Improving the accuracy of numerical methods remains a central challenge in many disciplines and is especially important for nonlinear simulation problems. A representative example of such problems is fluid flow, which has been thoroughly studied to arrive at efficient simulations of complex flow phenomena. This paper presents a data-driven approach that learns to improve the accuracy of numerical solvers. The proposed method utilizes an advanced numerical scheme with a fine simulation resolution to acquire reference data. We, then, employ a neural network that infers a correction to move a coarse thus quickly obtainable result closer to the reference data. We provide insights into the targeted learning problem with different learning approaches: fully supervised learning methods with a naive and an optimized data acquisition as well as an unsupervised learning method with a differentiable Navier-Stokes solver. While our approach is very general and applicable to arbitrary partial differential equation models, we specifically highlight gains in accuracy for fluid flow simulations. Numerical methods are a central component of many disciplines and widely used for solving a variety of linear and nonlinear problems. One of the long-standing targets is fluid flow, which is renowned for its great diversity and complexity in terms of dynamics. Studies in computational fluid dynamics have focused on numerical simulations for such problems and invested huge efforts in solving spatio-temporal partial differential equations (PDEs) such as the Navier-Stokes equations, which represent the well-established physical model for fluids. Traditional methods typically improve accuracy with fine discretizations both in space and time. While the methods and computing power for numerical simulation have seen advances in recent years, there is still a pressing need for better efficiency and accuracy. For most practical applications of computer simulations, we are still far away from fully resolving all necessary scales of nature around us (Verma et al., 2018; Cummins et al., 2018) . To tackle this problem, we propose a data-driven approach that "assists" a given numerical method to improve its accuracy. To this end, we introduce a first learning-based approach that puts special emphasis on the time dimension. We demonstrate two variants to achieve this goal in the context of fluids: a supervised version with an optimization algorithm for acquisition of temporally constrained correction data and an unsupervised version with a differentiable PDE solver that allows us to autonomously takes into account temporal information when training. We compare advantages and disadvantages of both approaches, and our experiments show that, using our trained models, the simulation accuracy of the given solver can be significantly improved. In all cases, our trained models yield improved dynamics, and the learned assistance function lets a coarse simulation reproduce the behavior of the reference data more closely. In particular, we demonstrate the improvements of our approach over ad-hoc learning approaches. A key benefit of our approach is the gain in performance resulting from our trained models. A correction velocity field for a given input is inferred only on the basic simulation grid, i.e., the lowresolution grid. This inference happens for each solving step to assist the underlying numerical solver and only requires a fixed O(n) cost for n degrees of freedom. For example, for our rising smoke test, a simulation involving the trained NN model took ca. 20 seconds for 1,000 steps whereas its corresponding high-resolution counterpart took ca. 104 seconds to compute. See Table 1 in Appx. A.4 for more details. At training time, the unsupervised learning approach leads to significantly longer training times since each recurrent step of the architecture can require evaluating a complex numerical procedure. A potential remedy and interesting topic for future work would be to use larger timestep size in the solver such that the model could directly learn the correction for longer horizons. As the excellent performance of the unsupervised model for the initial stages of our simulations suggests, this is a very promising avenue. To summarize, we introduced a novel approach that assists numerical methods by learning a correction function to improve the accuracy of the solution. We demonstrated that taking into account the temporal information is crucial for our goal and an optimization step can ensure that sequences of corrections can be learned accurately by a NN model. The model successfully improves the accuracy for previously unseen PDE solves. Additionally, an unsupervised training via a differentiable solver can be employed to further improve the learned correction for time spans that do not strongly exceed the number of steps seen during training. Despite focusing on fluids, we envision that our approach can be applied to a variety of other application domains that involve numerical methods for spatio-temporal problems, from plasma physics Lewis & Miller (1984) (He et al., 2015) . The two network models, which are used for our experiments, are shown in Fig. 9 . The models A and B consist of 405K and 265K training parameters, respectively. Hence, we use the model B for setups with a smaller amount of training data. Our models are implemented using the TensorFlow framework (Abadi et al., 2015) .
We introduce a neural network approach to assist partial differential equation solvers. The authors aim at improving the accuracy of numerical solvers by training a neural network on simulated reference data which corrects the numerical solver.