Equi-Euler GraphNet: An Equivariant, Temporal-Dynamics Informed Graph Neural Network for Dual Force and Trajectory Prediction in Multi-Body Systems

TL;DR

Equi-Euler GraphNet delivers a physics-informed graph neural network that jointly predicts internal contact forces and global trajectories in bearing systems. By leveraging an equivariant message-passing scheme and temporally-aware sub-time-step updates, the model achieves stable long-horizon rollouts and strong generalization to unseen speeds, loads, and configurations, while offering up to substantial speedups over high-fidelity solvers. The bearing-specific architecture decouples ring dynamics from rolling-element kinematics, enabling accurate force estimation alongside motion prediction, which supports fault detection and remaining-useful-life assessments. Evaluations on BEAST-generated data demonstrate superior interpolation and extrapolation performance compared with GNS, EGNN, and GMN baselines, with robust force distribution predictions and practical computational gains for digital twins and maintenance planning.

Abstract

Accurate real-time modeling of multi-body dynamical systems is essential for enabling digital twin applications across industries. While many data-driven approaches aim to learn system dynamics, jointly predicting internal loads and system trajectories remains a key challenge. This dual prediction is especially important for fault detection and predictive maintenance, where internal loads-such as contact forces-act as early indicators of faults, reflecting wear or misalignment before affecting motion. These forces also serve as inputs to degradation models (e.g., crack growth), enabling damage prediction and remaining useful life estimation. We propose Equi-Euler GraphNet, a physics-informed graph neural network (GNN) that simultaneously predicts internal forces and global trajectories in multi-body systems. In this mesh-free framework, nodes represent system components and edges encode interactions. Equi-Euler GraphNet introduces two inductive biases: (1) an equivariant message-passing scheme, interpreting edge messages as interaction forces consistent under Euclidean transformations; and (2) a temporal-aware iterative node update mechanism, based on Euler integration, to capture influence of distant interactions over time. Tailored for cylindrical roller bearings, it decouples ring dynamics from constrained motion of rolling elements. Trained on high-fidelity multiphysics simulations, Equi-Euler GraphNet generalizes beyond the training distribution, accurately predicting loads and trajectories under unseen speeds, loads, and configurations. It outperforms state-of-the-art GNNs focused on trajectory prediction, delivering stable rollouts over thousands of time steps with minimal error accumulation. Achieving up to a 200x speedup over conventional solvers while maintaining comparable accuracy, it serves as an efficient reduced-order model for digital twins, design, and maintenance.

Figures (9)

• Figure 1: Overview of Equi-Euler GraphNet Framework: (a) Train and test data generation from a multiphysics simulator, covering both interpolation and extrapolation cases. (b) Stacked GNN pipeline with equivariant message passing, comprising a dynamics prediction layer (for ring accelerations and forces) and a kinematics prediction layer (for roller motion), integrated over multiple sub-time steps using Euler integration. A loss function compares predicted accelerations and contact forces at initial and final sub-time steps. (c) Test-time trajectory rollout with iterative state updates over extended horizons, enabling accurate long-term predictions under unseen operating conditions.
• Figure 2: Graph Representation of a Ground-Mounted Bearing System: (a) Simulated cylindrical roller bearing model showing rollers (brown), with the inner ring (yellow) and outer ring (green) connected to the ground via reaction forces. (b) Graph representation of the bearing with nodes for rolling elements (RE), inner ring (IR), outer ring (OR), and a virtual ground (GND). Bidirectional edges (RE--IR, RE--OR) capture contact forces between rollers and rings, while directed edges from GND model external reactions on IR and OR. (c) Computation of the rolling element’s effective radius from IR and OR center distances for the relative position feature, encoding instantaneous roller deformation in contact edges.
• Figure 3: Equi-Euler GraphNet: Dynamics and Kinematics Prediction Layers. The model predicts bearing dynamics by sequentially updating forces, accelerations, and velocities. (a) Bearing Graph represents rolling elements (RE), inner ring (IR), outer ring (OR), and ground (GND) as nodes, with bidirectional rolling element–ring edges for contact forces and directed ground–ring edges for external reactions. (b) Dynamics Predictor Layer estimates internal forces and ring accelerations. (b.i) Encoder embeds node and edge features. (b.ii) Edge Vector-Message Decoder computes interaction forces from relative distance and velocity vectors. (b.iii) Node Dynamics Decoder aggregates forces, incorporates external reactions, and computes accelerations. (c) Euler Integration updates ring node velocities using predicted accelerations over a sub-time step. (d) Directed Graph represents the updated state, where directed edges propagate velocity updates from rings to rolling elements. (e) Kinematics Predictor Layer estimates rolling element velocities. (e.i) Encoder embeds edge features. (e.ii) Edge Vector-Message Decoder scales velocity components. (e.iii) Kinematics Decoder aggregates edge messages to compute rolling element velocities. (f) Euler Integration updates node positions, forming a new system state. Steps (a)–(f) repeat for each message-passing iteration. (g) Outputs include interaction forces and accelerations of the rolling elements, inner ring, and outer ring nodes at initial and final sub-time steps.
• Figure 4: Problem Setup for Cylindrical Roller Bearing Dynamics: (a) Exploded diagram of the simulated N209ECP bearing assembly with inner ring, outer ring, rolling elements, and a simple cage. (b) Schematic of the 2D, three-degree-of-freedom multi-body simulation in BEAST, where the outer ring is fixed, an external step load is applied at the 12 o’clock position, doubled at the 2500th time step, and subsequently relaxed at the 5000th time step --- inducing transient vibrations --- while the inner ring rotates at a prescribed speed. Springs and dampers connect the bearing to ground, modeling reaction forces. Rolling elements, centered by a simple cage, respond freely to these transient load variations. The resulting dataset serves as the training and testing basis for the Equi-Euler GraphNet framework.
• Figure 5: Comparison of GNN Architectures for Bearing Dynamics: (a)--(d) Schematic of four message-passing frameworks (GNSsanchez2020learning, EGNNsatorras2021n, GMNhuang2022equivariant, and Equi-Euler GraphNet). While all models follow an encode--process–-decode paradigm, they differ in their feature representations, message construction, and state update mechanisms. GNS (a) encodes stacked scalar and vector features into latent node and edge embeddings, which are iteratively updated and finally decoded as node accelerations and pairwise forces. EGNN (b) and GMN (c) construct invariant scalar edge embeddings, transform them into vector messages, and update node embeddings and states through equivariant message passing. Since these models do not natively predict forces, we append equivariant decoders to map final edge embeddings to interaction forces. In contrast, Equi-Euler GraphNet (d) directly interprets edge messages as physical forces and updates node positions and velocities via Euler integration, enabling simultaneous prediction of trajectories and forces.