Learning to Simulate Complex Physics with Graph Networks

1 Learning to Simulate Complex Physics with Graph NetworksAlvaro Sanchez-Gonzalez* 1 Jonathan Godwin* 1 Tobias Pfaff* 1 Rex Ying* 1 2 Jure Leskovec2 Peter W. Battaglia1 AbstractHere we present a machine Learning frameworkand model implementation that can learn tosimulate a wide variety of challenging physi-cal domains, involving fluids, rigid solids, anddeformable materials interacting with one an-other. Our framework which we term GraphNetwork-based Simulators (GNS) representsthe state of a physical system with particles, ex-pressed as nodes in a Graph , and computes dy-namics via learned message-passing. Our re-sults show that our model can generalize fromsingle-timestep predictions with thousands of par-ticles during training, to different initial condi-tions, thousands of timesteps, and at least anorder of magnitude more particles at test model was robust to hyperparameter choicesacross various evaluation metrics: the main de-terminants of long-term performance were thenumber of message-passing steps, and mitigat-ing the accumulation of error by corrupting thetraining data with noise.

2 Our GNS frameworkadvances the state-of-the-art in learned physicalsimulation, and holds promise for solving a widerange of Complex forward and inverse IntroductionRealistic simulators of Complex Physics are invaluable tomany scientific and engineering disciplines, however tradi-tional simulators can be very expensive to create and a simulator can entail years of engineering ef-fort, and often must trade off generality for accuracy in anarrow range of settings. High-quality simulators require*Equal contribution1 DeepMind, London, UK2 Departmentof Computer Science, Stanford University, Stanford, CA, to: of the37thInternational Conference on MachineLearning, Online, PMLR 119, 2020. Copyright 2020 by the au-thor(s).

3 Water Goop SandTimeFigure of our GNS model for ourWATER-3D, GOOP-3 DandSAND-3 Ddatasets. It learns to Simulate rich materials atresolutions sufficient for high-quality rendering [video].substantial computational resources, which makes scalingup prohibitive. Even the best are often inaccurate due to in-sufficient knowledge of, or difficulty in approximating, theunderlying Physics and parameters. An attractive alternativeto traditional simulators is to use machine Learning to trainsimulators directly from observed data, however the largestate spaces and Complex dynamics have been difficult forstandard end-to-end Learning approaches to we present a powerful machine Learning framework forlearning to Simulate Complex systems from data GraphNetwork-based Simulators (GNS).

4 Our framework imposesstrong inductive biases, where rich physical states are rep-resented by graphs of interacting particles, and complexdynamics are approximated by learned message-passingamong implemented our GNS framework in a single deep learn-ing architecture, and found it could learn to accurately sim-ulate a wide range of physical systems in which fluids, rigidsolids, and deformable materials interact with one model also generalized well to much larger systems andLearning to Simulated Updatexiv0ie0i,jv0jvmiemi,jem+1i,jvm+1i( b)(c)(e)(d)(a)Xt0 XtKConstruct graphPass messagesExtract dynamics infoLearned simulator,s ++YEncoderDecoderProcessorXGN1 GNMGMG0G1GM s Figure 2.(a)Our GNS predicts future states represented as particles using its learned dynamics model,d , and a fixed update procedure.

5 (b)Thed uses an encode-process-decode scheme, which computes dynamics information,Y, from input state,X.(c)TheENCODER constructs latent Graph ,G0, from the input state,X.(d)ThePROCESSOR performsMrounds of learned message-passing over the latentgraphs,G0, .. , GM.(e)The DECODER extracts dynamics information,Y, from the final latent Graph , time scales than those on which it was trained. Whileprevious Learning simulation approaches (Li et al., 2018;Ummenhofer et al., 2020) have been highly specialized forparticular tasks, we found our single GNS model performedwell across dozens of experiments and was generally robustto hyperparameter choices. Our analyses showed that perfor-mance was determined by a handful of key factors: its abilityto compute long-range interactions, inductive biases for spa-tial invariance, and training procedures which mitigate theaccumulation of error over long simulated Related WorkOur approach focuses onparticle-basedsimulation, whichis used widely across science and engineering, , compu-tational fluid dynamics, computer graphics.

6 States are rep-resented as a set of particles, which encode mass, material,movement, etc. within local regions of space. Dynamics arecomputed on the basis of particles interactions within theirlocal neighborhoods. One popular particle-based methodfor simulating fluids is smoothed particle hydrodynamics (SPH) (Monaghan, 1992), which evaluates pressure and vis-cosity forces around each particle, and updates particles velocities and positions accordingly. Other techniques, suchas position-based dynamics (PBD) (M uller et al., 2007)and material point method (MPM) (Sulsky et al., 1995),are more suitable for interacting, deformable materials. InPBD, incompressibility and collision dynamics involve re-solving pairwise distance constraints between particles, anddirectly predicting their position changes.

7 Several differen-tiable particle-based simulators have recently appeared, ,DiffTaichi (Hu et al., 2019), PhiFlow (Holl et al., 2020), andJax-MD (Schoenholz & Cubuk, 2019), which can backprop-agate gradients through the simulations from data (Grzeszczuk et al., 1998)has been an important area of study with applications inphysics and graphics. Compared to engineered simulators,a learned simulator can be far more efficient for predictingcomplex phenomena (He et al., 2019); , (Ladick`y et al.,2015; Wiewel et al., 2019) learn parts of a fluid simulatorfor faster Networks (GN) (Battaglia et al., 2018) a type ofgraph neural network (Scarselli et al., 2008) have recentlyproven effective at Learning forward dynamics in various set-tings that involve interactions between many entities.

8 A GNmaps an input Graph to an output Graph with the same struc-ture but potentially different node, edge, and Graph -levelattributes, and can be trained to learn a form of message-passing (Gilmer et al., 2017), where latent information ispropagated between nodes via the edges. GNs and theirvariants, , interaction Networks , can learn to simu-late rigid body, mass-spring, n-body, and robotic controlsystems (Battaglia et al., 2016; Chang et al., 2016; Sanchez-Gonzalez et al., 2018; Mrowca et al., 2018; Li et al., 2019;Sanchez-Gonzalez et al., 2019), as well as non-physical sys-tems, such as multi-agent dynamics (Tacchetti et al., 2018;Sun et al., 2019), algorithm execution (Veli ckovi c et al.,2020), and other dynamic Graph settings (Trivedi et al.)

9 ,2019; 2017; Yan et al., 2018; Manessi et al., 2020). Learning to SimulateOur GNS framework builds on and generalizes several linesof work, especially Sanchez-Gonzalez et al. (2018) s GN-based model which was applied to various robotic controlsystems, Li et al. (2018) s DPI which was applied to fluiddynamics, and Ummenhofer et al. (2020) s Continuous Con-volution (CConv) which was presented as a non- Graph -basedmethod for simulating fluids. Crucially, our GNS frame-work is ageneralapproach to Learning simulation, is simplerto implement, and is more accurate across fluid, rigid, anddeformable material GNS Model General Learnable SimulationWe assumeXt Xis the state of the world at timet. Applying physical dynamics overKtimesteps yieldsa trajectory of states,Xt0:K= (Xt0.

10 ,XtK).Asimulator,s:X X, models the dynamics by map-ping preceding states to causally consequent denote a simulated rollout trajectory as, Xt0:K= (Xt0, Xt1,.., XtK), which is computed itera-tively by, Xtk+1=s( Xtk)for each timestep. Simulatorscompute dynamics information that reflects how the currentstate is changing, and use it to update the current state toa predicted future state (see Figure 2(a)). An example is anumerical differential equation solver: the equations com-pute dynamics information, , time derivatives, and theintegrator is the update learnable simulator,s , computes the dynamics in-formation with a parameterized function approximator,d :X Y, whose parameters, , can be optimized forsome training objective.