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

2024-07-04

arXiv.org

AI Summary

High-resolution Lagrangian dynamical system simulation faces prohibitive computational cost and reliance on discretization schemes and explicit governing equations. To address this, we propose Graph-Informed Operator Reduction (GIOROM), a model reduction framework that integrates graph neural operators with learnable kernel parameterization to directly approximate the PDE solution operator end-to-end from sparse observational data—bypassing explicit equations and numerical time integrators. Its core innovation lies in constructing local latent-variable evolution dynamics on the Lagrangian manifold and enabling discretization-invariant, manifold-adaptive function approximation via sparse graphs. Experiments across fluid, granular, and elastoplastic dynamical systems demonstrate that GIOROM achieves 6.6×–32× input dimensionality reduction while preserving high-fidelity reconstruction, significantly improving simulation efficiency and cross-system generalizability.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 ( extit{G}raph extit{I}nf extit{O}rmed extit{R}educed extit{O}rder extit{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$ imes$-32$ imes$ 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}

Problem

• Accelerating Lagrangian simulations with reduced computational costs
• Learning PDE solution operators from sparse graph data
• Achieving high-fidelity reconstructions in diverse Lagrangian regimes

Innovation

• Data-driven graph-based neural PDE approximation
• Learned kernel parameterization for order reduction
• Discretization-invariant framework for sparse observations