Physics-informed Neural Networks Research Articles

Optimized physics-informed neural network for analyzing the radiative-convective thermal performance of an inclined wavy porous fin

The significance of radiation and inclination on the temperature dispersion of the wavy porous fin has been addressed in the present study. Also, the influence of convection and internal heat generation on the thermal dissipation of the inclined wavy porous fin (IWPF) is examined. The pertinent temperature expression of the fin is represented using basic laws, and this equation is reduced to a dimensionless form via dimensionless variables. Additionally, a mix-encoding Genetic algorithm and Particle swarm optimization technique is shown to optimize the network hyperparameters. This resolves the issue of arbitrarily identifying the PINN's ideal network and successfully limits local optimization during the training phase. Further, the equation is also resolved numerically using Runge-Kutta Fehlberg's fourth-fifth (RKF-45) scheme, and the solutions are subsequently used to verify the PINN model's applicability. The temperature results estimated by PINN and their associated RKF-45 values correlate excellently, which indicates the accuracy of the applied PINN model. The obtained findings denote that reduced measures of convective-conductive variables stimulate the IWPF's thermal distribution. An inclination angle of the fin has a significant impact on the thermal variation of the IWPF.

Data-driven prediction of vortex solitons and multipole solitons in whispering gallery mode microresonator

Deep learning incorporating physics knowledge has become a powerful tool for studying the dynamic behavior of high-dimensional nonlinear systems. In this paper, the two-stage mini-batch resampling of adaptive physics-informed neural network (TMA-PINN) method is proposed to solve the (2 + 1)-dimensional variable-coefficient Lugiato-Lefever equation (vLLE). The vortex soliton in the WGM microresonator with different external excitation is investigated by TMA-PINN. It is found that external excitation can cause the rotation of vortex solitons. In addition, the effect of topological charge and external excitation on the dynamical characteristics of spatial solitons including vortex solitons and multipole solitons are investigated. The results show that the final shape of the rotation of vortex solitons and the number of azimuth lobes of multipole solitons are controlled by topological charges. Compared with classical PINN, TMA-PINN can better handle the gradient balance of various loss terms in (2 + 1)-dimensional vLLE to reconstruct the dynamic behavior of WGM microresonator solitons, having potential applications in other nonlinear systems.

Deep Learning for Solving Partial Differential Equations: A Review of Literature

Partial Differential Equations (PDEs) are fundamental in modeling various phenomena in physics, engineering, and finance. Traditional numerical methods for solving PDEs, such as finite element and finite difference methods, often face limitations when applied to high-dimensional and complex systems. In recent years, deep learning has emerged as a promising alternative for approximating solutions to PDEs, offering potential improvements in both efficiency and scalability. This paper provides a comprehensive review of the literature on deep learning-based methods for solving PDEs, focusing on key approaches such as Physics-Informed Neural Networks (PINNs), deep Galerkin methods, and neural operators. These methods leverage the expressiveness of neural networks to capture underlying physics while avoiding the curse of dimensionality associated with classical techniques. We explore the theoretical foundations, advantages, and limitations of these deep learning models, along with their applications in diverse fields like fluid dynamics, quantum mechanics, and financial modeling. Additionally, this review examines recent advancements in hybrid models that combine traditional numerical methods with deep learning approaches to enhance accuracy and stability. Through this review, we highlight key trends and open challenges in the field, paving the way for future research at the intersection of deep learning and computational mathematics.

Predicting bifurcation and amplitude death characteristics of thermoacoustic instabilities from PINNs-derived van der Pol oscillators

Self-sustained thermoacoustic oscillations as observed in low-emission combustion- involved gas turbines and aero-engines involve complicated thermal fluid–acoustics interaction and rich nonlinear dynamics. Such pulsating oscillations are known as thermoacoustic instability. When it occurs, large-amplitude limit cycle oscillations (LCOs) of thermodynamic parameters are frequently observed. These LCOs could cause overheating, flame flashback, and even engine failures. Thus it is critical to understand and predict the generation mechanisms and nonlinear dynamics behaviours, and then develop corresponding control approaches to prevent or control the onset of such instabilities. In this work, we develop and extend the classical van der Pol oscillators by integrating a physics-informed neural networks (PINNs) algorithm with a modelled nonlinear Rijke-type thermoacoustic combustor. The theoretical Rijke tube system (with Galerkin expansion and modified King's law implemented) and a CFD simulation model are applied to provide ‘training/calibration data’ for the extended van der Pol (EVDP)-PINNs model. The optimized EVDP oscillators are confirmed to be capable of capturing the key nonlinear characteristics by comparing the transient growth behaviours of thermodynamic perturbations and LCO amplitude and frequency. Further investigations are conducted to obtain Hopf bifurcation and amplitude death (AD) characteristics. Comparison is then made to the coupled EVDP systems. Quite similar Hopf bifurcation features, but differences in regions of AD, are observed. In general, we demonstrate an applicable approach to intelligently ‘learn’ a nonlinear thermoacoustic system and to create reliable EVDP oscillator systems, which have great potential to contribute to the development and testing of control approaches, such as the coupling described in this work, which may replace costly experimental tests.

Junction conditions for one-dimensional network hemodynamic model for total cavopulmonary connection using physically informed deep learning technique

Abstract This paper presents a novel methodology utilizing physics-informed neural network (PINN) as a junction condition for a 1D network model of blood flow in total cavopulmonary connection generated by the Fontan procedure. The technique integrates a 3D mesh generation process based on the parameterization of the junction geometry, along with a sophisticated physically regularized neural network architecture. Synthetic datasets are produced using 3D steady Stokes simulations within fixed boundaries. We use a physically informed feedforward neural network that utilizes a physically regularized loss function, which incorporates the principle of mass conservation. Our PINN achieves a tolerance of 6% on the test set. We develop a 1D-PINN multiscale model based on a previously developed method for multiscale 1D–3D simulations. Comparison with 1D–3D Stokes based model and 3D Navier–Stokes based model verifies the 1D-PINN model. In the first and second comparison, the maximum deviations of the averaged pressures and flows do not exceed 1.48% and 12.26%, respectively.

A physics-informed neural network method for identifying parameters and predicting remaining life of fatigue crack growth

Predicting the remaining life of fatigue cracks is crucial for planning maintenance and repair strategies to prevent untoward incidents. This paper proposes a novel physics-informed neural network (PINN) method for identifying parameters and predicting remaining fatigue crack growth life (FCGL). Initially, the relationship between crack length and fatigue cycles is established through a neural network, and the gradient of fatigue cycles with respect to crack length is obtained by automatic differentiation. Subsequently, a composite loss function is designed to incorporate this gradient within the confines of physical knowledge, ensuring that the established relationship not only aligns with observed data but also adheres to physical knowledge. Furthermore, during the network training, the parameters in physical models are simultaneously updated to better conform to the individuality of the monitored subject. All predicted remaining FCGLs fall within the 1.5 times error band. Compared to purely data-driven or physics-based methods, the proposed method offers more robust and accurate predictions of remaining FCGLs.

Solving transport equations on quantum computers—potential and limitations of physics-informed quantum circuits

AbstractQuantum circuits with trainable parameters, paired with classical optimization routines can be used as machine learning models. The recently popularized physics-informed neural network (PINN) approach is a machine learning algorithm that solves differential equations by incorporating them into a loss function. Being a mesh-free method, it is a promising approach for computational fluid dynamics. The question arises whether the properties of quantum circuits can be leveraged for a quantum physics-informed machine learning model. In this study, we compare the classical PINN-ansatz and its quantum analog, which we name the physics-informed quantum circuit (PIQC). The PIQC simulations are performed on a noise-free quantum computing simulator. Studying various differential equations, we compare expressivity, accuracy and convergence properties. We find that one-dimensional problems, such as the linear transport of a Gaussian-pulse or Burgers’ equation, allow a successful approximation with the classical and the quantum ansatz. For these examples, the PIQC overall performs similarly to PINN and converges more consistently and for Burgers’ equations even faster. While this is promising, the chosen quantum circuit approach struggles to approximate discontinuous solutions which the classical PINN-ansatz can represent. Based on this comparison, we extrapolate that the required number of qubits for solving two-dimensional problems in aerodynamics may already be available in the next few years. However, the acceleration potential is currently unclear and challenges like noisy circuits and approximations of discontinuous solutions have to be overcome.

Enhanced PINNs with augmented Lagrangian method and transfer learning for hydrodynamic lubrication analysis

Purpose By seamlessly integrating physical laws, physics-informed neural networks (PINNs) have flexibly solved a wide variety of partial differential equations (PDEs). However, encoding PDEs and constraints as soft penalties in the loss function can cause gradient imbalances, leading to training and accuracy issues. This study aims to introduce the augmented Lagrangian method (ALM) and transfer learning to address these challenges and enhance the effectiveness of PINNs for hydrodynamic lubrication analysis. Design/methodology/approach The loss function was reformatted by ALM, adaptively adjusting the loss weights during training. Transfer learning was used to accelerate the convergence of PINNs under similar conditions. Additionally, the iterative process for load balancing was reframed as an inverse problem by extending film thickness as a trainable variable. Findings ALM-PINNs significantly reduced the maximum absolute boundary error by almost 80%. Transfer learning accelerated PINNs for solving the Reynolds equation, reducing training epochs by an order of magnitude. The iterative process for load balancing was effectively eliminated by extending the thickness as a trainable parameter, achieving a maximum percentage error of 2.31%. These outcomes demonstrated strong agreement with FDM results, analytical solutions and experimental data. Originality/value This study proposes a PINN-based approach for hydrodynamic lubrication analysis that significantly improves boundary accuracy and the training process. Additionally, it effectively replaces the load balancing procedure. This methodology demonstrates considerable potential for broader applications across various boundary value problems and iterative processes. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-07-2024-0277/

Inverse problems in hydrodynamics lubrication: Parameter identification in the Reynold equation by using physics-informed neural networks

In this work, the inverse problems for hydrodynamic lubrication were solved by using the physics informed neural networks (PINNs) method. Both the conditions without and with the cavitation effects are studied. For the journal bearings without considering the cavitation effects, the eccentricity values can be accurately inverted using only 10 data points with a maximum error of less than 6%. For the problems with the mass-conserving cavitation, the geometry parameters for 1D conditions were estimated and accurate results can be obtained by also using 10 data points. While for the journal bearings with incompressible lubricants, the error was less than 3% with only 50 data points. For the ones with compressible lubricants, the eccentricity ratios can be estimated with 50 data points again, and the maximum error is less than 10%. In addition, the experimental oil pressure data were used to predict the journal bearing eccentricity, and the results were acceptable with only 9 oil pressure data points. The PINNs method presented two significant strengths: its ability to operate effectively with very sparse data points and its precision in delivering results, and it can provide a promising method for identifying unknown parameters in hydrodynamics lubrication with small data.

Physics-informed neural networks approach for pollutant dispersion flow caused by magnetic dipole

Abstract Research on the liquid flow and distribution of waste discharge concentration over cylindrical surfaces might lead to the growth of efficient waste management and pollution controlling strategies. In view of this, the pollutant concentration and magnetic dipole impact on the liquid flow through the cylinder with a porous medium are explored in the present analysis. Moreover, Fourier’s and Fick’s laws are considered to explore heat transport features. The governing partial differential equations (PDEs) of the present study are converted into non-dimensional ordinary differential equations (ODEs) utilizing similarity variables. Also, the current study proposes a novel implementation of neural networks based on physics knowledge for the liquid flow past a cylinder. The self-updating weights and bias facilitate the network to satisfy the minimization objective function and provide accurate results. Additionally, the solution obtained by Runge Kutta Fehlberg's fourth-fifth order (RKF-45) method is utilized to validate the physics-informed neural network (PINN) results, presenting mean squared errors from 10-9 to 10-11. Furthermore, it is found that the increase in temperature relaxation time parameter decreases the Nusselt number gradually at the rate of 1 to 5%. An upsurge in the values of the pollutant external source parameter and the pollutant external source variation parameter decreases the concentration profile.

Analysis of ventilation and infiltration rates using physics-informed neural networks: Impact of space operation and meteorological factors

This study presents the development and application of a Physics-Informed Neural Network (PINN) model to estimate ventilation and infiltration rates using long-term observation data, addressing the challenge of dynamically varying space operations and meteorological conditions. A central research equestion is: How can we accurately estimate ventilation rates while accounting for these time-varying factors? Traditional tracer gas methods require numerous measurements to accurately characterize air change rates (ACR) under dynamic space operations and varying meteorological conditions. Our PINN model integrates these fluctuating factors, providing a more precise analysis of their transient effects on ACR. We employed Shapley Additive Explanations (SHAP) to interpret the sensitivity and contributions of each influencing factor. Our findings indicate that the state of windows and doors significantly affects spatial operations, while wind speed and direction are the most impactful meteorological factors. The interaction between open windows and doors results in higher ventilation rates compared to their individual effects. Wind-related factors cause ACR variations exceeding 200 %, with the wind direction relative to the office window playing a crucial role. Additionally, external temperature and indoor-outdoor temperature differences show a strong correlation with ACR. However, limitations include the lack of outdoor CO2 measurements and the assumption of uniform indoor CO2 levels, which may affect accuracy. Generalizability is also limited due to the specificity of the space studied. Future work should incorporate outdoor CO2 data and multiple spaces to enhance model applicability. This study contributes to optimizing ventilation strategies for better indoor air quality and energy efficiency.

Various optimized artificial neural network simulations of advection-diffusion processes

The aim of this research is to describe an artificial neural network (ANN) based method to approximate the solutions of the natural advection-diffusion equations. Although the solutions of these equations can be obtained by various effective numerical methods, feed forward neural network (FFNN) techniques combined with different optimization techniques offer a more practicable and flexible alternative than the traditional approaches to solve those equations. However, the ability of FFNN techniques to solve partial differential equations is a questionable issue and has not yet been fully concluded in the existing literature. The reliability and accuracy of computational results can be advanced by the choice of optimization techniques. Therefore, this study aims to take an effective step towards presenting the ability to solve the advection-diffusion equations by leveraging the inherent benefits of ANN methods while avoiding some of the limitations of traditional approaches. In this technique, the solution process requires minimizing the error generated by using a differential equation whose solution is considered as a trial solution. More specifically, this study uses a FFNN and backpropagation technique, one of the variants of the ANN method, to minimize the error and the adjustment of parameters. In the solution process, the loss function (error) needs to be minimized; this is accomplished by fitting the trial function into the differential equation using appropriate optimization techniques and obtaining the network output. Therefore, in this study, the commonly used techniques in the literature, namely gradient descent (GD), particle swarm optimization (PSO) and artificial bee colony (ABC), are selected to compare the effectiveness of gradient and gradient-free optimization techniques in solving the advection-diffusion equation. The calculations with all three optimization techniques for linear and nonlinear advection-diffusion equations have been run several times to obtain the optimum accuracy of the results. The computed results are seen to be very promising and in good agreement with the effective numerical methods and the physics-informed neural network (PINN) method in the literature. It is also concluded that the PSO-based algorithm outperforms other methods in terms of accuracy.

Physics-informed neutral network with physically consistent and residual learning for excavator precision operation control

The data-driven methodologies can establish accurate Inverse Dynamics Model (IDM) of the excavator thus improving control precisions. However, the inherent black-box nature of these models often results in overfitting to the dataset, leading to predictions that deviate from the constraints of physical system. Consequently, this can lead to controller failures, introducing unpredictable behavior that threatens operation precision. In addition, the uncertainty of the external disturbance poses great challenge to the precision of controller. This study presents a physics-informed neural network to build accurate IDM with physical consistency. The Rigid Body Dynamics (RBD) of the excavator are coupled within a Deep Lagrangian Network (DeLaN), while a Convolutional Neural Network (CNN) and a Long Short-Term Memory Network (LSTM) are employed to assimilate the residual nonlinear characteristics, such as hydraulic flexibilities and stick–slip friction. To the uncertainty of the external disturbance, the Prescribed Performance Inverse Dynamics Controller combination with the DeLaN-CNN-LSTM model (PPIDC-DCL) is constructed for precise control by constraining the control error within a finite region. The experimental results demonstrate that the model captures the underlying structure of the dynamic and builds the IDM with high accuracy and robustness. Moreover, the PPIDC-DCL controller effectively constrains the control error and realizes precision control. The proposed method has potential applications and provides novel insights for achieving precise operation control of excavators.

Multi-receptive-field physics-informed neural network for complex electromagnetic media

Acquiring the electromagnetic response of intricate media at the nanoscale constitutes a pivotal phase in the design intricacies of nanophotonic apparatuses. Conventional numerical algorithms often necessitate intricate and specialized treatments to accommodate the unique properties of the medium, coupled with substantial computational time and resource demands. In recent years, the advent of deep learning technology has heralded numerous advancements in the domain of computational electromagnetics, albeit with a scarcity of solvers tailored for versatile complex media. Consequently, this study introduces an innovative multi-receptive-field physics-informed neural network (MRF-PINN) designed to tackle nano optical scattering predicaments inherent in media exhibiting dispersion, inhomogeneity, anisotropy, nonlinearity, and chirality. This framework adeptly captures electromagnetic perturbations surrounding scatterers via variable-scale receptive fields, thereby enhancing prediction precision. Within the training regimen, a scale balancing algorithm is proposed to expedite network convergence. Empirical findings demonstrate that a fully trained MRF-PINN proficiently reconstructs electromagnetic field distributions within complex nanomaterials within a mere tens of milliseconds of inference time. Such quasi real-time capabilities herald a novel approach to supplant the arduous forward calculation processes inherent in nanomaterial design workflows.

Physics-informed neural network solution for thermo-elastic cavity expansion problem

ABSTRACT This paper proposes a physics-informed neural network (PINN) to solve the thermo-elastic coupled cavity expansion problem under plane strain conditions. The solid is modelled as a linear thermo-elastic material, and cavity expansion is driven by a constant temperature and a radial displacement subjected to the inner cavity wall. Governing partial differential equations (PDEs) for the problem is firstly presented, and an analytical solution is available to help benchmark the PINN approach. Then, the PDEs are normalised into dimensionless forms and neural network details for solving the coupled PDEs are demonstrated. Finally, the performance of the PINN approach is shown by comparison with the analytical benchmark solution and it is found that the PINN can predict thermo-elastic cavity expansion behaviour with high accuracy. The PINN approach and the analytical solution can also serve as a benchmark for solving thermo-elastic coupled problems in geotechnical engineering by more advanced PINN.

An Approach to Realize Generalized Optimal Motion Primitives Using Physics Informed Neural Networks

Abstract Autonomous manipulation is a challenging problem in field robotics due to uncertainty in object properties, constraints, and coupling phenomenon with robot control systems. Humans learn motion primitives over time to effectively interact with the environment. We postulate that autonomous manipulation can be enabled by basic sets of motion primitives as well, but do not necessitate mimicking human motion primitives. This work presents an approach to generalized optimal motion primitives using physics-informed neural networks. Our simulated and experimental results demonstrate that optimality is notionally maintained where the mean maximum observed final position percent error was 0.564% and the average mean error for all the trajectories was 1.53%. These results indicate that notional generalization is attained using a physics-informed neural network approach that enables near optimal real-time adaptation of primitive motion profiles.

Optimal approximations for the free boundary problems of the space-time fractional Black-Scholes equations using a combined physics-informed neural network

The combined physics-informed neural network is employed to deal with the free boundary problems of fractional Black-Scholes equations. The solution assumption and the loss function are determined, the transfer learning is borrowed, the combined neural network with data enhancement layer is designed, then the classical Black-Scholes model is numerically solved and the comparative analysis of numerical results under different neural networks is made. For further insight into the long-term memory of fluctuation, the free boundary problems of the space-time Black-Scholes equations under Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives are studied. The corresponding empirical analyses are presented and the optimal exercise boundaries of American put option are simulated. The market analysis shows that introducing fractional calculus tools and neural network algorithms into American put option pricing can yield more realistic prediction results. The work provides a viable method for subsequent researchers to study American option pricing using fractional calculus and neural networks combined with true market data and to deal with the free boundary problems in other research fields.

Physics-Informed Neural Networks for the Reynolds Equation with Transient Cavitation Modeling

Gaining insight into tribological systems is crucial for optimizing efficiency and prolonging operational lifespans in technical systems. Experimental investigations are time-consuming and costly, especially for reciprocating seals in fluid power systems. Elastohydrodynamic lubrication (EHL) simulations offer an alternative but demand significant computational resources. Physics-informed neural networks (PINNs) provide a promising solution using physics-based approaches to solve partial differential equations. While PINNs have successfully modeled hydrodynamics with stationary cavitation, they have yet to address transient cavitation with dynamic geometry changes. This contribution applies a PINN framework to predict pressure build-up and transient cavitation in sealing contacts with dynamic geometry changes. The results demonstrate the potential of PINNs for modeling tribological systems and highlight their significance in enhancing computational efficiency.

Closed-Boundary Reflections of Shallow Water Waves as an Open Challenge for Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs) in fluid dynamics. By using PDE-derived loss functions and auto-differentiation, PINNs can recover solutions without requiring costly simulation data, spatial gridding, or time discretization. However, PINNs often exhibit slow or incomplete convergence, depending on the architecture, optimization algorithms, and complexity of the PDEs. To address these difficulties, a variety of novel and repurposed techniques have been introduced to improve convergence. Despite these efforts, their effectiveness is difficult to assess due to the wide range of problems and network architectures. As a novel test case for PINNs, we propose one-dimensional shallow water equations with closed boundaries, where the solutions exhibit repeated boundary wave reflections. After carefully constructing a reference solution, we evaluate the performance of PINNs across different architectures, optimizers, and special training techniques. Despite the simplicity of the problem for classical methods, PINNs only achieve accurate results after prohibitively long training times. While some techniques provide modest improvements in stability and accuracy, this problem remains an open challenge for PINNs, suggesting that it could serve as a valuable testbed for future research on PINN training techniques and optimization strategies.

Predicting the fatigue life of additively manufactured AlSi10Mg alloy using a physics-informed neural network incorporating continuous damage mechanics

The material characteristics of additively manufactured AlSi10Mg alloy, including the random distribution of process-induced micro defects, microstructual anisotropy, and grain morphologies with complex diversity, poses a significant challenge in accurately predicting its fatigue life, limiting its application in the aircraft field. This work herein introduces a novel approach incorporating a physics-informed neural network (PINN), in which an artificial neural network (ANN) is embedded with a damage mechanics model (CDM) and the critical process parameters of laser powder bed fusion (L-PBF) additive manufacturing. The partial differential equations of the updated CDM model are introduced into the training procedures of the PINN to building a loss function, effectively “teaching” the PINN to learn the physical knowledge. A comparison of the fatigue life prediction results of ANN and the proposed PINN models shows that the PINN model outperforms its counterparts with a 38.71% higher prediction accuracy. The effects of L-PBF process parameters on the fatigue life of as-built AlSi10Mg is examined using both ANN and PINN, proving the better predictive performance and data-physics consistency of the PINN model. The scheme of this work can inform the efficient prediction of fatigue life of additively manufactured alloys and also reverse prediction or fast finding of optimized process parameters for additive manufacturing.