOF responses and graph representations in the GNN could be standardized earlier in the manuscript to aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We sincerely thank the referee for their constructive and detailed feedback on our manuscript. We have carefully reviewed each major comment and provide point-by-point responses below, outlining planned revisions where appropriate to strengthen the presentation and validation of our physics-informed framework.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the model achieves modal decomposition 'in a purely physics-informed and unsupervised manner' by leveraging 'modal decomposition theory and the independence of structural modes' to guide learning is load-bearing, but the loss function's ability to uniquely enforce correct decompositions (rather than other minima satisfying the same constraints) is not demonstrated, particularly under sparse sensors and varying configurations.

    Authors: We acknowledge the referee's valid concern that theoretical uniqueness of the minima is not formally proven. The loss function is constructed from established modal theory (orthogonality, independence, and SDOF characteristics), and the transformer-GNN architecture further constrains solutions through graph-based mode shape recovery. Empirical results across multiple configurations and sensor densities consistently recover physically consistent modes matching reference methods. In the revision we will add a dedicated discussion subsection with sensitivity analyses to random initializations and comparisons against relaxed loss variants to better illustrate convergence behavior under sparsity. revision: partial

  2. Referee: [Validation] Validation through numerical simulations and laboratory experiments: While comparisons to established techniques are mentioned, the absence of detailed quantitative metrics, error bars, data exclusion criteria, or ablation studies on loss terms leaves the support for superior performance and robustness across structural variations limited and hard to verify.

    Authors: We agree that the current validation section would benefit from greater quantitative rigor. The revised manuscript will include comprehensive tables reporting mean absolute errors with standard deviations across repeated trials, explicit statements of data exclusion criteria, and ablation experiments that systematically remove or weight individual loss terms to quantify their impact on accuracy and robustness across varying loads and configurations. revision: yes

  3. Referee: [Methods] Methods (physics-informed loss): The assumption that embedding modal independence and SDOF response characteristics into the loss reliably supervises the transformer and GNN without labeled data or explicit mode shape supervision risks circularity, as the loss may primarily reinforce the same modal assumptions the model is intended to discover rather than providing external grounding.

    Authors: The loss terms are derived directly from classical modal analysis principles (mode orthogonality via inner products and SDOF free-vibration equations) that predate and exist independently of the neural architecture. These constraints supply external physical supervision rather than presupposing the decomposition the model must learn. We will expand the methods section with a clearer step-by-step derivation of each loss component and an explicit statement that no labeled modal data or mode-shape targets are used, thereby clarifying the non-circular nature of the supervision. revision: partial

Circularity Check

0 steps flagged

No significant circularity: physics loss uses external modal theory

full rationale

The paper embeds standard modal decomposition theory (mode independence, SDOF responses per mode, orthogonality) into the unsupervised loss to train the GNN-transformer model on sparse measurements. This theory is drawn from established structural dynamics, not from the paper's own fitted outputs, self-citations, or ansatzes. The decomposition and identification emerge as model outputs consistent with the external physics constraints rather than being presupposed by definition or by renaming known results. No load-bearing self-citation chain or fitted-input-called-prediction pattern is present; validation on numerical and experimental cases further confirms the derivation remains self-contained against independent benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on modal decomposition theory and the independence of structural modes as guiding principles for unsupervised training; no explicit free parameters or new invented entities are described in the abstract.

axioms (1)
  • domain assumption Structural modes are independent and can be decomposed from multi-degree-of-freedom responses using modal theory.
    Invoked to enable purely physics-informed unsupervised learning without labeled data.

pith-pipeline@v0.9.0 · 5741 in / 1260 out tokens · 58627 ms · 2026-05-22T17:05:08.146382+00:00 · methodology

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