Recently, domestic media reported that outstanding young researchers in the field of mechanics have published groundbreaking research papers in international top journals, proposing a new method to solve historical challenges in mechanics for the first time. As described in the report, Physics-Informed Neural Networks (PINNs) have recently emerged as a novel and popular method for addressing both forward and inverse problems involving partial differential equations (PDEs). However, ensuring stable training and obtaining accurate results remain challenging in many cases, often attributed to the ill-posedness of PINNs.
Despite this, there is still a lack of in-depth analysis that hinders the progress and application of PINNs in complex engineering problems. Inspired by the ill-posed analysis found in traditional numerical methods, this study establishes a close relationship between the ill-posedness of PINNs and the Jacobian matrix of the PDE system. Specifically, for any given system of partial differential equations, we construct a controlled system that allows for the adjustment of the condition number of the Jacobian matrix while retaining the same solution as the original system.
Numerical experiments demonstrate that as the condition number of the Jacobian matrix decreases, PINNs exhibit faster convergence rates and higher accuracy. Based on this principle and the extension of the controlled system, a general method to mitigate the ill-posedness of PINNs is proposed, successfully simulating the three-dimensional flow around an M6 wing at a Reynolds number of 5000. To our knowledge, this is the first successful simulation of such a complex system using PINNs, providing a promising new technique for addressing industrial complexity issues. The research results also offer valuable insights for guiding the future development of PINNs.
To reduce the computational cost of computational fluid dynamics (CFD) simulations, deep learning and machine learning techniques have garnered considerable attention. This study uses a deep learning model to predict steady-state flow around multiple fixed cylinders and examines the accuracy of the predicted velocity fields. When the arrangement of cylinders is inputted, the deep learning model outputs the x and y components of the velocity field. The accuracy of the predicted velocity fields is investigated, focusing on the velocity distribution of the fluid flow and the fluid forces acting on the cylinders. The model can accurately predict the flow when the number of cylinders equals or is close to the configurations in the training dataset. Extrapolating predictions to a smaller number of cylinders results in errors, which can be explained by the internal dissipation of the fluid. The results of the fluid forces acting on the cylinders indicate that this deep learning model has good generalization performance for systems with a large number of cylinders.
We evaluated the ability of convolutional neural networks (CNNs) to predict velocity fields, particularly in relation to fluid flow around various obstacles within a two-dimensional rectangular channel. Our network architecture is based on the gated residual U-Net template and is trained on velocity fields generated by CFD simulations. We then assess the extent to which our model can accurately and effectively predict velocity fields associated with inlet velocities and obstacle configurations that are not included in our training dataset. Practical applications often require fluid flow predictions in larger and more complex domains containing more obstacles than those used during model training.
To address this issue, we propose a method that decomposes a domain into multiple subdomains, allowing our model to accurately predict fluid flow individually for each subdomain. We then apply smoothness and continuity constraints to reconstruct the velocity field for the entire original domain. This piecewise semi-continuous method is computationally more efficient than alternatives that involve generating CFD datasets necessary for retraining models on larger, spatially complex domains.
We introduce a Local Orientation Vector Field Entropy (LOVE) metric, which quantifies the decorrelation scales of the velocity field in geometries with one or more obstacles, and use it to design a strategy for decomposing complex domains into weakly interacting subsets suitable for our modeling approach. Finally, we evaluate the error propagation across increasingly larger modeling domains.