TransFlowNet: A physics-constrained Transformer framework for spatio-temporal super-resolution of flow simulations

Highlights

• TransFlowNet is a novel physics-constrained deep learning framework.

• TransFlowNet focuses on the spatio-temporal super-resolution of flow simulations.

• TransFlowNet has a newly designed implicit feature encoder based on Swin Transformer.

• Two flow simulations are studied to evaluate the performance of TransFlowNet.

Abstract

We propose TransFlowNet, a novel physics-constrained deep learning framework that focuses on the spatio-temporal super-resolution (STSR) of flow simulations. A key insight is how to combine both statistical and physical properties in the process of an STSR network. Therefore, we elaborately design stacked convolutional layers and Transformer blocks to extract shallow and deep features. Besides, we employ an automatic differentiation process for solving the physical constraints. Unlike existing physics-informed solutions, our method is able to solve flow processes with uncertain boundary and initial conditions. Based on two typical flow simulations, we compare our method with the state-of-the-art physics-constrained model and a CNN-based baseline model. Our framework outperforms these methods in both PSNR and SSIM metrics and produces visually the best results. We also test our method at the large spatio-temporal scale, and the high-resolution outputs present stable performances.

Introduction

It is a fundamental task in science and engineering to model the dynamical processes of high-resolution spatio-temporal data on continuous scales of space and time (e.g., climate system [1], [2], physical oceanography [3], fluid mechanics  [4], [5]). This work plays an important role in understanding and reasoning about the natural physical world. Scientists attempt to model these processes in a principled way by conservation laws and physical laws, i.e., partial differential equations (PDEs), and further simulate these behaviors with computational devices. To this end, computational fluid dynamics (CFD [6]) has become a tremendous tool for solving various PDEs numerically in the last decades. However, conventional CFD schemes are based purely on physical properties: known physical laws of PDEs are solved step by step through mesh-based numerical discretization or integration schemes, such as the finite difference method (FDM) and the finite element method (FEM) [7]. These methods are time-consuming and require significant computational resources and expertise [8]. In recent times, deep learning methods have demonstrated great success in accelerating the processing of compute-intensive scientific problems [9]. As a result, researchers start to replace part of the solving process of the CFD scheme with deep learning methods to improve the computational efficiency of numerical results and reduce computational costs.

In this paper, we focus on one such computational problem: how to reconstruct spatio-temporal high-resolution results with fine-scale features given the spatio-temporal low-resolution physical flow simulation volume data (which can be produced by conventional CFD schemes) and governing PDEs. This procedure is referred as Physics-constrained STSR.

At first glance, this problem seems can be solved directly using general video super-resolution methods such as [10], [11]. But most of them are data-driven methods that have not yet been proven to be successful in accurately producing the physical properties of flow systems [12]. In contrast, a method called physical-informed neural networks (PINNs) and its variants [13], [14], [15], [16], [17] incorporate physical constraints into deep neural networks by taking advantage of the automatic differentiation. The application of PINNs in engineering problems is also an active research topic [18], [19]. PINNs accept governing PDEs and a set of boundary and initial conditions as input and produce mesh-free flow feature reconstruction results. However, the training process of PINNs is completely different from the video super-resolution method. It does not require low-resolution images as input, but boundary and initial conditions are necessary. Otherwise, the training loss may not converge. Therefore, the network needs to be retrained once the boundary and initial conditions change. This significant limitation is not sufficient for STSR of flow dynamics with unknown boundary and initial conditions.

Our work has three challenges generally. To extract the inherent statistical correlations between paired low-resolution and high-resolution flow data, the neural network must first be intricately designed. Second, physical constraints must be added to the deep learning framework in order for it to adhere to the physical laws dictated by the PDEs. Last but not least, an effective STSR model should be able to handle flow dynamics processes with unknown boundary and initial conditions and effectively represent high-resolution outputs, for example, by showing a high match on widely used metrics and scaling stably to large spatio-temporal scales.

We tackle these challenges through a novel STSR flow reconstruction network based on Transformer [20] architecture, called TransFlowNet. Our network consists of an encoder (Feature Extraction Network) and a decoder (Physics-constrained Network). For the encoder, we intricately design stacked convolutional and Transformer blocks as shallow and deep features extractors, respectively, with skip connections. Transformer is a new backbone network architecture that, in comparison to Convolutional Neural Network (CNN), has a larger and broader perceptual field for better extraction of deep-level features (i.e, Vision Transformer [21], Swin Transformer [22]). To resolve the physical constraints in the decoder, we use a Multilayer Perception (MLP) with a differentiable nature. A hybrid loss function is also used, which considers both statistical difference and physical residue.

Specifically, our main technical contributions are as follows:

• We present TransFlowNet, a novel physics-constrained deep learning framework, which focuses on the STSR of flow simulations with unknown boundary and initial conditions in physical systems.

• We propose a newly designed implicit feature encoder based on Swin Transformer, which gives better results than the CNN-based models and shows stable performances at the large spatio-temporal scale.

• We conduct experiments and evaluations on the Rayleigh–Bénard convection problem and shallow water equations (SWEs), that show our method has the potential ability in accelerating the processing of the compute-intensive CFD simulation problems.

All experimental results can be found in Section 4. We verify the effectiveness of generating high-resolution data by our method using the PSNR and SSIM metrics. The experimental results demonstrate that TransFlowNet successfully addresses the three challenges mentioned above. First, the implicit feature encoder prompts TransFlowNet to extract the effective deep feature on both spatial and temporal scales. Second, the physics-constrained decoder makes the high-resolution features consistent with the real physical process as much as possible. Finally, our method achieves excellent and stable performances for the spatio-temporal modeling of flow simulations with uncertain boundary and initial conditions. We compare our method with the state-of-the-art physics-constrained model and a CNN-based baseline model using two different flow simulations. Our framework outperforms these methods in both PSNR and SSIM metrics and produces visually the best results (Sections 4.1 Rayleigh–Bénard convection problem, 4.2 SWEs). We also demonstrate the necessity of some important modules in TransFlowNet by the ablation study (Section 4.3). At last, we test our method at the large spatio-temporal scale, and the high-resolution outputs present stable performances (Section 4.4).