Constraint-based graph network simulator

In the rapidly advancing area of learned physical simulators , nearly all methods train a forward model that directly predicts future states from input states . However , many traditional simulation engines use a constraint-based approach instead of direct prediction . Here we present a framework for constraint-based learned simulation , where a scalar constraint function is implemented as a trainable function approximator , and future predictions are computed as the solutions to a constraint satisfaction problem . We implement our method using a graph neural network as the constraint function and gradient descent as the constraint solver . The architecture can be trained by standard backpropagation . We test the model on a variety of challenging physical domains , including simulated ropes , bouncing balls , colliding irregular shapes and splashing fluids . Our model achieves better or comparable performance to top learned simulators . A key advantage of our model is the ability to generalize to more solver iterations at test time to improve the simulation accuracy . We also show how hand-designed constraints can be added at test time to satisfy objectives which were not present in the training data , which is not possible with forward approaches . Our constraint-based framework is applicable to any setting in which forward learned simulators are used , and more generally demonstrates key ways that learned models can leverage popular methods in numerical methods . 1 INTRODUCTION . Consider a bowling ball colliding with a bowling pin . You might explain this event as involving a pair of forces being generated , one which causes the pin to move , and the other which causes the ball to careen away with a different direction and speed . This kind of intuitive cause-and-effect approach is analogous to physical simulators that apply an explicit forward model to calculate a future state directly from the current one , such as when numerically integrating discretized equations of motion . An alternative , but equally valid , way to explain the collision is in terms of constraint satisfaction : the ball and pin can not occupy the same location at the same time , and their combined energies and momenta must be conserved , so the post-collision trajectories are the only way the future can unfold without violating these constraints . This constraint-based approach is analogous to physical simulators that use an implicit function to model a system of constraints over the current and future states , and which generate a prediction by searching for a future state that respects all constraints . Both families of simulators—those based on explicit , forward functions versus those which define the dynamics implicitly , via constraints—are widely used in physics , engineering , and graphics . In principle they can model the same types of dynamics , however they differ in how their respective predictions are computed and in practice strike different trade-offs that determine why one or the other is preferred in different domains . For example , explicit methods are popular for large systems with ( mostly ) independent local effects whose space and time derivatives are relatively smooth , and their accuracy can often be increased by discretizing space and time more finely . Implicit approaches are often preferred for systems with strong interactions , such rigid and stiff dynamics , and more accurate solutions can often be found by using more sophisticated constraint solvers or by increasing the computational budget ( e.g. , solver iterations ) allocated to searching for solutions . In machine learning ( ML ) , there have been rapid advances recently in methods for learning to simulate complex dynamic processes , however almost all ( e.g. , Sanchez-Gonzalez et al . ( 2020 ) ; Pfaff et al . ( 2021 ) ) have focused on explicit forward model approaches , with few exceptions ( Yang et al. , 2020 ) . Here we present a framework for learning to simulate complex dynamics via constraint satisfaction . Our “ Constraint-based Graph Network Simulator ” ( C-GNS ) defines a single scalar-valued constraint function that represents whether a future state satisfies the physical constraints , conditioned on the current and previous states . The constraint function is implemented as a Graph Neural Network ( GNN ) ( Bronstein et al. , 2017 ; Battaglia et al. , 2018 ) , which can model systems with rich compositional structure—multiple bodies , complex meshes , etc . To predict the next state via the constraint function ’ s implicit representation of the dynamics , a gradient-based solver finds a proposed state which satisfies the constraints . We train it through the solver by backpropagation . We also introduce a hybrid approach that proposes and refines the future state using an explicit iterative predictor , rather than solving for learned constraints . We tested the C-GNS on a variety of challenging physical simulation domains generated by several different simulation engines : simulated rope , bouncing balls , and bouncing irregular rigid shapes ( MuJoCo , Todorov et al . ( 2012 ) ) and splashing fluids ( Flex , Macklin et al . ( 2014 ) ) . We found that the C-GNS ’ s simulated rollouts were more accurate than a state-of-the-art Graph Net Simulator ( GNS ) ( Sanchez-Gonzalez et al. , 2020 ) with comparable number of parameters . At test time , the C-GNS could use additional solver iterations to improve its predictive accuracy , striking desired speed-accuracy trade-offs . It could also satisfy new , hand-designed constraints jointly alongside its learned constraints . Neither of these capabilities are possible in explicit forward-style approaches . 2 BACKGROUND AND RELATED WORK . Constraint solvers are central to many physics simulators . Most rigid-body and game engines use constraints to model joints , collision and contact ( Baraff , 1994 ) . They are used for limiting strain in realistic cloth simulation ( Thomaszewski et al. , 2009 ) , and are a core component in Eulerian incompressible fluid solvers to solve for pressure ( Chorin , 1967 ) . Recently , position-based ( Müller et al. , 2007 ) and projective dynamics methods ( Bouaziz et al. , 2014 ) have become very popular for interactive simulation . These methods express dynamics purely as constraints , and can simulate a wide range of physical systems from rigids over soft-bodies to fluids ( Macklin et al. , 2014 ) . Machine learning methods for accelerating scientific simulation of complex systems , such as turbulence ( Kochkov et al. , 2021 ; Wang et al. , 2020 ) and aerodynamics ( Thuerey et al. , 2020 ; Zhang et al. , 2018 ) , have grown rapidly in recent years . GNN-based learned simulators , in particular , are a very flexible approach which can model a wide range of systems , from articulated dynamics ( Sanchez-Gonzalez et al. , 2018 ) to particle-based physics ( Mrowca et al. , 2018 ; Li et al. , 2019 ; Sanchez-Gonzalez et al. , 2020 ) and mesh-based continuum systems ( Pfaff et al. , 2021 ; De Avila Belbute-Peres et al. , 2020 ) , and generalize well to unseen scenarios . Combining learning algorithms with principles from physics and numerical methods , such as auxiliary loss terms and rich inductive biases , can improve sample complexity , computational efficiency , and generalization ( Wu et al. , 2018 ; Karniadakis et al. , 2021 ; Chen et al. , 2018 ; Rubanova et al. , 2019 ) . Imposing Hamiltonian ( Greydanus et al. , 2019 ; Sanchez-Gonzalez et al. , 2019 ; Chen et al. , 2019 ) and Lagrangian ( Lutter et al. , 2019 ; Cranmer et al. , 2020 ; Finzi et al. , 2020 ) mechanics in learned simulators offers unique speed/accuracy tradeoffs and can preserve symmetries more effectively . Recent methods have been proposed for learning constraint functions and solving them in a model ’ s forward pass ( Duvenaud et al . ( 2020 ) ’ s “ Deep Implicit Layers ” tutorial is an excellent hands-on survey ) . Such models can play games ( Amos & Kolter , 2017 ; Wang et al. , 2019 ) , optimize power flow ( Donti et al. , 2021 ) , support robotic planning ( Loula et al. , 2020 ) , and perform combinatorial optimization ( Bartunov et al. , 2020 ) . Solvers such as gradient descent and Newton ’ s method are differentiable , and support training by backpropagation , but this can be computationally expensive , so approaches such as Deep Equilibrium Models ( DEM ) ( Bai et al. , 2019 ; 2020 ) use implicit differentiation to compute gradients only at the solution point . Despite the popularity of constraint-based traditional simulators , only a single simulator which uses learned constraints has been reported ( Yang et al. , 2020 ) . Their “ Neural Projections ” method , based on Goldenthal et al . ( 2007 ) , iteratively proposes a future state with an Euler step , then projects the proposal onto a learned constraint manifold , implemented as a multilayer perceptron ( MLP ) . Crucially , their constraint function only measures how much an individual state violates the learned constraints , and thus is not an implicit representation of the dynamics . It is suitable for quasi-static regimes , but not scenarios such as the elastic collisions in the bowling ball example described above . 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Simulation basics A physical trajectory , measured at discrete time intervals , is a sequence of states , ( X1 , . . . , XT ) , where Xt represents properties such as the positions , velocities , masses , etc , of elements of the system . A physical simulator , s , is a function that maps current and/or previous state ( s ) , which we term the context , X≤t , to a predicted future state , X̂t+1 = s ( X≤t ) ( see Figure 1a ) 1 . A simulated physical trajectory , termed a rollout , ( Xt , X̂t+1 , X̂t+2 , . . . ) , can be generated by repeatedly applying s to its own predicted state , X̂t+1 = s ( X̂≤t ) . Simulators are often comprised of a PREDICTOR mechanism which maps the context X≤t to an update value Ŷ , that represents information about the system ’ s temporal evolution at the current time . Then Ŷ is used by an UPDATER mechanism to update the current state to the next state : X̂t+1 = UPDATER ( X≤t , Ŷ ) , e.g. , updating current positions and velocities represented by Xt with new velocities and accelerations represented by Ŷ , to predict the next state . Explicit simulators Across science , engineering , and graphics , a popular class of simulators are defined explicitly : the state update Ŷ is predicted directly fromX≤t using an explicit forward function , Ŷ = fD ( X≤t ) , as illustrated in Figure 1b . Among the rapidly growing family of learned simulators , the forward function fD is typically implemented using a neural network ( Sanchez-Gonzalez et al. , 2020 ; Pfaff et al. , 2021 ) . Constraint-based implicit simulators Here we explore learned simulators based on implicit formulations of the dynamics . Rather than predicting the desired state directly , as in explicit formulations , our implicit simulator uses a differentiable constraint function , c = fC ( X≤t , Ŷ ) , where c is a scalar that quantifies how well a proposed state update Ŷ agrees withX≤t . A future prediction is generated by applying a solver , such as an optimization or zero-finding algorithm , to find a Ŷ that satisfies the constraint function , and applying the UPDATER to update Xt to X̂t+1 . The fC can represent all the physical constraints in the system , including the time dynamics . 1Despite that physics is Markovian , we use X≤t as input because our framework can also apply to dynamic processes which are non-Markovian . Providing previous states can also often be helpful when there are hidden properties of the system which are only identifiable over a sequence of observed states , and when a state does not represent velocity or momentum information . As illustrated in Figure 1d , we formulate our constraint-solving procedure via an iterative method that starts with an initial proposal , Y ( 0 ) . On the i-th iteration , the solver uses the gradient of fC w.r.t . Y at the current proposal to compute a change to the proposal , δY = −λ ∇Y fC ( X≤t , Y ) |Y=Y ( i ) . This δY is then used to revise the proposal to , Y ( i+1 ) = Y ( i ) + δY . This process repeats for N steps , and the final proposal value is treated as the PREDICTOR ’ s output , Ŷ = Y ( N ) . Our constraint-based model ’ s fC is defined as a trainable function approximator which is real-valued and lower bounded at zero , and uses gradient descent to find Ŷ that minimizes it , where λ is a fixed step size . This induces the semantics that the desired Ŷ = arg minY fC ( X≤t , Y ) . We also explore a second constraint-solving procedure , inspired by Yang et al . ( 2020 ) ’ s Neural Projections ’ use of “ fast projection ” Goldenthal et al . ( 2007 ) . Specifically , λ = − fC ( X≤t , Y ( i ) ) ‖∇Y fC ( X≤t , Y ) | Y =Y ( i ) ‖2 . Unlike gradient descent , fast projection is a zero-finding algorithm , so in this case fC is not lower bounded . This induces the semantics that fC ( X≤t , Ŷ ) = 0 . This general formulation of constraint-based learned simulation can be trained by backpropagating loss gradients through the solver loop2 . The computational budget of the forward pass can be varied via the number of solver iterations N . Explicit iterative simulators As a hybrid between forward and constraint-based simulators , we introduced a model which iteratively refines a proposed state update , like in the constraint-based approach described above , but using an explicit function to directly output a δY at each iteration , rather than solving a constraint function ( see Figure 1c ) . See Section 4.3 for details .

Summary: This manuscript proposes to learn the numerical solutions to Lagrangian physical simulation with a constraint-based inference method on graph neural networks. This method involves an iterative update during inference and thus enables test-time dynamical correction. The contributions are: (1) this manuscript builds a scalar predictor to indicate how well the constraint is agreed. (2) this manuscript proposes using the graph neural networks as the backbone to deal with a variable length of the physical domain. They also examine the effectiveness with a bunch of experiments including the state prediction experiment on four different environments and multiple ablation studies towards different hyper-parameters.