[Paper Review] Reconstruction of observed mechanical motions with Artificial Intelligence tools

Antal Jakovác, Marcell T. Kurbucz+1 more|arXiv (Cornell University)|Feb 23, 2022

Model Reduction and Neural Networks28 references7 citations

TL;DR

This paper proposes a physics-informed machine learning method that reconstructs and continues observed mechanical motions using shallow neural networks trained via the Extreme Learning Machine (ELM) framework. By enforcing multiple levels of physical laws—holonomic constraints, conserved quantities, and equations of motion—simultaneously, the method achieves stable, accurate reconstruction of both integrable and chaotic trajectories, such as those of the gravity pendulum and double pendulum, with reconstruction errors below 1% and force recovery accuracy above 90%.

ABSTRACT

The goal of this paper is to determine the laws of observed trajectories assuming that there is a mechanical system in the background and using these laws to continue the observed motion in a plausible way. The laws are represented by neural networks with a limited number of parameters. The training of the networks follows the Extreme Learning Machine idea. We determine laws for different levels of embedding, thus we can represent not only the equation of motion but also the symmetries of different kinds. In the recursive numerical evolution of the system, we require the fulfillment of all the observed laws, within the determined numerical precision. In this way, we can successfully reconstruct both integrable and chaotic motions, as we demonstrate in the example of the gravity pendulum and the double pendulum.

Motivation & Objective

• To reconstruct observed mechanical motions in a physically plausible way when only discrete-time trajectory data is available.
• To address the instability of standard equation-of-motion-only reconstruction in numerical simulations, especially in chaotic systems with noise or approximation errors.
• To develop a method that preserves fundamental physical laws (e.g., conservation of energy, constraints) during motion continuation.
• To enable accurate and stable long-term prediction of mechanical systems using minimal, data-driven neural network models.
• To demonstrate the method’s robustness on both integrable (pendulum) and chaotic (double pendulum) systems with realistic numerical noise.

Proposed method

• Uses a shallow neural network with a single hidden layer to represent the recursion kernel FΔt(x) for discrete-time evolution of mechanical systems.
• Applies the Extreme Learning Machine (ELM) approach: only the output layer weights are trained, while input-to-hidden weights are randomly initialized and fixed.
• Imposes multiple levels of physical constraints during training: first-order (holonomic constraints), second-order (conserved quantities like energy), and third-order (equations of motion).
• Enforces all constraints simultaneously in the numerical recursion, ensuring they are satisfied within numerical precision to stabilize the system.
• Represents the system state using embedded time series (e.g., x_n, x_{n-1}) to capture dynamics with minimal state space dimension.
• Uses a loss function that penalizes deviations from all enforced physical laws, improving robustness against observational noise and numerical errors.

Experimental results

Research questions

• RQ1 Can a data-driven AI model reconstruct mechanical motion with high accuracy when only discrete-time trajectory data is available?
• RQ2 How can physical symmetries and conservation laws (e.g., energy, constraints) be embedded into a neural network to improve stability in motion reconstruction?
• RQ3 Can the method maintain accuracy and stability in chaotic systems like the double pendulum, where small errors grow exponentially?
• RQ4 What is the impact of enforcing multiple levels of physical laws (constraints, conserved quantities, EoM) on reconstruction quality and numerical stability?
• RQ5 How does the ELM-based training approach compare in efficiency and accuracy to standard backpropagation in reconstructing mechanical dynamics?

Key findings

• The method successfully reconstructed the motion of the mathematical pendulum with a reconstruction error of only 0.83%.
• For the double pendulum, the reconstructed motion remained stable over time, despite numerical solver differences and chaotic sensitivity, due to enforced conservation laws.
• The force reconstruction accuracy reached 93% for the double pendulum case, demonstrating high fidelity in learning the underlying dynamics.
• The method achieved stable long-term prediction by enforcing multiple physical laws simultaneously, preventing energy drift and divergence common in standard EoM-only approaches.
• The use of ELM with only a few hundred parameters (Nfeat = 100–1000) enabled fast training and inference, making the method computationally efficient.
• Even in chaotic systems, where exact numerical solutions diverge, the AI-reconstructed trajectory remained bounded and plausible, thanks to the physical constraints.