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 paper presents to simulate physics via a constraint-based approach instead of direct prediction. In particular, the authors first employ GNN taking as input the history positions and dynamics, to capture the interaction between different particles within the system, whose output is considered as the constraint satisfaction scalar. Then, the gradient with respect to the constraint function is applied as the update of the dynamics over a certain number of solver iterations. The experiments are conducted on a variety of challenging physical domains, including simulated ropes, bouncing balls, colliding irregular shapes and splashing fluids.