Physics-informed Neural Networks Research Articles

Physics-Informed Neural Network for Turbulent Flow Reconstruction in Composite Porous-Fluid Systems

Abstract This study explores the implementation of Physics-Informed Neural Networks (PINN) to analyze turbulent flow in composite porous-fluid systems. These systems are composed of a fluid-saturated porous medium and an adjacent fluid, where the flow properties are exchanged across the porous-fluid interface. The PINN model employs a novel approach combining supervised learning and enforces fidelity to flow physics through penalization by the Reynolds-Averaged Navier-Stokes (RANS) equations. Two cases were simulated for this purpose: solid block, i.e., porous media with zero porosity, and porous block with a defined porosity. The effect of providing internal training data on the accuracy of the PINN predictions for prominent flow features including leakage, channeling effect and wake recirculation were investigated. Additionally, L2 norm error, which evaluates the prediction accuracy for flow variables was studied. Furthermore, PINN training time in both cases with internal training data were considered in this study. The results showed that the PINN predictions achieved high accuracy for the prominent flow features compared to the reference RANS data. In addition, second-order internal training data in the wall-normal direction reduced the L2 norm error by 100% for the solid block case, while for the porous block case, providing training data at the porous-fluid interface, increased the prediction accuracy by nearly 40% for second-order statistics. The study elucidates the impact of the internal training data distribution on the PINN training in complex turbulent flow dynamics, underscoring the necessity of turbulent second-order statistics variables in PINN training and an additional velocity gradient treatment to enhance PINN prediction.

Physics-informed neural network estimation of material properties in soft tissue nonlinear biomechanical models

AbstractThe development of biophysical models for clinical applications is rapidly advancing in the research community, thanks to their predictive nature and their ability to assist the interpretation of clinical data. However, high-resolution and accurate multi-physics computational models are computationally expensive and their personalisation involves fine calibration of a large number of parameters, which may be space-dependent, challenging their clinical translation. In this work, we propose a new approach, which relies on the combination of physics-informed neural networks (PINNs) with three-dimensional soft tissue nonlinear biomechanical models, capable of reconstructing displacement fields and estimating heterogeneous patient-specific biophysical properties and secondary variables such as stresses and strains. The proposed learning algorithm encodes information from a limited amount of displacement and, in some cases, strain data, that can be routinely acquired in the clinical setting, and combines it with the physics of the problem, represented by a mathematical model based on partial differential equations, to regularise the problem and improve its convergence properties. Several benchmarks are presented to show the accuracy and robustness of the proposed method with respect to noise and model uncertainty and its great potential to enable the effective identification of patient-specific, heterogeneous physical properties, e.g. tissue stiffness properties. In particular, we demonstrate the capability of PINNs to detect the presence, location and severity of scar tissue, which is beneficial to develop personalised simulation models for disease diagnosis, especially for cardiac applications.

Enhancing weld line visibility prediction in injection molding using physics-informed neural networks

Enhancing weld line visibility prediction in injection molding using physics-informed neural networks

Physics-informed neural networks (P INNs): application categories, trends and impact

Purpose This study aims to explore the evolving field of physics-informed neural networks (PINNs) through an analysis of 996 records retrieved from the Web of Science (WoS) database from 2019 to 2022. Design/methodology/approach WoS database was analyzed for PINNs using an inhouse python code. The author’s collaborations, most contributing institutes, countries and journals were identified. The trends and application categories were also analyzed. Findings The papers were classified into seven key domains: Fluid Dynamics and computational fluid dynamics (CFD); Mechanics and Material Science; Electromagnetism and Wave Propagation; Biomedical Engineering and Biophysics; Quantum Mechanics and Physics; Renewable Energy and Power Systems; and Astrophysics and Cosmology. Fluid Dynamics and CFD emerged as the primary focus, accounting for 69.3% of total publications and witnessing exponential growth from 22 papers in 2019 to 366 in 2022. Mechanics and Material Science followed, with an impressive growth trajectory from 3 to 65 papers within the same period. The study also underscored the rising interest in PINNs across diverse fields such as Biomedical Engineering and Biophysics, and Renewable Energy and Power Systems. Furthermore, the focus of the most active countries within each application category was examined, revealing, for instance, the USA’s significant contribution to Fluid Dynamics and CFD with 319 papers and to Mechanics and Material Science with 66 papers. Originality/value This analysis illuminates the rapidly expanding role of PINNs in tackling complex scientific problems and highlights its potential for future research across diverse domains.

Gravitational duals from equations of state

Holography relates gravitational theories in five dimensions to four-dimensional quantum field theories in flat space. Under this map, the equation of state of the field theory is encoded in the black hole solutions of the gravitational theory. Solving the five-dimensional Einstein’s equations to determine the equation of state is an algorithmic, direct problem. Determining the gravitational theory that gives rise to a prescribed equation of state is a much more challenging, inverse problem. We present a novel approach to solve this problem based on physics-informed neural networks. The resulting algorithm is not only data-driven but also informed by the physics of the Einstein’s equations. We successfully apply it to theories with crossovers, first- and second-order phase transitions.

Incremental Distillation Physics-Informed Neural Network (IDPINN) Accurately Models the Evolution of Optical Solitons

Incremental Distillation Physics-Informed Neural Network (IDPINN) Accurately Models the Evolution of Optical Solitons

Physics-informed neural networks with hard linear equality constraints

Surrogate modeling is used to replace computationally expensive simulations. Neural networks have been widely applied as surrogate models that enable efficient evaluations over complex physical systems. Despite this, neural networks are data-driven models and devoid of any physics. The incorporation of physics into neural networks can improve generalization and data efficiency. The physics-informed neural network (PINN) is an approach to leverage known physical constraints present in the data, but it cannot strictly satisfy them in the predictions. This work proposes a novel physics-informed neural network, KKT-hPINN, which rigorously guarantees hard linear equality constraints through projection layers derived from KKT conditions. Numerical experiments on Aspen models of a continuous stirred-tank reactor (CSTR) unit, an extractive distillation subsystem, and a chemical plant demonstrate that this model can further enhance the prediction accuracy.

A Physics-Informed Neural Network Approach to Augmented Dynamics Visual Servoing of Multirotors.

This article presents a visual servoing strategy that integrates the capabilities of a physics-informed neural network (PINN) to estimate system uncertainties and inaccuracies with a dynamics-centered visual servoing technique for multirotors. The proposed method effectively combines these approaches, eliminating the need for inverse Jacobian calculations to determine multirotor motion by directly relating pixel variations to the multirotor's torque and thrust inputs, while also strengthening the method's robustness through the utilization of the PINN to model and address uncertainties in camera and multirotor parameters, as well as the modeling inaccuracies inherent in the dynamics-centered visual servoing technique. In contrast to existing state-of-the-art data-driven approaches, the proposed PINN approach requires, on average, 65% less labeled data to characterize uncertainties and inaccuracies. To ensure real-time implementation of the visual servoing model, the PINN-learned model is combined with an adaptive horizon monotonically weighted nonlinear model predictive controller (NMPC), capable of processing control efforts at rates 10 times faster than existing Tube MPC and Adaptive MPC strategies. These findings are validated through real-time trajectory tracking experiments, which not only highlight the effectiveness of the proposed approach in approximating modeling inaccuracies but also its capability in handling uncertainties upto 70% in camera parameters.

Assimilating Sparse 3D-PTV Data Of A Pulsed Jet Using Physics-Informed Neural Networks

Phase-resolved volumetric velocity measurements of a pulsed jet are conducted by means of three-dimensional particle tracking velocimetry (PTV). The resulting scattered and relatively sparse data are densely reconstructed by adopting physics-informed neural networks (PINNs), here regularized by the Navier-Stokes equations. It is shown that the assimilation yields a higher spatial resolution, and the process remains robust, even at low particle densities ( 𝑝𝑝𝑝 < 0.001). This is achieved by enforcing compliance with the governing equations, thus leveraging the spatiotemporal evolution of the measured flow field. The results indicate that the PINN reconstructs unambiguously velocity, vorticity and pressure fields with a level of detail not attainable with conventional methods (binning) or more advanced data assimilation techniques (vortex-in-cell). The results of this article support the findings of Clark di Leoni (2023) suggesting that the PINN methodology is inherently suited to the assimilation of PTV data, in particular under conditions of severe sparsity or during experiments with limited control of seeding concentration.

Tomographic Reconstruction Of Combustion Flow Fields From Density Measurements Based On Physics-Informed Neural Networks

Achieving stable, low-emission combustion with green hydrogen is crucial for climate-neutral ground-based power generation in turbomachinery. Lean combustion modes with green hydrogen reduce fuel consumption but increase unsteadiness. Thus, multimodal detection techniques for parameters like density and flow velocity are essential to understand the interconnected behavior of combustion, advection velocity and noise production. We previously introduced a high-speed camera-based laser interferometric vibrometer system to detect thermoacoustic oscillations and record advection velocities, using interferometric detection of density fluctuations and correlation-based velocity estimation. However, this method only estimates integral velocity fields, necessitating the solution of the inverse problem. This is relying on iterative optimization, which is computationally expensive and struggles with high dimensional, noisy or limited data. Physics-Informed Neural Networks offer an innovative and efficient approach to these problems, combining neural network flexibility with physical law constraints to unravel intricate cause-effect relationships. Here an approach for reconstructing local velocity fields from a single projection velocity calculated from integral density data is presented. Using a U-net and model assumptions for coupling local and integral velocity fields, the training minimizes errors between measured integral input fields and the predicted local field integrals. The comparison of measured local velocity and the network prediction achieved a relative mean squared error of 3 %.

A physics-informed neural network approach for predicting fatigue life of SLM 316L stainless steel based on defect features

Defects in additively manufactured materials severely limit the performance of parts in practical applications, often exposing them to the risk of fatigue failure. In order to improve the reliability and performance of additively manufactured parts, it becomes crucial to accurately predict the fatigue life of the material. Although traditional semi-empirical formulas can assess the effect of defects on the fatigue performance of parts, they still lack detailed research and consideration of defect morphology features. Therefore, this study proposes a method based on Physics-Informed Neural Networks (PINN). This method improves the predictive capability of the model and enhances its interpretability by extracting the sensitive features of critical defects and embedding known physical knowledge or fracture mechanics methods as loss functions into the training process of the neural network. Additionally, the method effectively captures the complex relationship between defect features and fatigue life, providing a deeper understanding of the model prediction results. The results show that the PINN model considering feature-related knowledge has higher prediction accuracy and reliability, and all predicted fatigue life are narrowed within 2-factor bands, enabling more accurate prediction of fatigue life for SLM 316L stainless steel under different processing conditions.

Quantifying local and global mass balance errors in physics-informed neural networks

Physics-informed neural networks (PINN) have recently become attractive for solving partial differential equations (PDEs) that describe physics laws. By including PDE-based loss functions, physics laws such as mass balance are enforced softly in PINN. This paper investigates how mass balance constraints are satisfied when PINN is used to solve the resulting PDEs. We investigate PINN’s ability to solve the 1D saturated groundwater flow equations (diffusion equations) for homogeneous and heterogeneous media and evaluate the local and global mass balance errors. We compare the obtained PINN’s solution and associated mass balance errors against a two-point finite volume numerical method and the corresponding analytical solution. We also evaluate the accuracy of PINN in solving the 1D saturated groundwater flow equation with and without incorporating hydraulic heads as training data. We demonstrate that PINN’s local and global mass balance errors are significant compared to the finite volume approach. Tuning the PINN’s hyperparameters, such as the number of collocation points, training data, hidden layers, nodes, epochs, and learning rate, did not improve the solution accuracy or the mass balance errors compared to the finite volume solution. Mass balance errors could considerably challenge the utility of PINN in applications where ensuring compliance with physical and mathematical properties is crucial.

Physics-Informed Online Learning for Temperature Prediction in Metal AM.

In metal additive manufacturing (AM), precise temperature field prediction is crucial for process monitoring, automation, control, and optimization. Traditional methods, primarily offline and data-driven, struggle with adapting to real-time changes and new process scenarios, which limits their applicability for effective AM process control. To address these challenges, this paper introduces the first physics-informed (PI) online learning framework specifically designed for temperature prediction in metal AM. Utilizing a physics-informed neural network (PINN), this framework integrates a neural network architecture with physics-informed inputs and loss functions. Pretrained on a known process to establish a baseline, the PINN transitions to an online learning phase, dynamically updating its weights in response to new, unseen data. This adaptation allows the model to continuously refine its predictions in real-time. By integrating physics-informed components, the PINN leverages prior knowledge about the manufacturing processes, enabling rapid adjustments to process parameters, geometries, deposition patterns, and materials. Empirical results confirm the robust performance of this PI online learning framework in accurately predicting temperature fields for unseen processes across various conditions. It notably surpasses traditional data-driven models, especially in critical areas like the Heat Affected Zone (HAZ) and melt pool. The PINN's use of physical laws and prior knowledge not only provides a significant advantage over conventional models but also ensures more accurate predictions under diverse conditions. Furthermore, our analysis of key hyperparameters-the learning rate and batch size of the online learning phase-highlights their roles in optimizing the learning process and enhancing the framework's overall effectiveness. This approach demonstrates significant potential to improve the online control and optimization of metal AM processes.

Multi-output multi-physics-informed neural network for learning dimension-reduced probability density evolution equation with unknown spatio-temporal-dependent coefficients

The Dimension-Reduced Probability Density Evolution Equation (DR-PDEE) offers a promising approach for evaluating probability density evolution in stochastic dynamical systems. Physics-Informed Neural Networks (PINNs) are well-suited for solving DR-PDEE due to their ability to encode physical laws into the learning process. However, challenges arise from the spatio-temporal-dependence of unknown intrinsic drift and diffusion coefficients, which drive DR-PDEE, along with their derivatives. To address these challenges, a novel framework called Multi-Output Multi-Physics-Informed Neural Network (MO-MPINN) is proposed to predict the evolution of time-varying coefficients and response probability density simultaneously. MO-MPINN features multiple output neurons, eliminating the necessity for distinct identification of unknown spatio-temporal-dependent coefficients separately. It uses parallel subnetworks to reduce training complexity and embeds multiple physical laws in the loss function to ensure an accurate representation of the underlying principles. Leveraging automatic differentiation, MO-MPINN efficiently computes derivatives of coefficients without resorting to numerical differentiation. The framework is applicable to high-dimensional stochastic nonlinear systems with double randomness in structural parameters and excitations. Several structures are presented to validate the performance of the MO-MPINN. This study introduces a new paradigm for solving partial differential equations involving differentiation of spatio-temporal-dependent coefficients.

E-PINN: extended physics informed neural network for the forward and inverse problems of high-order nonlinear integro-differential equations

Physics informed neural network (PINN) is a new deep learning paradigm, which embeds the physical information delineated by PDEs in the loss function and optimizes the weights in the neural network. Based on PINN, an extended PINN(E-PINN) is proposed, which is a mixture of the polynomial function approximation method and PINN's learning framework. A preprocess layer is added before the classical PINN, using Legendre polynomials as the polynomial basis function. Therefore, E-PINN not only has the excellent approximation ability of the polynomial basis function, but also inherits the learning framework of the neural network method. In numerical experiments, the proposed E-PINN algorithms have high accuracy in solving 1D, 2D high-order nonlinear Fredholm equations and equations system, including the forward and inverse problems.

E-PINN: extended physics informed neural network for the forward and inverse problems of high-order nonlinear integro-differential equations

Physics informed neural network (PINN) is a new deep learning paradigm, which embeds the physical information delineated by PDEs in the loss function and optimizes the weights in the neural network. Based on PINN, an extended PINN(E-PINN) is proposed, which is a mixture of the polynomial function approximation method and PINN's learning framework. A preprocess layer is added before the classical PINN, using Legendre polynomials as the polynomial basis function. Therefore, E-PINN not only has the excellent approximation ability of the polynomial basis function, but also inherits the learning framework of the neural network method. In numerical experiments, the proposed E-PINN algorithms have high accuracy in solving 1D, 2D high-order nonlinear Fredholm equations and equations system, including the forward and inverse problems.

Physics-informed neural network simulation of thermal cavity flow

Physics-informed neural networks (PINNs) are an emerging technology that can be used both in place of and in conjunction with conventional simulation methods. In this paper, we used PINNs to perform a forward simulation without leveraging known data. Our simulation was of a 2D natural convection-driven cavity using the vorticity-stream function formulation of the Navier-Stokes equations. We used both 2D simulations across the x and z domains at constant Rayleigh (Ra) numbers and 3D simulations across the x, z and Ra domains. The 3D simulation was tested for a PINN’s ability to learn solutions in a higher-dimensional space than standard simulations. The results were validated against published solutions at Ra values of 103\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{3}$$\\end{document}, 104\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{4}$$\\end{document}, 105\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{5}$$\\end{document}, and 106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{6}$$\\end{document}. Both the 2D simulations and 3D simulations successfully matched the expected results. For the 2D cases, more training iterations were needed for the model to converge at higher Ra values (105\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^5$$\\end{document} and 106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^6$$\\end{document}) than at lower Ra (103\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^3$$\\end{document} and 104\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^4$$\\end{document}) indicating increased nonlinear fluid-thermal coupling. The 3D case was also able to converge but, but it required more training than any of the 2D cases due to the curse of dimensionality. These results showed the validity of standard simulations via PINNs and the feasibility of higher-order parameter space solutions that are not possible using conventional methods. They also showcased the additional computational demand associated with increasing the dimensionality of the learned parameter space.

Review of Machine Learning Techniques in Soft Tissue Biomechanics and Biomaterials.

Advanced material models and material characterization of soft biological tissues play an essential role in pre-surgical planning for vascular surgeries and transcatheter interventions. Recent advances in heart valve engineering, medical device and patch design are built upon these models. Furthermore, understanding vascular growth and remodeling in native and tissue-engineered vascular biomaterials, as well as designing and testing drugs on soft tissue, are crucial aspects of predictive regenerative medicine. Traditional nonlinear optimization methods and finite element (FE) simulations have served as biomaterial characterization tools combined with soft tissue mechanics and tensile testing for decades. However, results obtained through nonlinear optimization methods are reliable only to a certain extent due to mathematical limitations, and FE simulations may require substantial computing time and resources, which might not be justified for patient-specific simulations. To a significant extent, machine learning (ML) techniques have gained increasing prominence in the field of soft tissue mechanics in recent years, offering notable advantages over conventional methods. This review article presents an in-depth examination of emerging ML algorithms utilized for estimating the mechanical characteristics of soft biological tissues and biomaterials. These algorithms are employed to analyze crucial properties such as stress-strain curves and pressure-volume loops. The focus of the review is on applications in cardiovascular engineering, and the fundamental mathematical basis of each approach is also discussed. The review effort employed two strategies. First, the recent studies of major research groups actively engaged in cardiovascular soft tissue mechanics are compiled, and research papers utilizing ML and deep learning (DL) techniques were included in our review. The second strategy involved a standard keyword search across major databases. This approach provided 11 relevant ML articles, meticulously selected from reputable sources including ScienceDirect, Springer, PubMed, and Google Scholar. The selection process involved using specific keywords such as "machine learning" or "deep learning" in conjunction with "soft biological tissues", "cardiovascular", "patient-specific," "strain energy", "vascular" or "biomaterials". Initially, a total of 25 articles were selected. However, 14 of these articles were excluded as they did not align with the criteria of focusing on biomaterials specifically employed for soft tissue repair and regeneration. As a result, the remaining 11 articles were categorized based on the ML techniques employed and the training data utilized. ML techniques utilized for assessing the mechanical characteristics of soft biological tissues and biomaterials are broadly classified into two categories: standard ML algorithms and physics-informed ML algorithms. The standard ML models are then organized based on their tasks, being grouped into Regression and Classification subcategories. Within these categories, studies employ various supervised learning models, including support vector machines (SVMs), bagged decision trees (BDTs), artificial neural networks (ANNs) or deep neural networks (DNNs), and convolutional neural networks (CNNs). Additionally, the utilization of unsupervised learning approaches, such as autoencoders incorporating principal component analysis (PCA) and/or low-rank approximation (LRA), is based on the specific characteristics of the training data. The training data predominantly consists of three types: experimental mechanical data, including uniaxial or biaxial stress-strain data; synthetic mechanical data generated through non-linear fitting and/or FE simulations; and image data such as 3D second harmonic generation (SHG) images or computed tomography (CT) images. The evaluation of performance for physics-informed ML models primarily relies on the coefficient of determination . In contrast, various metrics and error measures are utilized to assess the performance of standard ML models. Furthermore, our review includes an extensive examination of prevalent biomaterial models that can serve as physical laws for physics-informed ML models. ML models offer an accurate, fast, and reliable approach for evaluating the mechanical characteristics of diseased soft tissue segments and selecting optimal biomaterials for time-critical soft tissue surgeries. Among the various ML models examined in this review, physics-informed neural network models exhibit the capability to forecast the mechanical response of soft biological tissues accurately, even with limited training samples. These models achieve high values ranging from 0.90 to 1.00. This is particularly significant considering the challenges associated with obtaining a large number of living tissue samples for experimental purposes, which can be time-consuming and impractical. Additionally, the review not only discusses the advantages identified in the current literature but also sheds light on the limitations and offers insights into future perspectives.

Self-adaptive and time divide-and-conquer physics-informed neural networks for two-phase flow simulations using interface tracking methods

Physics-informed neural networks (PINNs) are emerging as a promising artificial intelligence approach for solving complex two-phase flow simulations. A critical challenge in these simulations is an accurate representation of the gas–liquid interface using interface tracking methods. While numerous studies in conventional computational fluid dynamics (CFD) have addressed this issue, there remains a notable absence of research within the context of PINNs-based two-phase flow simulations. Therefore, this study aims to develop a robust and generic PINNs for two-phase flow by incorporating the governing equations with three advanced interface tracking methods—specifically, the Volume of Fluid, Level Set, and Phase-Field method—into an improved PINN framework that has been previously proposed and validated. To further enhance the performance of the PINNs in simulating two-phase flow, the phase field constraints, residual connection and the time divide-and-conquer strategies are employed for restricting neural network training within the scope of physical laws. This self-adaptive and time divide-and-conquer (AT) PINNs is then optimized by minimizing both the residual and loss terms of partial differential equation. By incorporating the three different interface tracking methods, it efficiently handles high-order derivative terms and captures the phase interface. The case of single rising bubble in two-phase flow is simulated to validate the robustness and accuracy of the AT PINNs. The simulation's accuracy is evaluated by comparing its performance in terms of velocity, pressure, phase field, center of mass, and rising velocity with that of conventional PINNs and CFD benchmarks. The results indicate that the AT PINNs coupled with these interface tracking methods offers a satisfactory performance in simulating rising bubble phenomenon.

Interfacial conditioning in physics informed neural networks

Physics informed neural networks (PINNs) have effectively demonstrated the ability to approximate the solutions of a system of partial differential equations (PDEs) by embedding the governing equations and auxiliary conditions directly into the loss function using automatic differentiation. Despite demonstrating potential across diverse applications, PINNs have encountered challenges in accurately predicting solutions for time-dependent problems. In response, this study presents a novel methodology aimed at enhancing the predictive capability of PINNs for time-dependent scenarios. Our approach involves dividing the temporal domain into multiple subdomains and employing an adaptive weighting strategy at the initial condition and at the interfaces between these subdomains. By employing such interfacial conditioning in physics informed neural networks (IcPINN), we have solved several unsteady PDEs (e.g., Allen–Cahn equation, advection equation, Korteweg–De Vries equation, Cahn–Hilliard equation, and Navier–Stokes equations) and conducted a comparative analysis with numerical results. The results have demonstrated that IcPINN was successful in obtaining highly accurate results in each case without the need for using any labeled data.