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Towards a GraphNARX model for structural dynamics

George Tsialiamanis, Keith Worden

arXiv:2610.23103v1cs.LG

TL;DR

Machine-learning system-identification models can be tied to the structures and sensor locations used for data acquisition, limiting their broader application. This paper proposes GraphNARX, a graph-based NARX approach for structural dynamics, and reports accurate predictions, generalisation to larger structures, and virtual-sensing results. Its evaluation is on simulated structures, and the authors identify experimental validation as needed.

  • Problem

    Conventional NARX models predict for the specific structures and sensor positions represented in their data, limiting their applicability beyond those settings.

    Source: §2.1 · p. 3

  • Method

    GraphNARX combines graph neural networks and NARX modelling, using graphs to represent structural geometry and local relationships between structural degrees of freedom.

    Source: §1 · p. 1, §2.3 · p. 5

  • Results

    0.0204% one-step-ahead NMSE was achieved on the ten-degree-of-freedom system; virtual-sensing errors were around 5% on that system and around 2% on the 20-degree-of-freedom system.

    Source: §5 · p. 13

  • Takeaways & Limitations

    The results indicate that GraphNARX can train on structures of one size and generalise to larger structures, though further fine-tuning is needed.

    Source: §5 · p. 13

    The evaluation uses simulated structures, and the authors state that evaluation on experimental structures is needed.

    Source: §5 · p. 13

Abstract

from arXiv · show

The current paper introduces the GraphNARX model, a graph-based nonlinear autoregressive exogenous model for structural dynamics. The model is built to account for the geometry of structures and to model dynamic behaviour as the interaction between elements of the structure. Similar to other types of NARX models, GraphNARX is used for system identification. Nodal attributes of the graph encode past displacements and nodal forces, and the output of the model is the next-timestep displacement. The graph formulation of the model allows it to generalise to structures of different dimensions and degrees of freedom. Moreover, because graph-learning is performed locally between the nodes, the GraphNARX model can also learn to infer displacements of nodes for which data are not available, i.e.\ to perform virtual sensing. The proposed model's performance is evaluated in a system-identification framework and a virtual-sensing framework, and achieves good accuracy and exhibits generalisation capabilities to structures of different sizes.

1 Introduction

Structural modelling has advanced through approaches such as finite elements, while identifying models of existing structures is also useful. The paper proposes a graph-based system-identification method intended to account for structural geometry, generalise across sizes, and support virtual sensing.

1 Introduction

  • The inverse problem of identifying a model for an existing structure complements forward structural modelling and can be useful in engineering practice.System identification requires data from the physical structure and may involve calibrating physics-based models, fitting machine-learning models, or combining both approaches.Source: §1 · p. 1
  • Physics-based models may generalise beyond calibration when they encode system physics, whereas data-driven models mainly interpolate and may generalise less well.The paper frames its motivation around grey-box modelling that combines physics-based and data-driven approaches.Source: §1 · p. 1
  • GraphNARX combines graph neural networks and NARX models to represent structural geometry and attempt generalisation between structures of different sizes.The proposed method is intended to be scalable in size.Source: §1 · p. 1
  • Because graph learning is local, the method is also evaluated for virtual sensing: inferring displacements at points without sensors.The paper evaluates system identification on a simulated structure and tests extrapolation to a larger structure, alongside a separate virtual-sensing evaluation.Source: §1 · p. 2

2 The GraphNARX model

The GraphNARX model combines nonlinear autoregressive dynamics with graph-based local interactions to represent structural geometry. Its formulation uses graph aggregation to support different graph sizes and encodes lagged nodal displacements and forces as inputs for next-step displacement prediction.

2.1 Nonlinear autoregressive exogenous (NARX) modelling

  • NARX models predict future system states from past states and external inputs through a nonlinear transition, while their traditional single-structure formulation limits transfer across structures.Polynomial NARX models can become difficult to specify as terms proliferate, while GP- and neural-network NARX models have their own modeling and extrapolation constraints.Source: §2.1 · pp. 2–3

2.2 Graph neural network (GNN)

  • GNNs update edge, node, and global features through learned functions and aggregations, allowing inference on graphs of different sizes.Common aggregation choices include averaging, summation, maximum, and minimum.Source: §2.2 · p. 4

2.3 GNN as a system identification model

  • GraphNARX represents structural dynamics with graph nodes for degrees of freedom and edges for structural connectivity, using lagged nodal displacements and forces as node features.Edge features encode structural-segment characteristics in the basic formulation; the proposed variant instead uses relative displacements when material properties are unknown.Source: §2.3 · pp. 5–6

3 System identification application

The section evaluates GraphNARX on a ten-degree-of-freedom system and tests whether a model trained there extrapolates to a larger, 20-degree-of-freedom system. It reports low one-step-ahead errors and multi-step prediction accuracy that declines with longer horizons.

3.1 A ten-degree-of-freedom system

  • GraphNARX predicts each node’s next displacement from neighbouring nodal displacements, using connectivity as an inductive bias for extrapolation to different geometries.The selected model used GNN 2 with displacement lag ny = 6 and force lag nx = 2.Source: §3.1 · pp. 7–8
  • Training first used one-step-ahead loss, then a lower learning rate in a second stage, which significantly improved multi-step predictive error.Each stage lasted 100 epochs, with learning rates of 10^-3 and 10^-4.Source: §3.1 · p. 7
  • 0.0204% testing-data OSA error was achieved on the ten-degree-of-freedom system, with the predictions also denoising the 1% white-noise-corrupted data.The reported 1% noise was added for regularisation, while predictions were compared with noise-free data.Source: §3.1 · pp. 7–8
  • On the ten-degree-of-freedom system, average NMSE stayed below 5% for 400-step-ahead predictions and increased almost linearly afterward.Figure 6 shows average NMSE across prediction horizons on the testing dataset.Source: §3.1 · p. 8, Figure 6 · p. 9

3.2 Extrapolation to a 20-degree-of-freedom system

  • 0.02283% OSA NMSE was achieved on the 20-degree-of-freedom system using the model trained on the ten-degree-of-freedom system, without retraining.The larger system used the same mass, damping, and stiffness parameters.Source: §3.2 · p. 8
  • Below 5% average NMSE persisted through 400-step-ahead predictions on the 20-degree-of-freedom system, but error rose faster than for the smaller system.The authors suggest that more degrees of freedom may accumulate more errors.Source: §3.2 · p. 10

4 GraphNARX for virtual sensing

GraphNARX supports virtual sensing by masking unavailable nodal inputs and transferring information between neighbouring nodes. Tests report NMSE for missing-sensor configurations on both ten- and 20-degree-of-freedom systems.

4 GraphNARX for virtual sensing

  • Virtual sensing masks past displacements and forces at unsensed nodes, trains with randomly sampled 50%-missing masks, and computes loss only at available sensors.Information is transferred only between neighbouring nodes using the same neural-network model.Source: §4 · pp. 10–11
  • On the ten-degree-of-freedom system, NMSE was 4.994255% for the alternating-sensor mask and 6.35518% for a mask with 60% missing sensors.The authors attribute the increase potentially to more missing sensors and information transfer farther between known-displacement nodes; Figure 9 shows a missing-node prediction.Source: §4 · p. 11, Figure 9 · p. 12
  • On the 20-degree-of-freedom system, NMSE was 1.925% with one missing sensor between available sensors and 2.505196% with two missing sensors between them.The second mask represented two unavailable sensors between available ones; Figure 10 illustrates a missing-node prediction.Source: §4 · p. 11, Figure 10 · p. 12

5 Discussion and conclusions

GraphNARX combines graph neural networks with NARX to model structural dynamics, targeting transfer to larger structures and virtual sensing from incomplete nodal data. Simulations report low prediction errors, while the authors call for further tuning and experimental evaluation.

  • GraphNARX uses local graph relationships to target generalisation to larger structures of the same material and to infer displacements where sensor data are missing.Source: §5 · p. 13
  • 0.0204% NMSE on the 10-degree-of-freedom system and 0.02283% one-step-ahead error on the 20-degree system; multi-step error rose almost linearly with horizon.The model was trained on the 10-degree-of-freedom system and tested on a 20-degree-of-freedom system with the same structural-parameter values.Source: §5 · p. 13
  • Missing-sensor next-step prediction errors were around 5% on the 10-degree system and around 2% on the 20-degree system.Source: §5 · p. 13
  • The study uses limited GNN architectures and one-step validation selection; the authors call for more fine-tuning, multi-step-aware criteria or loss, and experimental evaluation.Source: §5 · p. 13

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