Physics-informed neural networks for high-speed flows

sented in these previous studies. Because the linearized Euler equations in conservative form are used in the present work, new source terms have to be derived for the conservative set of equations. A formal derivation of source terms for the linearized Euler equations in conservative form is presented, and the system of equations is compared to those of Goldstein and Bailly et al. Simplified versions of the derived source terms are also developed. To test the derived source terms, a direct simulation of a forced two-dimensional mixing layer is carried out. The direct simulation provides the mean flow and source field for the inhomogeneous linearized Euler equations. The solutions to the linearized Euler equations with source terms are compared to the solution of the direct simulation and show good agreement. All simulations are performed using Tam and Webb’s fourth-order dispersion-relation-preserving scheme and a four-step fourth-order Runge‐Kutta time-marching technique. Artificial selective damping introduced through the numerical scheme is used to avoid spurious waves. Absorbing boundary conditions based on characteristic variables are used at the free boundaries and a buffer layer is added at the outflow.

The present work studies upwind schemes applied to the solution of aeronautical and aerospace problems. The Harten, the Frink, Parikh and Pirzadeh, the Liou and Steffen and the Radespiel and Kroll algorithms, all first order accurate in space, are studied. The Euler equations in conservative form, employing a finite volume formulation and a structured spatial discretization, in the two-dimensional space, are solved. A time splitting method and a Runge-Kutta method of five stages are used to perform the time march of the numerical schemes. The steady state physical problems of the supersonic flow along a ramp and around a blunt body configuration are studied. All algorithms are accelerated to the steady state solution using a spatially variable time step. This technique has proved excellent gains in terms of convergence ratio as reported in Maciel. The results have demonstrated that the Liou and Steffen scheme has presented the most critical solutions, in both example-cases, in relation to the others schemes and a more accurate solution, in terms of the determination of the stagnation pressure in the blunt body case, than the Harten and the Radespiel and Kroll schemes. In the ramp problem, the Harten scheme predicts the best pressure distribution along the ramp wall in comparison with theoretical results. In the blunt body problem, the Liou and Steffen scheme presents the highest value of Cp at the configuration nose in relation to the other schemes. Values of cL and cD have been accurately predicted by all schemes, except by the Harten scheme.

Numerical simulation of the instability of an inviscid liquid jet in a coflowing compressible airstream

Investigations of flutter in transonic turbine cascades have shown that the movement of unsteady normal shocks has an important effect on the excitation of blades. In order to predict this phenomenon correctly, detailed studies concerning the response of unsteady blade pressures versus different parameters of an oscillating shock wave should be performed, if possible isolated from other flow effects in cascades. In the present investigation the correlation between an oscillating normal shock wave and the response of wall-mounted time-dependent pressure transducers was studied experimentally in a nozzle with fluctuating back pressure. Excitation frequencies between 0 Hz and 180 Hz were investigated. For the measurements, various measuring techniques were employed. The determination of the unsteady shock position was made by a line scan camera using the Schlieren flow visualization technique. This allowed the simultaneous use of unsteady pressure transducers to evaluate the behavior of the pressure under the moving shock. A numerical code, based on the fully unsteady Euler equations in conservative form, was developed to simulate the behavior of the shock and the pressures. The main results of this work were: (1) The boundary layer over an unsteady pressure transducer has a quasi-steady behavior with respect to the phase lag. The pressure amplitude depends on the frequency of the back pressure. (2) For the geometry investigated the shock amplitude decreased with increasing excitation frequency. (3) The pressure transducer sensed the arriving shock before the shock had reached the position of the pressure transducer. (4) The computed unsteady phenomena agree well with the results of the measurements.

Physics-Informed Neural Networks (PINNs) provide a powerful framework for solving partial differential equations (PDEs) across various scientific and engineering disciplines. Unlike traditional numerical methods, PINNs do not require extensive expertise in the underlying numerical techniques of the target domain, making them accessible for a broader range of users. While PINNs have shown promise for solving forward and inverse problems, their performance relative to classical methods remains a topic of active investigation, particularly in radiative transfer applications. In this study, we assess the application of PINNs to a one-dimensional radiative transfer equation. Through simple test cases, we evaluate their strengths and limitations compared to the discrete ordinate method. Our findings indicate that PINNs can achieve an error as low as 0.3 % within several minutes of computation, whereas traditional solvers are significantly faster. However, the ease of implementation and the ability of PINNs to address inverse and high-dimensional problems highlight their potential as a complementary approach, particularly for complex scenarios.

A physics-informed neural network (PINN) uses physics-augmented loss functions, e.g., incorporating the residual term from governing partial differential equations (PDEs), to ensure its output is consistent with fundamental physics laws. However, it turns out to be difficult to train an accurate PINN model for many problems in practice. In this article, we present a novel perspective of the merits of learning in sinusoidal spaces with PINNs. By analyzing behavior at model initialization, we first show that a PINN of increasing expressiveness induces an initial bias around <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">flat output functions</i> . Notably, this initial solution can be very close to satisfying many physics PDEs, i.e., falling into a local minimum of the PINN loss that only minimizes PDE residuals, while still being far from the true solution that jointly minimizes PDE residuals and the initial and/or boundary conditions. It is difficult for gradient descent optimization to escape from such a local minimum trap, often causing the training to stall. We then prove that the sinusoidal mapping of inputs—in an architecture we label as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sf</i> -PINN—is effective to increase input gradient variability, thus avoiding being trapped in such deceptive local minimum. The level of variability can be effectively modulated to match high-frequency patterns in the problem at hand. A key facet of this article is the comprehensive empirical study that demonstrates the efficacy of learning in sinusoidal spaces with PINNs for a wide range of forward and inverse modeling problems spanning multiple physics domains.

Physics-informed neural networks (PINNs) have been widely used to solve various scientific computing problems. However, large training costs limit PINNs for some real-time applications. Although some works have been proposed to improve the training efficiency of PINNs, few consider the influence of initialization. To this end, we propose a New Reptile initialization based Physics-Informed Neural Network (NRPINN). The original Reptile algorithm is a meta-learning initialization method based on labeled data. PINNs can be trained with less labeled data or even without any labeled data by adding partial differential equations (PDEs) as a penalty term into the loss function. Inspired by this idea, we propose the new Reptile initialization to sample more tasks from the parameterized PDEs and adapt the penalty term of the loss. The new Reptile initialization can acquire initialization parameters from related tasks by supervised, unsupervised, and semi-supervised learning. Then, PINNs with initialization parameters can efficiently solve PDEs. Besides, the new Reptile initialization can also be used for the variants of PINNs. Finally, we demonstrate and verify the NRPINN considering both forward problems, including solving Poisson, Burgers, and Schr\"odinger equations, as well as inverse problems, where unknown parameters in the PDEs are estimated. Experimental results show that the NRPINN training is much faster and achieves higher accuracy than PINNs with other initialization methods.

Identifying optimal architectures of physics-informed neural networks by evolutionary strategy

This paper presents a numerical simulation method developed for separated flow in cascades using the Euler equations and demonstrates the feasibility of this method. MacCormack’s two-steps explicit finite difference scheme is used to discretize the equations in conservation form, and the artificial viscosity is added to the discretized inviscid equations by means of the self-adapted filter technique. The initial separation boundary is given according to simple experimental results. The numerical simulation results including subsonic and transonic turbine cascades flow with or without separation show that the fundamental idea of this numerical method is reasonable and simple. The present study indicates that for solving certain engineering problems it is a simple and effective tool for adding some viscosity corrections to inviscid flow model, especially the current when the Navier-Stokes equations have not been solved very effectively for various complicated flows in turbomachinery.

1-D coupled surface flow and transport equations revisited via the physics-informed neural network approach

In recent years, successful applications of deep learning (DL) have inspired scientists to explore the possibilities of applying DL approaches to modeling scientific problems. Existing studies have revealed that to bake the physics into the DL makes a good supplement to the traditional numerical methods (e.g., finite element, finite volume method) which primarily rely on partial differential equations (PDEs). While DL models are ordinarily trained in a purely data-driven manner, integrating physics into them for simulating scientific problems has several benefits such as (i) physics constraints could regularize the over-parameterized model and hence mitigate the overfitting issue commonly seen in DL; (ii) physics information could also effectively reduce the amount of data needed for training the model; (iii) the resultant physics-informed DL models feature better interpretability and generalizability compared with the conventional black-box model. Furthermore, the powerful expressiveness of the deep network, guaranteed by the universal approximation theorem, makes it a suitable approximator for the solution to a physical system. In this dissertation, we develop two different DL architectures (or approaches), one being continuous scheme-based while the other discrete scheme-based, that leverage physics knowledge for modeling scientific problems. Through comprehensive numerical experiments, we demonstrate the proposed models can be used in solving general PDEs, establishing predictive data-driven models for dynamical systems, identifying the parameters in governing PDEs or even discovering the entire governing PDEs of dynamical systems from scarce and noisy measurements. The continuous model roots on the physics-informed neural network (PINN) which uses a fully connected neural network (FCNN) to approximate the physical fields of a system globally. This model is mesh-free as the residual of the physics (e.g., PDEs, initial/boundary values) is evaluated on a set of collocation points within the physical domain. Several applications including the forward simulations, data-driven simulations and solving inverse problems are presented to exemplify the advantages of PINN over traditional numerical methods. However, the original PINN suffers from inaccurate initial/boundary values due to the weak enforcement of the initial/boundary conditions (I/BCs). To overcome this issue, we propose an improved PINN model by utilizing multiple deep neural networks (DNNs) to construct the solution. Through a DNN pre-trained to represent the initial/boundary values, the approximated solution would obey the given I/BCs forcibly. With several numerical examples, we show that the improved PINN is characterized with much better accuracy on the I/BCs. Though the PINN shows great promise in data-driven modeling and solving inverse problems, some inherent limitations of PINN still exist, such as (i) the solution might lacks fine-scale details due to the global approximation of FCNN; (ii) high computational expense caused by the FCNN it roots on; (iii) incapability to incorporate existing PDE terms (e.g., $\Delta u$) into the network architecture. To overcome these drawbacks, this dissertation also proposes a discrete model - Physics-encoded Recurrent Convolutional Neural Network (PeRCNN) which recurrently updates the solution (or state variable) for time marching. Specifically, it utilizes convolutional neural network (CNN) to capture the spatial patterns of the solution while the recurrent network mimics the forward Euler scheme (or Runge-Kutta scheme) in numerical methods. PeRCNN is a mesh-based and discrete model due to the discretization in time and spatial dimension. The local connectivity of CNN makes PeRCNN more computationally efficient. In addition, the coercive encoding mechanism of physics in PeRCNN, fundamentally different from the PINN relying on soft penalty, ensures the network to rigorously obey given physics. The proposed PeRCNN is successfully applied to solving general PDEs, the data-driven modeling of dynamical systems and the data-driven discovery of governing PDEs from scarce and noisy measurements. Comparisons with the state-of-the-art DL models demonstrate that the proposed PeRCNN possesses excellent computational efficiency, accuracy and generalizability. --Author's abstract

Physics-informed neural networks (PINNs) represent a transformative approach to data models by incorporating known physical laws into neural network training, thereby improving model generalizability, reduce data dependency, and enhance interpretability. Like many other fields in engineering and science, the analysis of physiological signals has been influenced by PINNs in recent years. This manuscript provides a comprehensive overview of PINNs from various perspectives in the physiological signal analysis domain. After exploring the literature and screening the search results, more than 40 key studies in the related domain are selected and categorized based on both practically and theoretically significant perspectives, including input data types, applications, physics-informed models, and neural network architectures. While the advantages of PINNs in tackling forward and inverse problems in physiological signal contexts are highlighted, challenges such as noisy inputs, computational complexity, loss function types, and overall model configuration are discussed, providing insights into future research directions and improvements. This work can serve as a guiding resource for researchers exploring PINNs in biomedical and physiological signal processing, paving the way for more precise, data-efficient, and clinically relevant solutions.

Solving the two‐dimensional Shallow Water Equations (SWE) is a fundamental problem in flood simulation technology. In recent years, physics‐informed neural networks (PINNs) have emerged as a novel methodology for addressing this problem. Given their advantages in parallel computing, potential for data assimilation and parameter calibration, and the rapid advancement of artificial intelligence, it is crucial to investigate both the capabilities and limitations of PINNs. While current research has demonstrated the significant potential of PINNs, many aspects of this new approach remain to be explored. In this paper, we employ PINNs enhanced by dimensional transformation and Neuron‐wise Local Adaptive Activation Function techniques to validate its effectiveness in solving two‐dimensional free surface flow with rainfall on terrain topography. The SWE primarily exist in two forms: the primitive form and the conservative form. Through theoretical analysis and experimental validation, we demonstrate that a hybrid primitive‐conservative form offers superior performance. Additionally, in our cases, we find that incorporating the energy conservation law, specifically the entropy condition, does not improve accuracy and sometimes destabilizes training due to an over‐weighting of the mass conservation law. Furthermore, we have developed an open‐source module on the PINNacle platform for solving SWE using PINNs, which includes over 10 case studies and various equation forms, to promote research and application in this field.

Physics-informed neural networks for understanding shear migration of particles in viscous flow

In this work, physics-informed neural networks are applied to incompressible two-phase flow problems. We investigate the forward problem, where the governing equations are solved from initial and boundary conditions, as well as the inverse problem, where continuous velocity and pressure fields are inferred from scattered-time data on the interface position. We employ a volume of fluid approach, i.e. the auxiliary variable here is the volume fraction of the fluids within each phase. For the forward problem, we solve the two-phase Couette and Poiseuille flow. For the inverse problem, three classical test cases for two-phase modeling are investigated: (i) drop in a shear flow, (ii) oscillating drop and (iii) rising bubble. Data of the interface position over time is generated by numerical simulation. An effective way to distribute spatial training points to fit the interface, i.e. the volume fraction field, and the residual points is proposed. Furthermore, we show that appropriate weighting of losses associated with the residual of the partial differential equations is crucial for successful training. The benefit of using adaptive activation functions is evaluated for both the forward and inverse problem.