MBD-NODE: Physics-informed data-driven modeling and simulation of constrained multibody systems

TL;DR

MBD-NODE applies Neural Ordinary Differential Equations to learn continuous-time multibody dynamics while explicitly embedding physical constraints through constrained optimization. The framework extends NODE to handle second-order MBD systems, external inputs, and problem-specific parameters, and offers both soft (augmented Lagrangian) and hard constraint approaches (via minimal/dependent coordinates) to enforce physics during training and inference. Across seven numerical experiments, MBD-NODE consistently outperforms FCNN, LSTM, and energy-conserving HNN/LNN in ID and OOD regimes, and demonstrates strong long-horizon predictive capability, including MPC-oriented control in the cart-pole system and long-horizon predictions in the slider-crank mechanism. The work provides a reproducible open-source benchmark and highlights benefits of physics-informed NODEs for accurate, data-efficient, and constraint-respecting multibody simulations with practical implications for control and design of mechanical systems.

Abstract

We describe a framework that can integrate prior physical information, e.g., the presence of kinematic constraints, to support data-driven simulation in multi-body dynamics. Unlike other approaches, e.g., Fully-connected Neural Network (FCNN) or Recurrent Neural Network (RNN)-based methods that are used to model the system states directly, the proposed approach embraces a Neural Ordinary Differential Equation (NODE) paradigm that models the derivatives of the system states. A central part of the proposed methodology is its capacity to learn the multibody system dynamics from prior physical knowledge and constraints combined with data inputs. This learning process is facilitated by a constrained optimization approach, which ensures that physical laws and system constraints are accounted for in the simulation process. The models, data, and code for this work are publicly available as open source at https://github.com/uwsbel/sbel-reproducibility/tree/master/2024/MNODE-code.

Figures (26)

• Figure 1: The discretized forward pass for MBD-NODE for general MBD.
• Figure 2: The discretized forward pass for MBD-NODE for general MBD without hard constraints which means the external force/torque could be directly added to the acceleration from the NODE that models the internal acceleration without additional input channel for external force. Here $\mathbf{\ddot z}_{ext,n}$ is the acceleration caused by the external input and $\mathbf{\ddot z}_{int,n}$ is the internal acceleration predicted by MBD-NODE.
• Figure 3: Single mass-spring system; $k$ and $m$ denote the spring constant and the mass of the object, respectively. Only the motion along $x$-direction is considered.
• Figure 4: Comparison of $t$ vs $x$ (Left Column) and $t$ vs $v$ (Right Column) for the different model and integrator combinations (rows) for the single-mass-spring system. Notice the dashed lines represent performance on the training data set $t \in [0,3]$, after which the dotted lines represent performance on the testing data set $t \in [0,30]$. (a), (b) are for the MBD-NODE with RK4; (c), (d) are for the MBD-NODE with leapfrog integrator; (e), (f) are for the RK4 integrator; (g), (h) are for the LNN; (i), (j) are for the HNN.
• Figure 5: (a) The phase space $x$ vs $v$ and (b) the system energy for the test data for the single mass-spring system