An improved random projection-based integration method for differential-algebraic equations (DAEs) of constrained mechanical systems

Abstract

Accurately integrating stiff ordinary differential equations (ODEs) and index-1 differential-algebraic equations that govern constrained multibody systems remains computationally demanding, especially for real-time applications. This article proposes an improved parsimonious physics-informed random-projection neural-network (PIRPNN) integrator that embeds explicit velocity- and position-correction into a single-hidden-layer random-feature framework and employs a vectorized assembly of system matrices for computation efficiency. Two illustrative examples are utilized to demonstrate the proposed algorithm and advantages. For the multibody dynamics benchmark problems examined, the improved PIRPNN demonstrates improved accuracy in terms of L₂-trajectory error and energy drift, together with a notable reduction in computational cost compared with the original PIRPNN. At peculiar tolerances from 10⁻⁶ to 10⁻¹⁰, it also outperforms the implicit MATLAB solvers ode15s and ode23t, further lowering both trajectory error and total-energy drift. The results underscore the potential of random-projection neural integrators as lightweight, constraint-preserving integration method alternative to classical integrators or more complex learning-based approaches in real-time multibody simulation.