Reduced-Order Neural Operators: Learning Lagrangian Dynamics on Highly Sparse Graphs

Abstract: Simulating complex physical systems governed by Lagrangian dynamics often requires solving partial differential equations (PDEs) over high-resolution spatial domains, resulting in substantial computational costs. We present GIOROM (\textit{G}raph \textit{I}nf\textit{O}rmed \textit{R}educed \textit{O}rder \textit{M}odeling), a data-driven discretization invariant framework for accelerating Lagrangian simulations through reduced-order modeling (ROM). Previous discretization invariant ROM approaches rely on PDE time-steppers for spatiotemporally evolving low-dimensional reduced-order latent states. Instead, we leverage a data-driven graph-based neural approximation of the PDE solution operator. This operator estimates point-wise function values from a sparse set of input observations, reducing reliance on known governing equations of numerical solvers. Order reduction is achieved by embedding these point-wise estimates within the reduced-order latent space using a learned kernel parameterization. This latent representation enables the reconstruction of the solution at arbitrary spatial query points by evolving latent variables over local neighborhoods on the solution manifold, using the kernel. Empirically, GIOROM achieves a 6.6$\times$-32$\times$ reduction in input dimensionality while maintaining high-fidelity reconstructions across diverse Lagrangian regimes including fluid flows, granular media, and elastoplastic dynamics. The resulting framework enables learnable, data-driven and discretization-invariant order-reduction with reduced reliance on analytical PDE formulations. Our code is at \href{https://github.com/HrishikeshVish/GIOROM}{https://github.com/HrishikeshVish/GIOROM}