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

Flame prediction using deep learning technology could promote the research and development of flame propagation in scramjet combustors. A data-driven prediction model is proposed to effectively predict a future flame based on the flame sequence at the previous moments. A convolutional neural network is used to construct the prediction model, and the network training is performed using an experimental dataset. Ground experiments are conducted in a scramjet combustor using different equivalence ratio variation laws, and the flame evolution in the experiments is recorded and processed into a dataset. The flame prediction accuracy of the proposed model under different equivalence ratio variation laws is analyzed in detail. Moreover, both subjective and objective analysis results show that the flame prediction well agrees with the experimental result, and the flame boundary and area are accurately predicted to a certain extent. The influence of the prediction span on the flame prediction accuracy is also discussed.

Research on flame prediction in a scramjet combustor using data-driven model

Time- and rate-dependent material functions in non-Newtonian fluids in response to different deformation fields pose a challenge in integrating different constitutive models into conventional computational fluid dynamic platforms. Considering their relevance in many industrial and natural settings alike, robust data-driven frameworks that enable accurate modeling of these complex fluids are of great interest. The main goal is to solve the coupled Partial Differential Equations (PDEs) consisting of the constitutive equations that relate the shear stress to the deformation and fully capture the behavior of the fluid under various flow protocols with different boundary conditions. In this work, we present non-Newtonian physics-informed neural networks (nn-PINNs) for solving systems of coupled PDEs adopted for complex fluid flow modeling. The proposed nn-PINN method is employed to solve the constitutive models in conjunction with conservation of mass and momentum by benefiting from Automatic Differentiation (AD) in neural networks, hence avoiding the mesh generation step. nn-PINNs are tested for a number of different complex fluids with different constitutive models and for several flow protocols. These include a range of Generalized Newtonian Fluid (GNF) empirical constitutive models, as well as some phenomenological models with memory effects and thixotropic timescales. nn-PINNs are found to obtain the correct solution of complex fluids in spatiotemporal domains with good accuracy compared to the ground truth solution. We also present applications of nn-PINNs for complex fluid modeling problems with unknown boundary conditions on the surface, and show that our approach can successfully recover the velocity and stress fields across the domain, including the boundaries, given some sparse velocity measurements. 

nn-PINNs: Non-Newtonian physics-informed neural networks for complex fluid modeling

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

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.

Ruchit Doshi

Papers

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

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