Title: Constraint-based graph network simulator. Abstract: 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.