IEEE Transactions on Pattern Analysis and Machine Intelligence

Reduced order modeling (ROM) has been widely used to create lower order, computationally inexpensive representations of higher-order dynamical systems. Using these representations, ROMs can efficiently model flow fields while using significantly lesser parameters. Conventional ROMs accomplish this by linearly projecting higher-order manifolds to lower-dimensional space using dimensionality reduction techniques such as proper orthogonal decomposition (POD). In this work, we develop a novel deep learning framework DL-ROM (deep learning—reduced order modeling) to create a neural network capable of non-linear projections to reduced order states. We then use the learned reduced state to efficiently predict future time steps of the simulation using 3D Autoencoder and 3D U-Net-based architectures. Our model DL-ROM can create highly accurate reconstructions from the learned ROM and is thus able to efficiently predict future time steps by temporally traversing in the learned reduced state. All of this is achieved without ground truth supervision or needing to iteratively solve the expensive Navier–Stokes (NS) equations thereby resulting in massive computational savings. To test the effectiveness and performance of our approach, we evaluate our implementation on five different computational fluid dynamics (CFD) datasets using reconstruction performance and computational runtime metrics. DL-ROM can reduce the computational run times of iterative solvers by nearly two orders of magnitude while maintaining an acceptable error threshold.

Deep learning for reduced order modelling and efficient temporal evolution of fluid simulations

Graph neural network-accelerated Lagrangian fluid simulation

Data-driven learning of partial differential equations' solution operators has recently emerged as a promising paradigm for approximating the underlying solutions. The solution operators are usually parameterized by deep learning models that are built upon problem-specific inductive biases. An example is a convolutional or a graph neural network that exploits the local grid structure where functions' values are sampled. The attention mechanism, on the other hand, provides a flexible way to implicitly exploit the patterns within inputs, and furthermore, relationship between arbitrary query locations and inputs. In this work, we present an attention-based framework for data-driven operator learning, which we term Operator Transformer (OFormer). Our framework is built upon self-attention, cross-attention, and a set of point-wise multilayer perceptrons (MLPs), and thus it makes few assumptions on the sampling pattern of the input function or query locations. We show that the proposed framework is competitive on standard benchmark problems and can flexibly be adapted to randomly sampled input.

Transformer for Partial Differential Equations' Operator Learning

A flow field reconstruction algorithm based on deep learning is an effective method to detect the evolution of wave structure in a scramjet combustor, which is of great significance for accurately predicting the operating performance of the scramjet. This paper proposes a dual-branch fusion model based on a multi-head attention mechanism to reconstruct the flow field schlieren image in a supersonic combustor. The proposed model adopts a dual-path fusion mode. One branch is composed of transposed convolution and conventional convolution, forming a symmetrical structure for dimension enhancement and feature extraction. The other is formed by a multi-head attention mechanism and full connection layer in series. It utilizes the same attention mechanism to obtain different sensitive features and enhance the global model perception. The proposed model was trained and tested on a dataset constructed from hydrogen-fueled scramjet experiments in a direct-connected supersonic pulse combustion wind tunnel at Mach 2.5. Numerous experiments prove that the model can effectively reconstruct the basic wave system structure of complex flow field, and it is in good agreement with the original flow field. The average peak signal-to-noise ratio, structural similarity, and average linear correlation coefficient of the proposed model are reached 20.92, 0.602, and 0.943, respectively, which verify the effectiveness of the proposed model in reconstructing the supersonic flow field.

Dual-path flow field reconstruction for a scramjet combustor based on deep learning

Within the domain of Computational Fluid Dynamics, Direct Numerical Simulation (DNS) is used to obtain highly accurate numerical solutions for fluid flows. However, this approach for numerically solving the Navier-Stokes equations is extremely computationally expensive mostly due to the requirement of greatly refined grids. Large Eddy Simulation (LES) presents a more computationally efficient approach for solving fluid flows on lower-resolution (LR) grids but results in an overall reduction in solution fidelity. Through this paper, we introduce a novel deep learning framework SR-DNS Net, which aims to mitigate this inherent trade-off between solution fidelity and computational complexity by leveraging deep learning techniques used in image super-resolution. Using our model, we wish to learn the mapping from a coarser LR solution to a refined high-resolution (HR) DNS solution so as to eliminate the need for performing DNS on highly refined grids. Our model efficiently reconstructs the high-fidelity DNS data from the LES like low-resolution solutions while yielding good reconstruction metrics. Thus our implementation improves the solution accuracy of LR solutions while incurring only a marginal increase in computational cost required for deploying the trained deep learning model.

Deep Learning for Efficient Reconstruction of High-Resolution Turbulent DNS Data.

Reduced Order Modelling (ROM) has been widely used to create lower order, computationally inexpensive representations of higher-order dynamical systems. Using these representations, ROMs can efficiently model flow fields while using significantly lesser parameters. Conventional ROMs accomplish this by linearly projecting higher-order manifolds to lower-dimensional space using dimensionality reduction techniques such as Proper Orthogonal Decomposition (POD). In this work, we develop a novel deep learning framework DL-ROM (Deep Learning - Reduced Order Modelling) to create a neural network capable of non-linear projections to reduced order states. We then use the learned reduced state to efficiently predict future time steps of the simulation using 3D Autoencoder and 3D U-Net based architectures. Our model DL-ROM is able to create highly accurate reconstructions from the learned ROM and is thus able to efficiently predict future time steps by temporally traversing in the learned reduced state. All of this is achieved without ground truth supervision or needing to iteratively solve the expensive Navier-Stokes(NS) equations thereby resulting in massive computational savings. To test the effectiveness and performance of our approach, we evaluate our implementation on five different Computational Fluid Dynamics (CFD) datasets using reconstruction performance and computational runtime metrics. DL-ROM can reduce the computational runtimes of iterative solvers by nearly two orders of magnitude while maintaining an acceptable error threshold.

Pranshu Pant

Papers

Deep learning for reduced order modelling and efficient temporal evolution of fluid simulations

Deep Learning for Efficient Reconstruction of High-Resolution Turbulent DNS Data.

Deep Learning for Reduced Order Modelling and Efficient Temporal Evolution of Fluid Simulations