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Traditional numerical methods, such as the finite element method and finite volume method, adress partial differential equations (PDEs) by discretizing them into algebraic equations and solving these iteratively. However, this process is often computationally expensive and time-consuming. An alternative approach involves transforming PDEs into integral equations and solving them using Green's functions, which provide analytical solutions. Nevertheless, deriving Green's functions analytically is a challenging and non-trivial task, particularly for complex systems. In this study, we introduce a novel framework, termed GreensONet, which is constructed based on the strucutre of deep operator networks (DeepONet) to learn embedded Green's functions and solve PDEs via Green's integral formulation. Specifically, the Trunk Net within GreensONet is designed to approximate the unknown Green's functions of the system, while the Branch Net are utilized to approximate the auxiliary gradients of the Green's function. These outputs are subsequently employed to perform surface integrals and volume integrals, incorporating user-defined boundary conditions and source terms, respectively. The effectiveness of the proposed framework is demonstrated on three types of PDEs in bounded domains: 3D heat conduction equations, reaction-diffusion equations, and Stokes equations. Comparative results in these cases demonstrate that GreenONet's accuracy and generalization ability surpass those of existing methods, including Physics-Informed Neural Networks (PINN), DeepONet, Physics-Informed DeepONet (PI-DeepONet), and Fourier Neural Operators (FNO)., Comment: no comments

A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these solution generators after training on high-fidelity ground truth data (e.g. numerical simulations). However, in order to generalize well to unseen spatial domains, neural operators must be trained on an extensive amount of geometrically varying data samples that may not be feasible to acquire or simulate in certain contexts (e.g., patient-specific medical data, large-scale computationally intensive simulations.) We propose that in order to learn a PDE solution operator that can generalize across multiple domains without needing to sample enough data expressive enough for all possible geometries, we can train instead a latent neural operator on just a few ground truth solution fields diffeomorphically mapped from different geometric/spatial domains to a fixed reference configuration. Furthermore, the form of the solutions is dependent on the choice of mapping to and from the reference domain. We emphasize that preserving properties of the differential operator when constructing these mappings can significantly reduce the data requirement for achieving an accurate model due to the regularity of the solution fields that the latent neural operator is training on. We provide motivating numerical experimentation that demonstrates an extreme case of this consideration by exploiting the conformal invariance of the Laplacian

With growing investigations into solving partial differential equations by physics-informed neural networks (PINNs), more accurate and efficient PINNs are required to meet the practical demands of scientific computing. One bottleneck of current PINNs is computing the high-order derivatives via automatic differentiation which often necessitates substantial computing resources. In this paper, we focus on removing the automatic differentiation of the spatial derivatives and propose a spectral-based neural network that substitutes the differential operator with a multiplication. Compared to the PINNs, our approach requires lower memory and shorter training time. Thanks to the exponential convergence of the spectral basis, our approach is more accurate. Moreover, to handle the different situations between physics domain and spectral domain, we provide two strategies to train networks by their spectral information. Through a series of comprehensive experiments, We validate the aforementioned merits of our proposed network.

Background: Invasive coronary arteriography (ICA) is recognized as the gold standard for diagnosing cardiovascular diseases, including unstable angina (UA). The challenge lies in determining the optimal timing for ICA in UA patients, balancing the need for revascularization in high-risk patients against the potential complications in low-risk ones. Unlike myocardial infarction, UA does not have specific indicators like ST-segment deviation or cardiac enzymes, making risk assessment complex. Objectives: Our study aims to enhance the early risk assessment for UA patients by utilizing machine learning algorithms. These algorithms can potentially identify patients who would benefit most from ICA by analyzing less specific yet related indicators that are challenging for human physicians to interpret. Methods: We collected data from 640 UA patients at Shanghai General Hospital, including medical history and electrocardiograms (ECG). Machine learning algorithms were trained using multi-modal demographic characteristics including clinical risk factors, symptoms, biomarker levels, and ECG features extracted by pre-trained neural networks. The goal was to stratify patients based on their revascularization risk. Additionally, we translated our models into applicable and explainable look-up tables through discretization for practical clinical use. Results: The study achieved an Area Under the Curve (AUC) of $0.719 \pm 0.065$ in risk stratification, significantly surpassing the widely adopted GRACE score's AUC of $0.579 \pm 0.044$. Conclusions: The results suggest that machine learning can provide superior risk stratification for UA patients. This improved stratification could help in balancing the risks, costs, and complications associated with ICA, indicating a potential shift in clinical assessment practices for unstable angina.

The Green's function, serving as a kernel function that delineates the interaction relationships of physical quantities within a field, holds significant research implications across various disciplines. It forms the foundational basis for the renowned Biot-Savart formula in fluid dynamics, the theoretical solution of the pressure Poisson equation, and et al. Despite their importance, the theoretical derivation of the Green's function is both time-consuming and labor-intensive. In this study, we employed DISCOVER, an advanced symbolic regression method leveraging symbolic binary trees and reinforcement learning, to identify unknown Green's functions for several elementary partial differential operators, including Laplace operators, Helmholtz operators, and second-order differential operators with jump conditions. The Laplace and Helmholtz operators are particularly vital for resolving the pressure Poisson equation, while second-order differential operators with jump conditions are essential for analyzing multiphase flows and shock waves. By incorporating physical hard constraints, specifically symmetry properties inherent to these self-adjoint operators, we significantly enhanced the performance of the DISCOVER framework, potentially doubling its efficacy. Notably, the Green's functions discovered for the Laplace and Helmholtz operators precisely matched the true Green's functions. Furthermore, for operators without known exact Green's functions, such as the periodic Helmholtz operator and second-order differential operators with jump conditions, we identified potential Green's functions with solution error on the order of 10^(-10). This application of symbolic regression to the discovery of Green's functions represents a pivotal advancement in leveraging artificial intelligence to accelerate scientific discoveries, particularly in fluid dynamics and related fields., Comment: arXiv admin note: text overlap with arXiv:2210.16016 by other authors

The recent mainstream reinforcement learning control for quadruped robots often relies on privileged information, demanding meticulous selection and precise estimation, thereby imposing constraints on the development process. This work proposes a Self-learning Latent Representation (SLR) method, which achieves high-performance control policy learning without the need for privileged information. To enhance the credibility of the proposed method's evaluation, SLR was directly compared with state-of-the-art algorithms using their open-source code repositories and original configuration parameters. Remarkably, SLR surpasses the performance of previous methods using only limited proprioceptive data, demonstrating significant potential for future applications. Ultimately, the trained policy and encoder empower the quadruped robot to traverse various challenging terrains. Videos of our results can be found on our website: https://11chens.github.io/SLR/

The rapid expansion of wind power worldwide underscores the critical significance of engineering-focused analytical wake models in both the design and operation of wind farms. These theoretically-derived ana lytical wake models have limited predictive capabilities, particularly in the near-wake region close to the turbine rotor, due to assumptions that do not hold. Knowledge discovery methods can bridge these gaps by extracting insights, adjusting for theoretical assumptions, and developing accurate models for physical processes. In this study, we introduce a genetic symbolic regression (SR) algorithm to discover an interpretable mathematical expression for the mean velocity deficit throughout the wake, a previously unavailable insight. By incorporating a double Gaussian distribution into the SR algorithm as domain knowledge and designing a hierarchical equation structure, the search space is reduced, thus efficiently finding a concise, physically informed, and robust wake model. The proposed mathematical expression (equation) can predict the wake velocity deficit at any location in the full-wake region with high precision and stability. The model's effectiveness and practicality are validated through experimental data and high-fidelity numerical simulations.

Helical magnetohydrodynamic turbulence with Hall effects is ubiquitous in heliophysics and plasma physics, such as star formation and solar activities, and its intrinsic mechanisms are still not clearly explained. Direct numerical simulations reveal that when the forcing scale is comparable to the ion inertial scale, Hall effects induce remarkable cross helicity. It then suppresses the inverse cascade efficiency, leading to the accumulation of large-scale magnetic energy and helicity. The process is accompanied by the breaking of current sheets via filaments along magnetic fields. Using the Ulysses data, the numerical findings are separately confirmed. These results suggest a novel mechanism wherein small-scale Hall effects could strongly affect large-scale magnetic fields through cross helicity., Comment: 5 figures, 6 pages

High-entropy alloys (HEAs) refer to alloys composed of five or more elements in equal or near-equal amounts or in an atomic concentration range of 5 to 35 atomic percent (at%). Different elemental ratios will affect the microstructures of HEAs and provide them with unique properties. Based on past research, HEAs have exhibited superior performance, relative to most conventional alloys, with respect to many properties, such as strength, toughness, corrosion resistance, magnetic behavior, etc. Among them, fatigue behavior has been a topic of focus, due to its importance in industrial applications. In this article, we summarized the research progress in the HEA-fatigue behavior in the past ten years, including experimental results and theoretical studies in subdivisions, such as high-cycle fatigue, low-cycle fatigue, fatigue-crack growth, fatigue mechanisms, etc. The influence of the processing and test methods on HEAs is described. The accuracy of several commonly used prediction models is also outlined. Finally, unresolved issues and suggestions on the direction of future research efforts are presented., Comment: Submitted to Progress in Material Science on January 8, 2024. Resubmitted to Progress in Material Science on November 10, 2024

Embedding physical knowledge into neural network (NN) training has been a hot topic. However, when facing the complex real-world, most of the existing methods still strongly rely on the quantity and quality of observation data. Furthermore, the neural networks often struggle to converge when the solution to the real equation is very complex. Inspired by large eddy simulation in computational fluid dynamics, we propose an improved method based on filtering. We analyzed the causes of the difficulties in physics informed machine learning, and proposed a surrogate constraint (filtered PDE, FPDE in short) of the original physical equations to reduce the influence of noisy and sparse observation data. In the noise and sparsity experiment, the proposed FPDE models (which are optimized by FPDE constraints) have better robustness than the conventional PDE models. Experiments demonstrate that the FPDE model can obtain the same quality solution with 100% higher noise and 12% quantity of observation data of the baseline. Besides, two groups of real measurement data are used to show the FPDE improvements in real cases. The final results show that FPDE still gives more physically reasonable solutions when facing the incomplete equation problem and the extremely sparse and high-noise conditions. For combining real-world experiment data into physics-informed training, the proposed FPDE constraint is useful and performs well in two real-world experiments: modeling the blood velocity in vessels and cell migration in scratches.

The wall cooling effect on the spectra and structures of thermodynamic variables are investigated in hypersonic turbulent boundary layers. The density and temperature can be divided into the acoustic and entropic modes based on the Kovasznay decomposition. The intensities of the pressure and the acoustic modes of density and temperature attain the maximum values near the wall, while those of the entropy and the entropic modes of density and temperature achieve their primary peaks near the edge of boundary layer. In the near-wall region, the pressure and the acoustic modes of density and temperature are significantly enhanced when the wall is strongly cooled, which can be attributed to the appearance of the travelling-wave-like alternating positive and negative structures. Moreover, the intensities of the entropy and the entropic modes of density and temperature become stronger near the wall as the wall temperature decreases, due to the appearance of the streaky entropic structures. The streaky entropic structures are mainly caused by the advection effect of the strong positive wall-normal gradient of the mean temperature associated with ejection and sweep events. {It is also found that the profiles of the intensities of the entropy, density and temperature are similar to each other far from the wall, which are mainly due to the reason that the entropic modes are dominant in the fluctuating density and temperature in the far-wall region. The acoustic modes of density and temperature only have significant contributions in the near-wall region., Comment: 38 pages, 35 figures

Extended thermodynamics commonly uses polynomial moments to model non-equilibrium transportation, but faces a crisis due to sub-shocks, which are anomalous discontinuities in gas properties when predicting shock waves. The cause of sub-shocks is still unclear, challenging the validity of extended thermodynamics. This paper reveals, for the first time, that sub-shocks arise from intrinsic limitations of polynomials leading to a discontinuous phase transition. Therefore extended thermodynamics necessitates alternative moments beyond polynomials to avoid sub-shocks., Comment: 5 pages, 4 figures

Localized turbulence is common in geophysical flows, where the roles of rotation and stratification are paramount. In this study, we investigate the evolution of a stratified turbulent cloud under rotation. Recognizing that a turbulent cloud is composed of vortices of varying scales and shapes, we start our investigation with a single eddy using analytical solutions derived from a linearized system. Compared to an eddy under pure rotation, the stratified eddy shows the physical manifestation of a known potential vorticity mode, appearing as a static stable vortex. In addition, the expected shift from inertial waves to inertial-gravity waves is observed. In our numerical simulations of the turbulent cloud, carried out at a constant Rossby number over a range of Froude numbers, stratification causes columnar structures to deviate from vertical alignment. This deviation increases with increasing stratification, slowing the expansion rate of the cloud. The observed characteristics of these columnar structures are consistent with the predictions of linear theory, particularly in their tilt angles and vertical growth rates, suggesting a significant influence of inertial-gravity waves. Using Lagrangian particle tracking, we have identified regions where wave activity dominates over turbulence. In scenarios of milder stratification, these inertial-gravity waves are responsible for a significant energy transfer away from the turbulent cloud, a phenomenon that attenuates with increasing stratification.

The maximal entropy moment method (MEM) is systematic solution of the challenging problem: generating extended hydrodynamic equations valid for both dense and rarefied gases. However, simulating MEM suffers from a computational expensive and ill-conditioned maximal entropy problem. It causes numerical overflow and breakdown when the numerical precision is insufficient, especially for flows like high-speed shock waves. It also prevents modern GPUs from accelerating MEM with their enormous single floating-point precision computation power. This paper aims to stabilize MEM, making it possible to simulating very strong normal shock waves on modern GPUs at single precision. We improve the condition number of the maximal entropy problem by proposing gauge transformations, which moves not only flow fields but also hydrodynamic equations into a more optimal coordinate system. We addressed numerical overflow and breakdown in the maximal entropy problem by employing the canonical form of distribution and a modified Newton optimization method. Moreover, we discovered a counter-intuitive phenomenon that over-refined spatial mesh beyond mean free path degrades the stability of MEM. With these techniques, we accomplished single-precision GPU simulations of high speed shock wave up to Mach 10 utilizing 35 moments MEM, while previous methods only achieved Mach 4 on double-precision., Comment: 56 pages, 8 figures

Data reconstruction of rotating turbulent snapshots is investigated utilizing data-driven tools. This problem is crucial for numerous geophysical applications and fundamental aspects, given the concurrent effects of direct and inverse energy cascades, which lead to non-Gaussian statistics at both large and small scales. Data assimilation also serves as a tool to rank physical features within turbulence, by evaluating the performance of reconstruction in terms of the quality and quantity of the information used. Additionally, benchmarking various reconstruction techniques is essential to assess the trade-off between quantitative supremacy, implementation complexity, and explicability. In this study, we use linear and non-linear tools based on the Proper Orthogonal Decomposition (POD) and Generative Adversarial Network (GAN) for reconstructing rotating turbulence snapshots with spatial damages (inpainting). We focus on accurately reproducing both statistical properties and instantaneous velocity fields. Different gap sizes and gap geometries are investigated in order to assess the importance of coherency and multi-scale properties of the missing information. Surprisingly enough, concerning point-wise reconstruction, the non-linear GAN does not outperform one of the linear POD techniques. On the other hand, supremacy of the GAN approach is shown when the statistical multi-scale properties are compared. Similarly, extreme events in the gap region are better predicted when using GAN. The balance between point-wise error and statistical properties is controlled by the adversarial ratio, which determines the relative importance of the generator and the discriminator in the GAN training. Robustness against the measurement noise is also discussed.

Reservoir computing is a powerful tool for forecasting turbulence because its simple architecture has the computational efficiency to handle large systems. Its implementation, however, often requires full state-vector measurements and knowledge of the system nonlinearities. We use nonlinear projector functions to expand the system measurements to a high dimensional space and then feed them to a reservoir to obtain forecasts. We demonstrate the application of such reservoir computing networks on spatiotemporally chaotic systems, which model several features of turbulence. We show that using radial basis functions as nonlinear projectors enables complex system nonlinearities to be captured robustly even with only partial observations and without knowing the governing equations. Finally, we show that when measurements are sparse or incomplete and noisy, such that even the governing equations become inaccurate, our networks can still produce reasonably accurate forecasts, thus paving the way towards model-free forecasting of practical turbulent systems., Comment: 15 pages, 7 figures, currently submitted to neural networks

This paper pioneers the direct numerical simulation (DNS) and physical analysis in supersonic three-temperature carbon dioxide (CO2) turbulent channel flow. CO2 is a linear and symmetric triatomic molecular, with the thermal non-equilibrium three-temperature effects arising from the interactions among translational, rotational and vibrational modes under room temperature. Thus, the rotational and vibrational modes of CO2 are addressed. Thermal non-equilibrium effect of CO2 has been modeled in an extended three-temperature BGK-type model, with the calibrated translational, rotational and vibrational relaxation time. To solve the extended BGK-type equation accurately and robustly, non-equilibrium high-accuracy gas-kinetic scheme is proposed within the well-established two-stage fourth-order framework. Compared with the one-temperature supersonic turbulent channel flow, supersonic three-temperature CO2 turbulence enlarges the ensemble heat transfer of the wall by approximate 20%, and slightly decreases the ensemble frictional force. The ensemble density and temperature fields are greatly affected, and there is little change in Van Driest transformation of streamwise velocity. The thermal non-equilibrium three-temperature effects of CO2 also suppress the peak of normalized root-mean-square of density and temperature, normalized turbulent intensities and Reynolds stress. The vibrational modes of CO2 behave quite differently with rotational and translational modes. Compared with the vibrational temperature fields, the rotational temperature fields have the higher similarity with translational temperature fields, especially in temperature amplitude. Current thermal non-equilibrium models, high-accuracy DNS and physical analysis in supersonic CO2 turbulent flow can act as the benchmark for the long-term applicability of compressible CO2 turbulence., Comment: Carbon dioxide flow, Vibrational modes, Three-temperature effects, Supersonic turbulent channel flows

The breakup of a spherical droplet in a decaying homogeneous isotropic turbulence is studied by solving the Cahn-Hilliard-Navier-Stokes equations, using the discrete unified gas kinetic scheme combined with the free-energy-based phase-field model. We focus on the combined effects of turbulence and surface tension on the breakup process by assuming that the two fluid phases have the same density and same viscosity. The key physical parameters of the system include the volume fraction (6.54%), the initial Weber number (21.7), and the initial Taylor microscale Reynolds number (58). Three distinct stages of droplet evolution are identified, namely, the deformation stage when the initially spherical droplet evolves into an irregular geometric shape with complex structures, the breakup stage when many daughter droplets are formed, and the restoration stage when the droplets relax towards spherical shape. These three stages are analyzed systematically from several perspectives: (1) a geometric perspective concerning the maximum equivalent diameter, the total number of droplets, total interface area, and probability distribution of droplet diameters, (2) a dynamic perspective concerning the evolution of local velocity and vorticity at the fluid-fluid interface, (3) a global perspective concerning the evolution of average kinetic energy / dissipation rate and their Fourier spectra, (4) spherical harmonics based energetics concerning simultaneous transfer of kinetic energy across different length scales and different radii relative to initial droplet center, and (5) the time evolution of global kinetic energy and free energy of the system.