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 solvers for partial differential equations take a lot of time to get high resolution results since they have to explore high dimensional grid in function domain. Thus, it is important to interpolate between grids to get high resolution results. In this paper, the authors propose the model that assists PDE solver by correcting residuals in a data-driven way. Specifically, they try to approximate NN to correction function in supervised and unsupervised manners. They also propose a temporal regularization method that smooths behavior of fluid between times. As a result, proposed method can generate high resolution results in efficient way with smoke rising simulation dataset.
Significance
- The field that the paper target is too narrow so that it may not be significant to general machine learning community.
Novelty
- I think this approach is somewhat novel since it introduces new direction in the field where the model assists PDE solver to improve performance with learned correction function.
Clarity
- I think this paper is well written in most places, but the paper is not intended to make it easy for general machine learning researchers to understand and follow the problem thoroughly.
Pros
- The proposed method is orthogonal to other methods such as super-resolution and may be complemented to other existing approaches.
Cons
My main concern is that the evaluation of proposed method is quite limited in the sense that- The method is evaluated with only single dataset.- Only basic model is used to compare to proposed model. What about other baseline models use similar deep learning approaches?
- There should be various PDE problems other than Navier-Stokes to evaluate the general effectiveness of the proposed model.
- I think time comparison/analysis should be provided since the paper targets to get efficient approximation (using NN) in numerical analysis field.
Questions and comments
- I wonder the affect of hyper-parameter in temporal regularization.
- I wonder performance comparison to existing NN based approaches.