Surrogate Modeling for Fluid Flows Based on Physics-Constrained Deep Learning Without Simulation Data (1906.02382v2)

Abstract: Numerical simulations on fluid dynamics problems primarily rely on spatially or/and temporally discretization of the governing equation into the finite-dimensional algebraic system solved by computers. Due to complicated nature of the physics and geometry, such process can be computational prohibitive for most real-time applications and many-query analyses. Therefore, developing a cost-effective surrogate model is of great practical significance. Deep learning (DL) has shown new promises for surrogate modeling due to its capability of handling strong nonlinearity and high dimensionality. However, the off-the-shelf DL architectures fail to operate when the data becomes sparse. Unfortunately, data is often insufficient in most parametric fluid dynamics problems since each data point in the parameter space requires an expensive numerical simulation based on the first principle, e.g., Naiver--Stokes equations. In this paper, we provide a physics-constrained DL approach for surrogate modeling of fluid flows without relying on any simulation data. Specifically, a structured deep neural network (DNN) architecture is devised to enforce the initial and boundary conditions, and the governing partial differential equations are incorporated into the loss of the DNN to drive the training. Numerical experiments are conducted on a number of internal flows relevant to hemodynamics applications, and the forward propagation of uncertainties in fluid properties and domain geometry is studied as well. The results show excellent agreement on the flow field and forward-propagated uncertainties between the DL surrogate approximations and the first-principle numerical simulations.

• The paper introduces a label-free, physics-constrained deep learning method to model fluid flows, bypassing traditional CFD data requirements.
• It employs structured DNN architectures that integrate initial and hard boundary conditions using automatic differentiation to enforce Navier-Stokes equations.
• The approach achieves significant computational cost reductions and accurate predictions in challenging scenarios such as pipe and aneurysmal flows.

Surrogate Modeling for Fluid Flows via Physics-Constrained Deep Learning

The paper presents a paper on surrogate modeling for fluid flows by leveraging deep learning (DL) models devoid of simulation data in the training phase. Traditional methods for fluid dynamics simulations, particularly computational fluid dynamics (CFD), require extensive computational resources due to the complexity of solving multi-scale partial differential equations such as the Navier-Stokes equations. Moreover, generating accurate meshes for intricate geometries further complicates CFD, limiting its utility in real-time applications and scenarios necessitating multiple queries like optimization and uncertainty quantification. Therefore, cost-effective surrogate modeling has significant scientific and practical interest.

The authors introduce a physics-informed deep learning approach designed to construct surrogate models for fluid flows, which circumvents the reliance on costly CFD-generated training data. This novel approach constructs a structured deep neural network (DNN) to incorporate both initial and boundary conditions directly into the network architecture. Moreover, the training of the DNN is guided by enforcing the governing physics—a strategy that aligns with the principles of physics-informed neural networks (PINNs) but extends this idea further by relying entirely on label-free configurations for surrogate modeling.

Methodology and Implementation

The core of the paper describes using DNNs to capture the solutions of parametric Navier-Stokes equations. Unlike conventional surrogate modeling approaches, where simulation data is pivotal for DNN training, the method here uses perturbation strategies that integrate physics constraints directly into the training process. This eliminates the need for traditional data-labeling processes.

Enforcing hard constraints directly within the network’s construction, they employ a structured-network design integrating boundary conditions—a methodological choice that distinctly sets apart their approach—contrasting with off-the-shelf deep learning models that typically impose these conditions as additional loss penalties. Moreover, the residuals of the equations themselves contribute to the training process, facilitated by automatic differentiation capabilities within modern DL frameworks such as TensorFlow and PyTorch.

Numerical Results and Performance

The numerical experiments leverage domain-standard test cases pertinent to hemodynamics, including scenarios such as pipe, stenotic, and aneurysmal flows. These cases evaluate the surrogate model's ability to generalize across different viscosity and geometry parameterizations.

Results indicate that the DNN successfully learns these fluid dynamics scenarios with significant accuracy, even capturing complex non-linear interactions in challenging geometries. Additionally, the method is demonstrated to offer substantial computational cost reductions, achieving faster uncertainty quantification relative to traditional CFD simulations.

Notably, an empirical observation highlights that soft-constrained boundary condition enforcement could lead to suboptimal solutions, emphasizing the importance of enforcing boundary conditions as hard constraints for ensuring unique and accurate solutions.

Implications and Future Developments

The implications of label-free DNN methods are noteworthy, suggesting a pathway for deploying surrogate models in real-time applications where speed is critical and computational resources are limited. The advancements in physics-constrained learning offer promise for tackling high-dimensional, parameter-varying systems in engineering and applied physics fields.

Future work could explore extending these frameworks to more complex 3D fluid dynamics systems and hybrid modeling paradigms that partially integrate simulation data to refine and validate surrogate models in scenarios where partial data accessibility exists. Moreover, enhancing the handling of higher-dimensional problems and exploring the scalability of these methods across diverse computational architectures remain fertile grounds for continued exploration.

Overall, the proposed method delineates a practical avenue for overcoming traditional surrogate modeling challenges, empowering more efficient multi-query realizations integral to modern engineering challenges.