[Paper Review] Accurately Solving Physical Systems with Graph Learning

Han Shao, Tassilo Kugelstadt+5 more|arXiv (Cornell University)|Jun 6, 2020

Model Reduction and Neural Networks55 references4 citations

TL;DR

This paper proposes using graph networks (GNs) to predict initial guesses for iterative solvers in physical simulations, significantly accelerating convergence while preserving long-term stability. By replacing the standard initial guess in position-based dynamics (PBD) with a GN-predicted one, the method reduces iterations by up to 17% in complex rod simulations, including collisions and large-scale tree models, without sacrificing accuracy or stability.

ABSTRACT

Iterative solvers are widely used to accurately simulate physical systems. These solvers require initial guesses to generate a sequence of improving approximate solutions. In this contribution, we introduce a novel method to accelerate iterative solvers for physical systems with graph networks (GNs) by predicting the initial guesses to reduce the number of iterations. Unlike existing methods that aim to learn physical systems in an end-to-end manner, our approach guarantees long-term stability and therefore leads to more accurate solutions. Furthermore, our method improves the run time performance of traditional iterative solvers. To explore our method we make use of position-based dynamics (PBD) as a common solver for physical systems and evaluate it by simulating the dynamics of elastic rods. Our approach is able to generalize across different initial conditions, discretizations, and realistic material properties. Finally, we demonstrate that our method also performs well when taking discontinuous effects into account such as collisions between individual rods. Finally, to illustrate the scalability of our approach, we simulate complex 3D tree models composed of over a thousand individual branch segments swaying in wind fields. A video showing dynamic results of our graph learning assisted simulations of elastic rods can be found on the project website available at http://computationalsciences.org/publications/shao-2021-physical-systems-graph-learning.html .

Motivation & Objective

• To improve the runtime efficiency of iterative solvers in physical simulations without compromising long-term stability.
• To address the instability and poor generalization common in end-to-end deep learning approaches for physics simulation.
• To enable fast, stable, and generalizable simulation of complex physical systems such as elastic rods and dynamic tree models.
• To demonstrate that GNs can effectively accelerate traditional solvers by predicting high-quality initial guesses.

Proposed method

• A graph network (COPINGNet) is trained to predict the outcome of the constraint projection step in position-based dynamics (PBD), serving as an initial guess for iterative solvers.
• The method leverages the stability of classic PBD solvers while accelerating convergence via GN-predicted initial states.
• Training data is generated using vanilla PBD simulations of elastic rods under various initial conditions, discretizations, and material properties.
• Collision handling is implemented via line-to-line and half-space constraints with hierarchical spatial hashing for efficient detection.
• The GN is not used end-to-end but only to predict initial guesses, preserving the solver’s inherent stability.
• The approach is evaluated on rod bending, rod collisions, and large-scale 3D tree simulations with over 1,000 segments.

Experimental results

Research questions

• RQ1 Can graph networks be used to accelerate iterative solvers in physical simulations without sacrificing long-term stability?
• RQ2 How well does a GN-predicted initial guess generalize across different rod discretizations, material properties, and initial conditions?
• RQ3 Can the proposed method handle discontinuous events such as rod collisions and complex topologies like dynamic trees?
• RQ4 What performance gains are achievable in terms of iteration reduction and runtime speedup?

Key findings

• The GN-based initial guess reduced the number of required iterations, resulting in a 17.0% runtime speedup for a 10-rod tree model and 11.1% for a 70-rod model, despite including neural network inference time.
• The method achieved a 10% speedup in helix-on-ground simulations and a 6% speedup in rod collision scenarios, demonstrating robustness to discontinuous events.
• Generalization across different rod discretizations and topologies was observed even when the GN was not trained on those configurations, indicating strong inductive bias for physical consistency.
• Unlike end-to-end learning, the method maintained long-term stability and avoided divergence, even under extreme deformations or out-of-distribution conditions.
• The approach preserved accuracy and physical plausibility in complex simulations, including 3D swaying trees with over 1,000 rod segments.
• The GN-predicted initial guess outperformed standard initial guesses (e.g., previous time step values) in convergence speed and stability, even when the network was trained only on a subset of scenarios.