Simulating large scenes with many rigid objects is crucial for a variety of applications, such as robotics, engineering, film and video games. Rigid interactions are notoriously hard to model: small changes to the initial state or the simulation parameters can lead to large changes in the final state. Recently, learned simulators based on graph networks (GNNs) were developed as an alternative to hand-designed simulators like MuJoCo and Bullet. They are able to accurately capture dynamics of real objects directly from real-world observations. However, current state-of-the-art learned simulators operate on meshes and scale poorly to scenes with many objects or detailed shapes. Here we present SDF-Sim, the first learned rigid-body simulator designed for scale. We use learned signed-distance functions (SDFs) to represent the object shapes and to speed up distance computation. We design the simulator to leverage SDFs and avoid the fundamental bottleneck of the previous simulators associated with collision detection.For the first time in literature, we demonstrate that we can scale the GNN-based simulators to scenes with hundreds of objects and up to 1.1 million nodes, where mesh-based approaches run out of memory. Finally, we show that SDF-Sim can be applied to real world scenes by extracting SDFs from multi-view images.

We introduce $r$-loopy Weisfeiler-Leman ($r$-$\ell$WL), a novel hierarchy of graph isomorphism tests and a corresponding GNN framework, $r$-$\ell$MPNN, that can count cycles up to length $r{+}2$. Most notably, we show that $r$-$\ell$WL can count homomorphisms of cactus graphs. This extends 1-WL, which can only count homomorphisms of trees and, in fact, is incomparable to $k$-WL for any fixed $k$. We empirically validate the expressive and counting power of $r$-$\ell$MPNN on several synthetic datasets and demonstrate the scalability and strong performance on various real-world datasets, particularly on sparse graphs.

This paper pertains to an emerging machine learning paradigm: learning higher- order functions, i.e. functions whose inputs are functions themselves, particularly when these inputs are Neural Networks (NNs). With the growing interest in architectures that process NNs, a recurring design principle has permeated the field: adhering to the permutation symmetries arising from the connectionist structure ofNNs. However, are these the sole symmetries present in NN parameterizations? Zooming into most practical activation functions (e.g. sine, ReLU, tanh) answers this question negatively and gives rise to intriguing new symmetries, which we collectively refer to as scaling symmetries, that is, non-zero scalar multiplications and divisions of weights and biases. In this work, we propose Scale Equivariant Graph MetaNetworks - ScaleGMNs, a framework that adapts the Graph Metanetwork (message-passing) paradigm by incorporating scaling symmetries and thus rendering neuron and edge representations equivariant to valid scalings. We introduce novel building blocks, of independent technical interest, that allow for equivariance or invariance with respect to individual scalar multipliers or their product and use them in all components of ScaleGMN. Furthermore, we prove that, under certain expressivity conditions, ScaleGMN can simulate the forward and backward pass of any input feedforward neural network. Experimental results demonstrate that our method advances the state-of-the-art performance for several datasets and activation functions, highlighting the power of scaling symmetries as an inductive bias for NN processing. The source code is publicly available at https://github.com/jkalogero/scalegmn.