Physics-informed neural networks for the repercussions of angular velocity of the cone and the free flow of chemically reactive ternary nanofluid

This study focuses on the optimization of a Physics-Informed Neural Network (PINN) to address Partial Differential Equation (PDE) problems associated with fluid flow. Specifically, the stationary, one-dimensional classical Reynolds equation is solved using the PINN. Within the conducted studies, a comparison is made between the solutions obtained using the PINN, the numerical Finite Difference Method (FD), and the analytical solution. We study various scenarios with diverse hyper-parameters such as learning rate, epochs, number of training points, etc., for constructing the neural network to identify the optimal setup. The PINN accurately approximated the solution to the Reynolds equation (up to O(10−2 ). This suggests that PINNs can be used to address diverse problems in fluid dynamics. We proposed a PINN configuration that outperformed the PINN presented in the literature. The finite differences method obtains a better approximation than the PINNs, however, the full potential of the PINNs is yet to be determined, as it can include data from the problem, that finite difference method (FD) can not. Further studies are planned to investigate the capabilities of PINNs.

We propose a new approach to the solution of the wave propagation and full waveform inversions (FWIs) based on a recent advance in deep learning called physics‐informed neural networks (PINNs). In this study, we present an algorithm for PINNs applied to the acoustic wave equation and test the method with both forward models and FWI case studies. These synthetic case studies are designed to explore the ability of PINNs to handle varying degrees of structural complexity using both teleseismic plane waves and seismic point sources. PINNs' meshless formalism allows for a flexible implementation of the wave equation and different types of boundary conditions. For instance, our models demonstrate that PINN automatically satisfies absorbing boundary conditions, a serious computational challenge for common wave propagation solvers. Furthermore, a priori knowledge of the subsurface structure can be seamlessly encoded in PINNs' formulation. We find that the current state‐of‐the‐art PINNs provide good results for the forward model, even though spectral element or finite difference methods are more efficient and accurate. More importantly, our results demonstrate that PINNs yield excellent results for inversions on all cases considered and with limited computational complexity. We discuss the current limitations of the method with complex velocity models as well as strategies to overcome these challenges. Using PINNs as a geophysical inversion solver offers exciting perspectives, not only for the full waveform seismic inversions, but also when dealing with other geophysical datasets (e.g., MT, gravity) as well as joint inversions because of its robust framework and simple implementation.

In recent engineering applications using deep learning, physics-informed neural network (PINN) is a new development as it can exploit the underlying physics of engineering systems. The novelty of PINN lies in the use of partial differential equations (PDE) for the loss function. Most PINNs are implemented using automatic differentiation (AD) for training the PDE loss functions. A lesser well-known study is the use of finite difference method (FDM) as an alternative. Unlike an AD based PINN, an immediate benefit of using a FDM based PINN is low implementation cost. In this paper, we propose the use of finite difference method for estimating the PDE loss functions in PINN. Our work is inspired by computational analysis in electromagnetic systems that traditionally solve Laplace’s equation using successive over-relaxation. In the case of Laplace’s equation, our PINN approach can be seen as taking the Laplacian filter response of the neural network output as the loss function. Thus, the implementation of PINN can be very simple. In our experiments, we tested PINN on Laplace’s equation and Burger’s equation. We showed that using FDM, PINN consistently outperforms non-PINN based deep learning. When comparing to AD based PINNs, we showed that our method is faster to compute as well as on par in terms of error reduction.

Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networks from additional information obtained by employing the physical laws and evaluating them at random points in the space–time domain. Such PIML integrates multimodality and multifidelity data with mathematical models, and implements them using neural networks or graph networks. Here, we review some of the prevailing trends in embedding physics into machine learning, using physics-informed neural networks (PINNs) based primarily on feed-forward neural networks and automatic differentiation. For more complex systems or systems of systems and unstructured data, graph neural networks (GNNs) present some distinct advantages, and here we review how physics-informed learning can be accomplished with GNNs based on graph exterior calculus to construct differential operators; we refer to these architectures as physics-informed graph networks (PIGNs). We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.

In the fields of physics and engineering, it is crucial to understand phase transition dynamics. This field involves fundamental partial differential equations (PDEs) such as the Allen–Cahn, Burgers, and two-dimensional (2D) wave equations. In alloys, the evolution of the phase transition interface is described by the Allen–Cahn equation. Vibrational and wave phenomena during phase transitions are modeled using the Burgers and 2D wave equations. The combination of these equations gives comprehensive information about the dynamic behavior during a phase transition. Numerical modeling methods such as finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) are often applied to solve phase transition problems that involve many partial differential equations (PDEs). However, physical problems can lead to computational complexity, increasing computational costs dramatically. Physics-informed neural networks (PINNs), as new neural network algorithms, can integrate physical law constraints with neural network algorithms to solve partial differential equations (PDEs), providing a new way to solve PDEs in addition to the traditional numerical modeling methods. In this paper, a hard-constraint wide-body PINN (HWPINN) model based on PINN is proposed. This model improves the effectiveness of the approximation by adding a wide-body structure to the approximation neural network part of the PINN architecture. A hard constraint is used in the physically driven part instead of the traditional practice of PINN constituting a residual network with boundary or initial conditions. The high accuracy of HWPINN for solving PDEs is verified through numerical experiments. One-dimensional (1D) Allen–Cahn, one-dimensional Burgers, and two-dimensional wave equation cases are set up for numerical experiments. The properties of the HWPINN model are inferred from the experimental data. The solution predicted by the model is compared with the FDM solution for evaluating the experimental error in the numerical experiments. HWPINN shows great potential for solving the PDE forward problem and provides a new approach for solving PDEs.

Foehn winds are dry, warm winds that are caused by the adiabatic warming of air as it descends down the leeward side of a mountain. This phenomenon has substantial effects on the melting of ice and snow on icy mountains. In this research, we present a physics informed neural network (PINN) model for simulating the transient heating of icy mountains under the influence of Foehn winds. We initially created a mathematical model of this transient heating process. First, we inserted the icy mountain into a wind tunnel to compute the wind velocity profile. Second, we used the velocities to compute the convective heat transfer coefficient along the mountain. Third, we computed the wind temperature taking into account adiabatic cooling and heating. Finally, we computed the temperature profile of the mountain. We solved this mathematical model for 24000 time steps to generate a collocation dataset for supervised training of the PINN. We also developed a custom loss function for unsupervised training of the PINN. After training, the PINN solved the transient heating of three 2D icy mountains in 0.893 sec, 0.875 sec, and 1.026 sec, which were 6351, 6649, and 6883 times faster than the finite difference method, respectively. The PINN solved the transient heating of a 3D icy mountain in 5 sec, which was 3974 times faster than the finite difference method. This model has the potential to better our understanding of Foehn winds and has potential applications in predicting avalanches, erosion, and ice shelf collapse.

Physics Informed Neural Network application on mixing and heat transfer in combined electroosmotic-pressure driven flow

Leveraging physics-informed neural computing for transport simulations of nuclear fusion plasmas

Nonlinear Schrodinger equation (NLSE) has important applications in quantum mechanics, nonlinear optics, plasma physics, condensed matter physics, optical fiber communication and laser system design, and its accurate solution is very important for understanding complex physical phenomena. Here, the traditional finite difference method (FDM), the split-step Fourier method (SSFM) and the physics-informed neural network (PINN) method are studied, aiming to analyze in depth the solving mechanisms of various algorithms, and then realize the efficient and accurate solution of complex NLSE in optical fiber. Initially, the steps, process and results of PINN in solving the NLSE for pulse under the condition of short-distance transmission are described, and the errors of these methods are quantitatively evaluated by comparing them with the errors of PINN, FDM and SSF. On this basis, the key factors affecting the accuracy of NLSE solution for pulse under long-distance transmission are further discussed. Then, the effects of different networks, activation functions, hidden layers and the number of neurons in PINN on the accuracy of NLSE solution are discussed. It is found that selecting a suitable combination of activation functions and network types can significantly reduce the error, and the combination of FNN and tanh activation functions is particularly good. The effectiveness of ensemble learning strategy is also verified, that is, by combining the advantages of traditional numerical methods and PINN, the accuracy of NLSE solution is improved. Finally, the evolution characteristics of Airy pulse with different chirps in fiber and the solution of vector NLSE corresponding to polarization-maintaining fiber are studied by using the above algorithm. This study explores the solving mechanisms of FDM, SSF and PINN in complex NLSE, compares and analyzes the error characteristics of those methods in various transmission scenarios, proposes and verifies the ensemble learning strategy, thus providing a solid theoretical basis for studying pulse transmission dynamics and data-driven simulation.

This study focuses on solving the Allen-Cahn (AC) equation with consideration of varying phase boundary thickness using Physics-Informed Neural Networks (PINN). Traditional numerical methods often exhibit declining computational efficiency and accuracy as problem complexity increases when handling complex phase boundary conditions; PINNs offer a novel approach to addressing such challenges. The study first conducts a detailed derivation of the AC equation and constructs a PINN-based solution model, including three key steps: designing a specific neural network architecture; building a loss function composed of partial differential equation (PDE) residual terms, boundary condition residual terms, and initial condition residual terms; and performing model training. The computational results demonstrate that the established PINN model can effectively solve the AC equation with varying phase boundary thicknesses, accurately revealing the distribution patterns of the order parameter under different phase boundary thicknesses, and adapt to the challenges brought by changes in phase boundary thickness by adjusting the number of iterations to achieve stable and efficient numerical solutions. The phase boundary thickness parameter has a significant impact on order parameter distribution, computational stability, and convergence. Additionally, a discussion is conducted between PINN and the finite difference method. This research provides support for in-depth understanding of the physical connotation of the AC equation and optimization of numerical calculation methods, and also lays a foundation for the application of PINN in solving similar complex problems in fields such as materials science and condensed matter physics. Future work can further expand the application of PINN in high-dimensional phase-field models and multi-physics coupling problems, combine other technologies to improve the accuracy and efficiency of phase-field simulations, and carry out more studies on practical application cases. 本研究聚焦于物理信息神经网络（PINN）求解不同相界厚度变化的Allen-Cahn（AC）方程。传统数值方法在处理复杂相边界条件时，其计算效率与精确度常随问题复杂度而下降；PINN为解决此类问题提供了新途径。本研究首先详细推导了AC方程，进而构建了基于PINN的求解模型，包括设计特定的神经网络架构、构建由偏微分方程残差项、边界条件残差项和初始条件残差项所组成的损失函数，并进行模型训练。计算结果表明，所建立的PINN模型能有效求解不同相界厚度变化的AC方程，准确揭示序参数在不同相界厚度下的分布规律，且能通过改变迭代次数适应相界厚度变化带来的挑战，实现稳定高效的数值求解。此外，将PINN计算结果与有限差分法进行了对比。本研究为深入理解AC方程物理内涵、优化数值计算方法提供支持，也为PINN在材料科学、凝聚态物理等领域解决类似复杂问题奠定基础。未来可进一步拓展PINN在高维相场模型、多物理场耦合问题中的应用，结合其他技术提高相场模拟精确度和效率，并开展更多实际应用案例研究。

Traditional numerical methods, such as finite difference methods (FDM), finite element methods (FEM), and spectral methods, often face meshing challenges and high computational cost for solving nonlinear coupled differential equations. Machine learning techniques, specifically Physics‐informed machine learning, address these obstacles by embedding the governing equations and boundary conditions directly into the learning process, providing a mesh‐free, efficient, and generalizable framework for solving such complex problems. In this regard, the objective of this work is to demonstrate the effectiveness of physics‐informed neural network (PINN) in modeling flow dynamics and heat transfer in the magnetohydrodynamic (MHD) past a horizontal permeable stretching surface with velocity‐slip effect. The governing nonlinear partial differential equations, which describe the flow dynamics and heat transfer phenomena, are transformed into coupled nonlinear ordinary differential equations using similarity transformation variables. This process introduces various physical parameters. A physics‐informed neural network with four hidden layers of 50 neurons each is used to train the model. The input layer takes and physical parameters, and the target outputs are the coefficient of skin friction, the coefficient of heat transfer rate, fluid temperature distribution, and fluid velocity distribution. The accuracy of PINN is validated against a benchmark numerical method, showing excellent agreement. The results indicate that increasing the velocity‐slip parameters reduces the skin‐friction coefficient while enhancing the heat transfer rate. Higher second‐order slip parameters decrease the fluid velocity and temperature, whereas first‐order slip increases both. These findings demonstrate that the PINN effectively captures the flow and thermal characteristics of the MHD problem, offering a reliable alternative for analyzing complex fluid and thermal systems.

Accurate modeling of heat transfer in biological tissues is essential for biomedical applications such as thermal therapies. The Pennes' bioheat equation provides a fundamental framework for understanding thermal dynamics in tissues; however, solving it in non-homogeneous domains remains computationally challenging. In this paper, we employ Physics-Informed Neural Networks (PINNs) to solve the non-dimensionalized Pennes' bioheat equation in a non-homogeneous tissue environment, incorporating variations between muscle and fat through a smooth transition function. To increase stability and efficiency, we introduce a non-dimensionalization process that scales spatial, temporal, and thermal parameters based on characteristic values. A custom PINN framework is implemented to simulate the Pennes' bioheat equation using NVIDIA Modulus, and different neural architectures are evaluated across various collocation densities. Model performance is benchmarked against a Finite Difference Method (FDM) solution by assessing different metrics. Our findings reveal that PINNs demonstrate superior training stability especially with Fourier-based architectures, and reduced loss compared with other architectures. These results show the effectiveness of non-dimensionalization and PINNs in advancing computational models for biomedical simulations and therapeutic applications.Clinical Impact- Optimizing thermal treatments, such as laser-induced thermotherapy (LITT), cryosurgery, and hyperthermia treatment, requires an understanding of heat transmission in biological tissues. This study offers an efficient method for modeling tissue architectures using PINNs to solve the Pennes' bioheat equation in non-homogeneous environments.

This paper presents data-driven simulations of two-phase fluid processes with heat transfer. A Physics-Informed Neural Network (PINN) was applied to capture the behaviour of phase interfaces in two-phase flows and model the hydrodynamics and heat transfer of flow configurations representative of established numerical test cases. The developed PINN approach was trained on simulation data derived from physically based Computational Fluid Dynamics (CFD) simulations with interface capturing. The present study considers fundamental problems, including tracking the rise of a single gas bubble in a denser fluid and exploring the heat transfer in the wake of a bubble rising close to a heated wall. Tracking of a rising bubble phase interface of fluids with disparate properties was performed, revealing a maximum error of only 5.2% at the interface edge and a maximum error of 2.8% at the position of the centre of mass. Inferred (hidden variable) flows are studied in addition to a purely extrapolative inverse isothermal bubble case. When no velocity data was supplied, velocity field predictions remained accurate. Rise of an inferred isothermal bubble with unseen fluid properties was found to produce a maximum mean-squared error of 0.28 and centre of mass error of 1.25%. For the case of the rising bubble with a hot wall, the maximum error in the temperature domain using specified boundary conditions was 6.8%, while the bubble position analysis reveals a maximum positional error of 3.6%. These results demonstrate that PINN is agnostic to geometry and fluid properties when studying the combined effects of convection and buoyancy on two-phase flows for the first time. This work serves as a starting point for PINN in multiphase cases involving heat transfer over a range of geometries. Eventually, PINN will be used in such cases to provide solutions for forward, inverse, and extrapolative cases. Each of which represent a dramatic saving in computational cost compared to traditional CFD.

formula omitted]-PINNs: Physics-informed neural networks on complex geometries

In the present work, single- and segregated-network physics-informed neural network (PINN) architectures are applied to predict momentum, species, and temperature distributions of a dry air humidification problem in a simple two-dimensional (2D) rectangular domain. The created PINN models account for variable fluid properties, species- and heat-diffusion, and convection. Both the mentioned PINN architectures were trained using different hyperparameter settings, such as network width and depth, to find the best-performing configuration. It is shown that the segregated-network PINN approach results in on-average 62% lower losses when compared to the single-network PINN architecture for the given problem. Furthermore, the single-network variant struggled to ensure species mass conservation in different areas of the computational domain, whereas the segregated approach successfully maintained species conservation. The PINN predicted velocity, temperature, and species profiles for a given set of boundary conditions were compared to results generated using OpenFOAM software. Both the single- and segregated-network PINN models produced accurate results for temperature and velocity profiles, with average percentage difference relative to the computational fluid dynamics results of approximately 7.5% for velocity and 8% for temperature. The mean error percentages for the species mass fractions are 9% for the single-network model and 1.5% for the segregated-network approach. To showcase the applicability of PINNs for surrogate modeling of multi-species problems, a parameterized version of the segregated-network PINN is trained that could produce results for different water vapor inlet velocities. The normalized mean absolute percentage errors, relative to the OpenFOAM results, across three predicted cases for velocity and temperature are approximately 7.5% and 2.4% for water vapor mass fraction.