Physics-informed Neural Networks Research Articles

Optical interferometry based acoustic field characterization using physics informed neural networks

The acousto-optic effect offers a non-contact way of measuring and visualizing acoustic fields in transparent fluids. Acoustic density fluctuations induce changes in the media's refractive index that can be measured using interferometry. While sensing methods based on acousto-optics have gathered increasing attention, many of such methods are only sensitive to projections of the acoustic field along the optical path, limiting the fields that can be characterized to one or two dimensions, or requiring complex experimental setups. Recently, reconstruction methods that abide by the physics of acoustic wave propagation showed unparalleled performance compared with classical reconstruction methods, reducing the sampling requirements and enabling to characterize three-dimensional acoustic fields. Nonetheless, such reconstruction methods rely on discrete approximations which might be poor at describing spatially complex fields. In this work we use physics-informed neural networks (PINNs) to estimate the acoustic pressure over space from interferometer measurements. The network is flexible enough to represent complex acoustic fields, while its output is constrained to fulfill the wave equation. Once trained on the data available for a given experiment, the PINN can estimate the acoustic pressure at arbitrary positions. The approach is tested in an experimental study, showing an improved performance compared with state-of-the-art reconstruction methods.

Comparison of standard Bayesian methods and PINNs for the reconstruction of acoustic fields in 'in situ' combustion chambers

Acoustic measurements, utilizing sensors such as microphones strategically placed, are commonly employed to monitor acoustic systems and extract quantities of interest, such as acoustic reflection coefficients. Typically, a specialized test rig must be constructed to allow for external harmonic acoustic forcing via loudspeakers. In this study, we propose two different methods: physics-informed neural networks (PINNs) and the extended Kalman filter, to reconstruct the acoustic field of a system and assess the acoustic reflection coefficient at a defined boundary. We demonstrate numerically that a single external broadband forcing signal suffices for these methods. This implies that these methods can be applied to 'in situ' configurations, such as industrial combustion chambers, where the broadband nature of combustion noise serves as the acoustic forcing element. Furthermore, we investigate the performance of the two aforementioned methods in reconstructing the acoustic systems and corresponding acoustic boundaries.

Neural Impedance Boundary (NeIB): a neural-network based framework for acoustic surface impedance estimation utilizing sparse measurement data

Amidst recent advancements in the 3D digital representation that have significantly enhanced the modeling of geometric attributes of pre-existing environments, accurate estimation of acoustic boundary conditions remains a complex challenge. This paper presents a novel way to determine what we refer to as a neural boundary field, using physics-informed neural networks (PINN). The aim is to estimate the surface impedance measured in-situ by utilizing several points of sound field pressure. This approach couples the Helmholtz equation with automatic differentiation in the PINN framework to estimate accurately the surface impedance using a hybrid modeling approach where measurement data and domain knowledge in the form of equations for the acoustic waves are combined. As a proof-of-concept, we train the neural networks using 2D sound field data obtained from high-fidelity numerical acoustical simulations that incorporate actual surface materials parameters. We discuss the measurement techniques associated with this method and outline our vision for its application to 3D scene reconstructions in the future.

Physics-informed neural networks for localized assessment in Euler-Bernoulli beams

This study employs the Physics-Informed Neural Networks (PINNs) to address an inverse problem, specifically evaluating local changes in material properties within an Euler-Bernoulli beam through numerical simulations. In our numerical model, a viscoelastic material is strategically introduced to a localized zone of the beam, inducing non-constant and complex alterations in the storage modulus. The training process incorporates the partial differential equation into the neural network's loss function, coupled with essential stability conditions for the machine learning model, enhancing the efficiency of the training process. Notably, this methodology excels in characterizing structural behavior influenced by these intricate local changes in material properties, bypassing the need to identify the entire structure and consequently to deal with uncertainties related to boundary conditions. The effectiveness of the approach is validated through numerical simulations, underscoring its potential for applications in efficiently assessing and understanding local material alterations within complex structural systems.

Predicting fatigue life of multi-defect materials using the fracture mechanics-based physics-informed neural network framework

To address the limitations of traditional methods (such as the S-N curves and Paris’s law) in evaluating the fatigue life of multi-defect materials, this study developed a fracture mechanics-based physics-informed neural network (PINN) to predict the lifetime of multi-defect materials using cyclic loading (Δσ) and the equivalent damage area (AD). Influences of multiple defects were unified characterized through the equivalent damage area, which was calculated based on the M−integral fatigue model. This model reflects the energy evolution of multi-defect fatigue damage. By embedding the prior knowledge of fracture mechanics derived from the M−integral fatigue model into the loss function of PINN, crucial physical information was captured during the training progress, enhancing the interpretability of the neural network. By integrating the advantage of the M−integral fatigue model in characterizing the fatigue performance of multi-defect materials and the nonlinear fitting ability of neural networks, the proposed approach effectively improves the generalization ability and predictive accuracy of limited fatigue data. The presented PINN models accurately forecast the fatigue life of multi-defect materials, with a squared correlation coefficient (R2) exceeding 0.9. The presented methodological framework addresses the existing gap in methods for evaluating the fatigue performance of multi-defect materials and reliance on fatigue testing.

Physics-informed deep learning for structural dynamics under moving load

Physics-informed deep learning has emerged as a promising approach that incorporates physical constraints into the model, reduces the amount of data required, and demonstrates robustness and potential in dealing with limited datasets for a variety of studies. However, several key challenges still exist, with one being the spectral bias problem of deep learning in the simulation of functions with multi-frequency features. To overcome the challenge, this study proposes a novel physics-informed deep learning method, which integrates physics-informed neural network with Fourier transform so as to solve partial differential equations in the frequency domain, thus alleviating the problem of spectral bias of neural networks in the simulation of multi-frequency functions. In addition, the proposed method is used to focus on the forward simulation and parameter inverse identification issues in structural dynamics under moving loads. To illustrate the superiority of the method, the issues of dynamic response of simply supported beams under moving loads are presented as case studies, and the performance of the method in multiple cases is analysed and discussed. The research results demonstrate the feasibility and effectiveness of the method for structural dynamics simulation and parameter inverse identifications using limited datasets.

Assessing physics-informed neural network performance with sparse noisy velocity data

The utilization of data in physics-informed neural network (PINN) may be considered as a necessity as it allows the simulation of more complex cases with a significantly lower computational cost. However, doing so would also make it prone to any issue with the data quality, including its noise. This study would primarily focus on developing a special loss function in the PINN to allow an effective utilization of noisy data. However, a study regarding the data location and amount was also conducted in order to allow a better data utilization in PINN. This study was conducted on a lid-driven cavity flow at Re = 200, 1000, and 5000 with a dataset of less than 100 velocity data and a maximum noise of 10% of the maximum velocity. The results show that by ensuring the data are distributed in a certain configuration, it has zero noise, and by using as much data as possible, the computational cost of PINN can be significantly reduced compared to without using any data at all. For Re = 200, it is 7.4 faster by using data, and this speedup is potentially higher for higher Re cases. For the noise in particular, it does not only make the PINN more inaccurate but also necessitate the usage of more data as this is the only way to make it more accurate. This issue though is capable to be solved with our new method, which only uses the data as an approximate solution, and the governing equation would figure out the details. This method was also shown to be capable to improve the PINN accuracy with the potential to almost completely eliminating the noise effect.

A comprehensive review of advances in physics-informed neural networks and their applications in complex fluid dynamics

Physics-informed neural networks (PINNs) represent an emerging computational paradigm that incorporates observed data patterns and the fundamental physical laws of a given problem domain. This approach provides significant advantages in addressing diverse difficulties in the field of complex fluid dynamics. We thoroughly investigated the design of the model architecture, the optimization of the convergence rate, and the development of computational modules for PINNs. However, efficiently and accurately utilizing PINNs to resolve complex fluid dynamics problems remain an enormous barrier. For instance, rapidly deriving surrogate models for turbulence from known data and accurately characterizing flow details in multiphase flow fields present substantial difficulties. Additionally, the prediction of parameters in multi-physics coupled models, achieving balance across all scales in multiscale modeling, and developing standardized test sets encompassing complex fluid dynamic problems are urgent technical breakthroughs needed. This paper discusses the latest advancements in PINNs and their potential applications in complex fluid dynamics, including turbulence, multiphase flows, multi-field coupled flows, and multiscale flows. Furthermore, we analyze the challenges that PINNs face in addressing these fluid dynamics problems and outline future trends in their growth. Our objective is to enhance the integration of deep learning and complex fluid dynamics, facilitating the resolution of more realistic and complex flow problems.

Physics-informed neural networks coupled with flamelet/progress variable model for solving combustion physics considering detailed reaction mechanism

In recent years, physics-informed neural networks (PINNs) have shown potential as a method for solving combustion physics. However, current efforts using PINNs for the direct predictions of multi-dimensional flames only use global reaction mechanisms. Considering detailed chemistry is crucial for understanding detailed combustion physics, and how to accurately and efficiently consider detailed mechanisms under the framework of PINNs has not been explored yet and is still an open question. To this end, this paper proposes a PINN/flamelet/progress variable (FPV) approach to accurately and efficiently solve combustion physics, considering detailed chemistry. Specifically, the combustion thermophysical properties are tabulated using several control variables, with the FPV model considering detailed chemistry. Then, PINNs are used to solve the governing equations of continuity, momentum, and control variables with the thermophysical properties extracted from the FPV library. The performance of the proposed PINN/FPV approach is assessed for diffusion flames in a two-dimensional laminar mixing layer by comparing it with the computational fluid dynamics (CFD) results. It has been found that the PINN/FPV model can accurately reproduce the flow and combustion fields, regardless of the presence or absence of observation points. The quantitative statistics demonstrated that the mean relative error was less than 10%, and R2 values were all higher than 0.94. The applicability and stability of this model were further verified on other unseen cases with variable parameters. This study provides an efficient and accurate method to consider detailed reaction mechanisms in solving combustion physics using PINNs.

Prediction of spatiotemporal dynamics using deep learning: Coupled neural networks of long short-terms memory, auto-encoder and physics-informed neural networks

Several classic reaction-diffusion models using partial differential equations (PDEs) have been established to elucidate the formation mechanism of vegetation patterns. However, predictive modeling of complex spatiotemporal dynamics using traditional numerical methods can be significantly challenging in many practical scenarios. Physics-Informed Neural Networks (PINNs) provide a new approach to predict the solution of PDEs. However, the generalization of PINNs is not satisfactory when pretrained PINNs is directly used in non-trained space (defined as explorations). This may be attributed to the lack of training in the time dimension. Therefore, a framework (LA-PINNs) is proposed to predict the evolutionary solution of the non-dimensional vegetation-sand model. The framework couples neural networks of Long-Short Terms Memory, Auto-Encoder and Physics-Informed Neural Networks. The predictions of LA-PINNs are much better than those of PINNs. Then we studied the effects of hyperparameters on the accuracy of predictions. Based on training in time dimension by LSTM module and pretraining for quick-training strategy, LA-PINNs can improve the accuracy of explorations.

The data-driven solutions and inverse problems of some nonlinear diffusion convection-reaction equations based on Physics-Informed Neural Network

The Physics-Informed Neural Network (PINN) has achieved remarkable results in solving partial differential equations (PDEs). This paper aims to solve the forward and inverse problems of some specific nonlinear diffusion convection-reaction equations, thereby validating the practical efficacy and accuracy of data-driven approaches in tackling such equations. In the forward problems, four different solutions of the studied equations are reproduced effectively and the approximation errors can be reduced to 10−5. Experiments indicate that the PINNs method based on adaptive activation functions (PINN-AAF), outperforms the standard PINNs in dealing with inverse problems. The unknown parameters are estimated effectively and the approximation errors can lower to 10−4. Additionally, training rules for both PINN and PINN-AAF are summarized. The results of this study validate the exceptional performance of the data-driven approach in solving the complex nonlinear diffusion convection-reaction equation problems, and provide an effective mechanism for dealing with analogous, intricate nonlinear problems.

Mesh-Free Solution of 2D Poisson Equation with High Frequency Charge Patterns Using Data-Free Physics Informed Neural Network

Abstract In this paper, we present a data-free physics-informed neural networks (PINNs) approach for solving two-dimensional (2D) Poisson equation, which is pivotal in fields such as electromagnetics, mechanical engginering, and thermodynamics. Traditional numerical method for solving this equation often require structured mesh generation such as Finite Element Method (FEM), which can be computationally expensive when dealing with high resolution Poisson Equation Solution. To address this challenge, we leverage the capabilities of PINNs capturing pattern of complex system by incorporating physical law and boundary condition as part of loss function on training model. While PINNs provide a robust framework for solving differential equations within boundary condition, they have struggle with capturing high-frequency pattern due to smooth nature of typical activation function used in neural networks. To evercome this issue, we enhance our model by incorporating Fourier Features Networks, which map inputs through a series of sinusoidal functions before feeding the input into the neural network. The result show that Fourier feature network can enhance convergence of training of PINNs model faster and obtained better result than PINNs without Fourier feature networks.

Data Assimilation and Parameter Identification for Water Waves Using the Nonlinear Schrödinger Equation and Physics-Informed Neural Networks

The measurement of deep water gravity wave elevations using in situ devices, such as wave gauges, typically yields spatially sparse data due to the deployment of a limited number of costly devices. This sparsity complicates the reconstruction of the spatio-temporal extent of surface elevation and presents an ill-posed data assimilation problem, which is challenging to solve with conventional numerical techniques. To address this issue, we propose the application of a physics-informed neural network (PINN) to reconstruct physically consistent wave fields between two elevation time series measured at distinct locations within a numerical wave tank. Our method ensures this physical consistency by integrating residuals of the hydrodynamic nonlinear Schrödinger equation (NLSE) into the PINN’s loss function. We first showcase a data assimilation task by employing constant NLSE coefficients predetermined from spectral wave properties. However, due to the relatively short duration of these measurements and their possible deviation from the narrow-band assumptions inherent in the NLSE, using constant coefficients occasionally leads to poor reconstructions. To enhance this reconstruction quality, we introduce the base variables of frequency and wavenumber, from which the NLSE coefficients are determined, as additional neural network parameters that are fine tuned during PINN training. Overall, the results demonstrate the potential for real-world applications of the PINN method and represent a step toward improving the initialization of deterministic wave prediction methods.

Physics-informed neural networks in support of modal wavenumber estimation.

A physics-informed neural network (PINN) enables the estimation of horizontal modal wavenumbers using ocean pressure data measured at multiple ranges. Mode representations for the ocean acoustic pressure field are derived from the Hankel transform relationship between the depth-dependent Green's function in the horizontal wavenumber domain and the field in the range domain. We obtain wavenumbers by transforming the range samples to the wavenumber domain, and maintaining range coherence of the data is crucial for accurate wavenumber estimation. In the ocean environment, the sensitivity of phase variations in range often leads to degradation in range coherence. To address this, we propose using OceanPINN [Yoon, Park, Gerstoft, and Seong, J. Acoust. Soc. Am. 155(3), 2037-2049 (2024)] to manage spatially non-coherent data. OceanPINN is trained using the magnitude of the data and predicts phase-refined data. Modal wavenumber estimation methods are then applied to this refined data, where the enhanced range coherence results in improved accuracy. Additionally, sparse Bayesian learning, with its high-resolution capability, further improves the modal wavenumber estimation. The effectiveness of the proposed approach is validated through its application to both simulated and SWellEx-96 experimental data.

Physics-Informed Neural Network (PINN) model for predicting subgrade settlement induced by shield tunnelling beneath an existing railway subgrade

For the shield tunnelling beneath an existing railway subgrade, factors such as disturbance during shield tunnel excavation can cause settlements of the existing subgrade above. Traditional prediction methods for subgrade settlement may not fully reflect the construction site conditions. Recently, machine learning has been widely applied due to its fast computation speed and accurate results. However, data-driven machine learning is considered a “black-box” algorithm, lacking a mechanical theory framework and having weak generalization capabilities. To address the limitations of machine learning and enhance the efficiency and accuracy of subgrade settlement prediction during shield tunnel construction, this paper proposes a prediction model based on a physics-informed neural network (PINN). Firstly, by integrating the non-uniform displacement mode function of the tunnel, the physical equations are constructed for the shield tunnel excavation process. Subsequently, the physical equations are coupled with the machine learning framework to develop a predictive model for shield tunnelling beneath an existing railway subgrade based on the PINN model. The performance of the proposed PINN model is evaluated through the application of two engineering cases and compared with traditional numerical simulations and data-driven machine learning. The results indicate that the proposed model can not only predict the settlement of the existing railway subgrade under scant and noisy monitoring data but also have accuracy in the inversion of tunnel displacement mode parameters. Finally, parametric analyses are conducted to reveal the influence of critical variables on the prediction performance of the PINN model.

Application for Predicting Vessel Motion Enhances Operability in Offshore Drilling

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper OTC 35341, “Enhancing Energy Efficiency and Operability in Offshore Drilling Operations: Validation and Benefits of a Machine-Learning-Based Vessel-Motion-Prediction Application,” by Daniel A.C. Canache, Massimiliano Russo, and Rune Haakonsen, Kongsberg Maritime, et al. The paper has not been peer reviewed. Copyright 2024 Offshore Technology Conference. _ This paper delves into the framework of a new application that leverages advanced machine-learning (ML) techniques in conjunction with metocean forecasts to predict vessel motions and thruster loads. The discussion encompasses the key role of diverse input factors in shaping prediction accuracy and outlines strategies to address inherent uncertainties associated with methods, weather forecasts, and the stochastic nature of the underlying problem. Methods, Procedures, and Process Traditional Physics-Based Vessel Model. A physics-based model of the vessel is used to train the ML model. As such, the accuracy of the physics-based model is critical to the success of the ML model. The model is built using vessel-specific hydrodynamic data, mass data, and thruster-control software. The thruster-control software is extracted from the actual vessel to ensure that the same settings are being used in the model. The model is used to execute time-domain simulations, which include second-order effects, from which prediction targets [six degrees-of-freedom (DOF) motions and thruster usage] are extracted. As shown in Fig. 1, four environmental components are typically considered: wind waves, swell waves, wind, and current. To train an ML model capable of predicting in any given weather condition, environmental parameters are varied over thousands of simulations. Physics-Informed Neural Network (PINN) Architecture. The adopted architecture for the ML model is a PINN, a model architecture that offers several advantages. By integrating vessel-specific hydrodynamic data, the model becomes grounded firmly in the underlying physical phenomena. Furthermore, the architecture facilitates the model’s comprehension and assessment of the environment by enabling the compilation of individual loads into an overall load acting on the vessel. Ultimately, the analysis of resultant loads on the vessel provides a means to compare various environmental load cases directly in relation to the vessel’s response. Optimizing the Training Matrix. The optimal training matrix should span the environmental parameter space up to the vessel’s station-keeping capacity while minimizing the number of necessary load cases. Given the computational expense and prolonged execution times associated with time-domain simulations, it is important to maintain a manageable number of cases. While resultant loads facilitate the comparison across different environment load cases, a systematic algorithm is needed for case selection. To address this requirement, a similarity metric incorporating cosine similarity and Euclidean distance was implemented. The downside to the Euclidean distance is that, at low magnitudes, the distance will be low regardless of direction. This is mitigated to some extent by combining the measure with the cosine similarity. The cosine similarity will be high if two vectors are close in direction regardless of their magnitudes. For the specific problem described in the complete paper (vessel responses), it is desirable that the similarities at low loads stay high because the response also will likely be low.

A physics-informed neural network enhanced importance sampling (PINN-IS) for data-free reliability analysis

Reliability analysis of highly sensitive structures is crucial to prevent catastrophic failures and ensure safety. Therefore, these safety-critical systems are to be designed for extremely rare failure events. Accurate statistical quantification of these events associated with a very low probability of occurrence requires millions of evaluations of the limit state function (LSF) involving computationally expensive numerical simulations. Variance reduction techniques like importance sampling (IS) reduce such repetitions to a few thousand. The use of a data-centric metamodel can further cut it down to a few hundred. In data-centric metamodeling approaches, the actual complex numerical analysis is performed at a few points to train the metamodel for approximating the structural response. On the other hand, a physics-informed neural network (PINN) can predict the structural response based on the governing differential equation describing the physics of the problem, without a single evaluation of the complex numerical solver, i.e., data-free. However, the existing PINN models for reliability analysis have been effective only in estimating a large range of failure probabilities (10-1∼10-3). To address this issue, the present study develops a PINN-based data-free reliability analysis for low failure probabilities (<10-5). In doing so, a two-stage PINN integrated with IS (PINN-IS) is proposed. The first stage is employed to approximate the most probable failure point (MPP) appropriately while the second stage enhances the accuracy of LSF predictions at the IS population centred on the approximated MPP. The effectiveness of the proposed approach is numerically illustrated by three structural reliability analysis examples.

On acoustic solitary waves in a multispecies degenerate relativistic magnetized plasma using physics informed neural networks

In this paper, we investigate the nonlinear electrostatic wave propagation in a two-dimensional magnetized plasma. The plasma consists of electron and positron components with relativistic degeneracy and stationary ions for neutralizing its background. Using the basic equations for this type of plasma in combination with the reductive perturbation method, we derived the Zakharov–Kuznetsov equation using the Lorentz transformation stretching method (LT). For the first time, we compared the results of the Galilean transformation stretching method (GT) and the LT method to investigate the effect of plasma parameters, such as the relativistic degeneracy parameter of electron particles (re0), the density ratio of ion to electrons (δ), and the normalized electron cyclotron (Ωe), on the amplitude and width of the wave solutions. The plasma parameters used in this research are representative of compact astrophysical objects. Numerical results showed that the amplitude of wave solutions obtained by the LT method is smaller than the GT method, but the width is greater. We provide a physical explanation for these differences. Furthermore, we present a physics-informed neural network (PINN) approach to directly recover the intrinsic nonlinear dynamics from spatiotemporal data. The PINN model uses a deep neural network constrained by the governing equations to learn the optimal parameters, with the aim of enhancing the predictive capabilities of the system. The results of this study provide valuable insight into the propagation of nonlinear waves in white dwarfs, where relativistic effects are significant. These findings could substantially advance the development of emerging machine learning applications in astrophysics.

Addressing the non-perturbative regime of the quantum anharmonic oscillator by physics-informed neural networks

Abstract The use of deep learning in physical sciences has recently boosted the ability of researchers to tackle physical systems where little or no analytical insight is available. Recently, the Physics−Informed Neural Networks (PINNs) have been introduced as one of the most promising tools to solve systems of differential equations guided by some physically grounded constraints. In the quantum realm, such an approach paves the way to a novel approach to solve the Schrödinger equation for non-integrable systems. By following an unsupervised learning approach, we apply the PINNs to the anharmonic oscillator in which an interaction term proportional to the fourth power of the position coordinate is present. We compute the eigenenergies and the corresponding eigenfunctions while varying the weight of the quartic interaction. We bridge our solutions to the regime where both the perturbative and the strong coupling theory work, including the pure quartic oscillator. We investigate systems with real and imaginary frequency, laying the foundation for novel numerical methods to tackle problems emerging in quantum field theory.

Entropy production minimization in channel flows using computational fluid dynamics and physics-informed neural networks

Entropy production minimization in channel flows using computational fluid dynamics and physics-informed neural networks