Physics-informed Neural Networks Research Articles

The Potential for Material Property Characterization Using Physics-Informed Neural Networks and Ultrasonic Wave Data

ABSTRACT There is a need for reliable nondestructive test methods that can collect data from structural members and analyze the results in a rapid and efficient manner. Large amounts of test data are needed to achieve such characterization, which provides additional challenges because of their heterogeneity and complexity. Advances in machine learning, in particular physics-informed neural networks (PINN), offer potential to address these problems. PINN is a particular form of artificial neural networks (ANN) and portends notable advantages over traditional measurand analysis or purely data-driven approaches. Here, we explore the potential of heterogeneous material property characterization using PINN and ultrasonic wave data. First, several types of 1-D ultrasonic wave data are numerically simulated for a spatially heterogeneous material, and then PINN is applied to predict wave velocity, defect location, and energy dissipation. Then, three different types of defects are simulated and all defects are detected using the corresponding 2-D ultrasonic wave data and PINN. The presented results demonstrate the promise of PINN to assist with heterogeneous material characterization methods.

Improving the Efficiency of Training Physics-Informed Neural Networks Using Active Learning

Improving the Efficiency of Training Physics-Informed Neural Networks Using Active Learning

A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations

Abstract Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs). To this end, we suggest to use a weighted sum of expansion coefficients of the residual in terms of an adaptive wavelet expansion both for the loss function and an error bound. This approach is shown here for elliptic PPDEs using both the standard variational and an optimally stable ultra-weak formulation. Numerical examples show a very good quantitative effectivity of the wavelet-based error bound.

Prediction of Dynamic Ni Morphology Changes in Patterned Ni-YSZ Anode with Physics-Informed Neural Networks

Abstract Nickel (Ni) film on patterned Ni- yttria-stabilized zirconia (YSZ) anode shows dynamic spreading and splitting during solid oxide fuel cell (SOFC) operation, where wettability of Ni on YSZ is greatly enhanced (Z. Jiao, N. Shikazono, J. Power Sources 396 119–123, 2018). In the present study, a physics-informed neural network (PINN) constrained by Cahn-Hilliard equation of phase field model is proposed to estimate the unknown parameters for predicting dynamic Ni movements of the patterned Ni-YSZ anode. The unknown parameters such as interface thickness and mobility are inversely inferred by PINN using top-view images obtained from the operando experiments. Obtained excess surface diffusivity values were three to four orders of magnitude larger than the values reported for surface diffusion in the literature. It is therefore considered that Ni spreading and splitting of patterned anode cannot be simply explained by surface diffusion, and other mechanisms should be introduced.

Solving forward and inverse problems of contact mechanics using physics-informed neural networks

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush–Kuhn–Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer–Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.

Physics-Informed Machine Learning of Argon Gas-Driven Melt Pool Dynamics

Abstract Melt pool dynamics in metal additive manufacturing (AM) is critical to process stability, microstructure formation, and final properties of the printed materials. Physics-based simulation including computational fluid dynamics (CFD) is the dominant approach to predict melt pool dynamics. However, the physics-based simulation approaches suffer from the inherent issue of very high computational cost. This paper provides a physics-informed machine learning method by integrating the conventional neural networks with the governing physical laws to predict the melt pool dynamics such as temperature, velocity, and pressure without using any training data on velocity. This approach avoids solving numerically the non-linear Navier-Stokes equation, which significantly reduces the computational cost (if including the cost of velocity data generation). The difficult-to-determine parameters' values of the governing equations can also be inferred through data-driven discovery. In addition, the physics-informed neural network (PINN) architecture has been optimized for efficient model training. The data-efficient PINN model is attributed to the extra penalty by incorporating governing PDEs, initial conditions, and boundary conditions in the PINN model.

Analyzing magnetic dipole impact in fluid flow with endothermic/exothermic reactions: neural network simulation

The consequence of exothermic/endothermic chemical reactions and Arrhenius activation on the heat and mass transport of the liquid flow past a cylinder in the incidence of a magnetic dipole is considered in the current investigation. Magnetic dipoles are used in medical applications such as magnotherapy and spectroscopy, to produce static magnetic fields. Scientists and engineers can improve the effectiveness of chemical reactions or heat transfer operations by analyzing the impact of reactions on flow and building systems with optimized flows. The modelled equations are converted into non-dimensional ordinary differential equations (ODEs) by using similarity variables. The resultant equations are solved by employing the physics-informed neural network (PINN) technique. Additionally, the comparison of PINN with the numerical method Runge–Kutta Fehlberg’s fourth-fifth order (RKF-45) is studied. The effects of different parameters on the temperature, concentration, and velocity profiles for endothermic/exothermic instances are shown graphically. The thermal, velocity, and concentration profiles get stronger as the curvature parameter values increase for both endothermic and exothermic cases. The influence of activation energy parameters, chemical reaction parameters, and endothermic/exothermic reaction parameters on the thermal and concentration is also depicted.

Modified failproof physics-informed neural network framework for fast and accurate optical fiber transmission link modeling

Physics-informed neural networks (PINNs) have recently emerged as an important and ground-breaking technique in scientific machine learning for numerous applications including in optical fiber communications. However, the vanilla/baseline version of PINNs is prone to fail under certain conditions because of the nature of the physics-based regularization term in its loss function. The use of this unique regularization technique results in a highly complex non-convex loss landscape when visualized. This leads to failure modes in PINN-based modeling. The baseline PINN works very well as an optical fiber model with relatively simple fiber parameters and for uncomplicated transmission tasks. Yet, it struggles when the modeling task becomes relatively complex, reaching very high error, for example, numerous modeling tasks/scenarios in soliton communication and soliton pulse development in special fibers such as erbium-doped dispersion compensating fibers. We implement two methods to circumvent the limitations caused by the physics-based regularization term to solve this problem, namely, the so-called scaffolding technique for PINN modeling and the progressive block learning PINN modeling strategy to solve the nonlinear Schrödinger equation (NLSE), which models pulse propagation in an optical fiber. This helps PINN learn more accurately the dynamics of pulse evolution and increases accuracy by two to three orders of magnitude. We show in addition that this error is not due to the depth or architecture of the neural network but a fundamental issue inherent to PINN by design. The results achieved indicate a considerable reduction in PINN error for complex modeling problems, with accuracy increasing by up to two orders of magnitude.

Physics-informed neural networks for solving flow problems modeled by the 2D Shallow Water Equations without labeled data

This paper investigates the application of physics-informed neural networks (PINNs) to solve free-surface flow problems governed by the 2D shallow water equations (SWEs). Two types of PINNs are developed and analyzed: a physics-informed fully connected neural network (PIFCN) and a physics-informed convolutional neural network (PICN). The PINNs eliminate the need for labeled data for training by employing the SWEs, initial and boundary conditions as components of the loss function to be minimized. Results from a set of idealized and real-world tests showed that the prediction accuracy and computation time (i.e., training time) of both PINNs may be less affected by the resolution of the domain discretization when compared against solutions by a Finite Volume (FV) model. Overall, the PICN shows a better trade-off between computational speed and accuracy than the PIFCN. Also, our results for the idealized problems indicated that PINNs can provide more than 5 times higher prediction accuracy than the FV model, while the FV simulation with coarse resolution (e.g., 10 m) can provide sub-centimeter accurate (RMSE) solutions at least one order of magnitude faster than the PINNs.Results from a river flood simulation showed that PINNs delivered better speed-accuracy trade-off than the FV model in terms of predicting the water depth, while FV models outperformed the PINNs for predictions of total flow discharge.

Physics-informed neural networks in groundwater flow modeling: Advantages and future directions

In recent years, there has been enormous development in soft computing, especially artificial intelligence (AI), which has developed robust methods for solving complex engineering problems. Researchers in the field of water resources engineering have applied these AI methods to solve a variety of hydrological problems. Despite their widespread use in the surface and atmospheric hydrology fields, groundwater hydrologists have not widely used AI methods in their routine field-scale modeling efforts. This is because AI models have been primarily considered black box models that lack physical meaning. Furthermore, using AI models to generate the space-time distribution of transient groundwater level variations is challenging and requires further flux balance and mass transport analyses. More recently, a new type of physics-informed neural network (PINN) model has been developed to address several limitations by integrating governing physics (groundwater flow equations) into the AI tools. This study presents the systematic advantages of the PINN algorithm for solving groundwater problems using a set of classic test problems. As discussed in detail in the article, these advantages and potentials are associated with the meshless nature of PINN, its continuous time and space dimensions, its independence from time-stepping and incremental marching in space, and its efficiency in running time. However, despite PINN's promising attributes, it is important to acknowledge its nascent stage of development and the inherent limitations of all neural network models, such as training challenges and hyperparameter selection. Thus, collaborative efforts between groundwater modelers and computer scientists are imperative to explore and exploit the full potential of PINN in tackling increasingly complex groundwater problems and nurturing PINN into a dependable modeling tool in industry and academia.

NG-DPSM: A neural green-distributed point source method for modelling ultrasonic field emission near fluid-solid interface using physics informed neural network

Distributed point source method (DPSM) is a collocation point-based semi-analytical method to model the scattered ultrasonic fields in isotropic and anisotropic materials. It requires the evaluation of Green's function solutions and its rigorous differentiation for complex geometry problems having a large set of distributed point sources. DPSM is much faster than the finite element method (FEM) however, it is analytically demanding as it requires differentiation of the Green's function. The intricacy associated with differentiation operation hinders the potential application of DPSM in large-scale automation and real-time structural health monitoring. The current paper introduces machine-learning-based strategies within DPSM and yields the proposed Neural Green-DPSM (NG-DPSM) framework. A physics-informed neural network (PINN) is incorporated in the DPSM framework for evaluating the Green's function and its gradients. To train the PINN, displacement Green's function solutions of wave propagation in isotropic solid is used as the governing physics. Once the PINN is trained for displacement Green's function solutions, the NG-DPSM framework leverages the trained network and its automatic differentiation capability to predict displacement and stress fields annihilating the rigorous differentiation of Green's functions. The accuracy and efficacy of NG-DPSM are demonstrated using two numerical experiments. First, the ultrasonic fields are evaluated for the problem geometry of a plate immersed in fluid excited at different angles of incidence. Further, the efficacy of the proposed approach is demonstrated for wave scattering through a circular hole in the plate. The results show a strong agreement between the ultrasonic fields computed using both NG-DPSM and conventional DPSM.

A macroscopic traffic flow modelling and collision avoidance using B-spline and Physics-Informed Neural Network approaches

A macroscopic traffic flow modelling and collision avoidance using B-spline and Physics-Informed Neural Network approaches

Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks

In this paper, we apply a machine learning approach to learning traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture called the Separable Gaussian Neural Network (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs, which treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into what is called the co-traveling wave frame. This adaptation effectively addresses the issue of propagation failure in PINNs when applied to large computational domains. Here, the SGNN architecture demonstrates robust approximation capabilities for single-peakon, multi-peakon, and stationary solutions (known as “leftons”) within the (1 + 1)-dimensional b-family of PDEs. In addition, we expand our investigation and explore not only peakon solutions in the ab-family but also compacton solutions in the (2 + 1)-dimensional Rosenau–Hyman family of PDEs. A comparative analysis with a multi-layer perceptron (MLP) reveals that the SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader applications in solving complex nonlinear PDEs.

Exploring Physics-Informed Neural Networks for the Generalized Nonlinear Sine-Gordon Equation

The nonlinear sine-Gordon equation is a prevalent feature in numerous scientific and engineering problems. In this paper, we propose a machine learning-based approach, physics-informed neural networks (PINNs), to investigate and explore the solution of the generalized non-linear sine-Gordon equation, encompassing Dirichlet and Neumann boundary conditions. To incorporate physical information for the sine-Gordon equation, a multiobjective loss function has been defined consisting of the residual of governing partial differential equation (PDE), initial conditions, and various boundary conditions. Using multiple densely connected independent artificial neural networks (ANNs), called feedforward deep neural networks designed to handle partial differential equations, PINNs have been trained through automatic differentiation to minimize a loss function that incorporates the given PDE that governs the physical laws of phenomena. To illustrate the effectiveness, validity, and practical implications of our proposed approach, two computational examples from the nonlinear sine-Gordon are presented. We have developed a PINN algorithm and implemented it using Python software. Various experiments were conducted to determine an optimal neural architecture. The network training was employed by using the current state-of-the-art optimization methods in machine learning known as Adam and L-BFGS-B minimization techniques. Additionally, the solutions from the proposed method are compared with the established analytical solutions found in the literature. The findings show that the proposed method is a computational machine learning approach that is accurate and efficient for solving nonlinear sine-Gordon equations with a variety of boundary conditions as well as any complex nonlinear physical problems across multiple disciplines.

Applications of Symmetry-Enhanced Physics-Informed Neural Networks in High-Pressure Gas Flow Simulations in Pipelines

Applications of Symmetry-Enhanced Physics-Informed Neural Networks in High-Pressure Gas Flow Simulations in Pipelines

Physics-Informed Neural Networks for an optimal counterdiabatic quantum computation

Abstract A novel methodology that leverages Physics-Informed Neural Networks to optimize quantum circuits in systems with NQ qubits by addressing the counterdiabatic protocol is introduced. The primary purpose is to employ physics-inspired deep learning techniques for accurately modeling the time evolution of various physical observables within quantum systems. To achieve this, we integrate essential physical information into an underlying neural network to effectively tackle the problem. Specifically, the imposition of the solution to meet the principle of least action, along with the hermiticity condition on all physical observables, among others, ensuring the acquisition of appropriate counterdiabatic terms based on underlying physics. This approach provides a reliable alternative to previous methodologies relying on classical numerical approximations, eliminating their inherent constraints. The proposed method offers a versatile framework for optimizing physical observables relevant to the problem, such as the scheduling function, gauge potential, temporal evolution of energy levels, among others. This methodology has been successfully applied to 2-qubit representing H2 molecule using the STO-3G basis, demonstrating the derivation of a desirable decomposition for non-adiabatic terms through a linear combination of Pauli operators. This attribute confers significant advantages for practical implementation within quantum computing algorithms.

Adaptive Deep Fourier Residual method via overlapping domain decomposition

The Deep Fourier Residual (DFR) method is a specific type of variational physics-informed neural network (VPINN). It provides a robust neural network-based solution to partial differential equations (PDEs). The DFR strategy is based on minimizing the dual norm of the weak residual of a PDE, which is equivalent to minimizing the energy norm of the error. To compute the dual norm of the weak residual, the DFR method employs an orthonormal spectral basis of the test space, known for rectangles or cuboids for multiple function spaces.In this work, we extend the DFR method with ideas of traditional domain decomposition (DD). This enables two improvements: (a) to solve problems in more general polygonal domains and (b) to develop an adaptive refinement technique in the test space using a Döfler marking algorithm. In the former case, we retain the desirable equivalence between the employed loss function and the H1 error under non-restrictive assumptions, numerically demonstrating adherence to explicit bounds in the case of the L-shaped domain problem. In the latter, we show how refinement strategies lead to potentially significant improvements against a reference, classical DFR implementation with a test function space of significantly lower dimensionality, allowing us to better approximate singular solutions at a more reasonable computational cost.

Dynamically configured physics-informed neural network in topology optimization applications

The integration of physics-informed neural network (PINN) and topology optimization (TO) is an attractive issue because PINN can avoid the prohibitive data acquisition of solving forward problems compared to traditional machine learning. To enhance the efficiency of this integrated optimization framework, a dynamically configured PINN-based topology optimization (DCPINN-TO) method is proposed in conjunction with an active sampling strategy. The DCPINN comprises two sub-networks with distinct training costs, capable of dynamically adjusting the trainable parameters based on the optimization state of TO. Moreover, the active sampling strategy selectively samples collocations based on the pseudo-densities, which can significantly reduce training costs by decreasing the number of input collocations. Additionally, the Gaussian integral is used to calculate the strain energy of elements to decouple the mapping of the material at the collocations. The proposed method is extended to various scenarios, including those with high resolution, multiple loads, and displacement constraints. Its efficiency and generalization are validated by several illustrative examples. Furthermore, the accuracy of DCPINN versus finite element analysis-based TO (FEA-TO) was also investigated.

PINN-CHK: physics-informed neural network for high-fidelity prediction of early-age cement hydration kinetics

AbstractCement hydration kinetics, characterized by heat generation in early-age concrete, poses a modeling challenge. This work proposes a physics-informed neural network (PINN) named PINN-CHK designed for cement hydration kinetics, to predict early-age temperature rises in cement paste. PINN-CHK leverages data-driven solutions to craft a high-fidelity prediction model, encompassing material properties and maturity functions in cement hydration. Trained on heated cement paste data, it simultaneously fits experimental results and underlying physics, yielding a mesh-free simulation. Incorporating governing partial differential equations (PDEs), and initial and boundary conditions into its loss function, PINN-CHK architecture undergoes rigorous benchmark testing, demonstrating unparalleled predictive accuracy compared to conventional deep-learning methods. It excels in predicting complete temperature fields during spatial–temporal cement hydration, achieving a remarkable relative L2 error as low as 0.00341. PINN-CHK achieves exceptional convergence and accuracy with only 5% of the training data, ushering in a new era in this crucial field. This innovative approach bridges the gap between theory and practice, offering an attractive alternative to conventional finite element solvers for enhanced comprehension of cement hydration kinetics and concrete maturity and strength development in cement-based materials.

Physics-informed deep-learning applications to experimental fluid mechanics

High-resolution reconstruction of flow-field data from low-resolution and noisy measurements is of interest due to the prevalence of such problems in experimental fluid mechanics, where the measurement data are in general sparse, incomplete and noisy. Deep-learning approaches have been shown suitable for such super-resolution tasks. However, a high number of high-resolution examples is needed, which may not be available for many cases. Moreover, the obtained predictions may lack in complying with the physical principles, e.g. mass and momentum conservation. Physics-informed deep learning provides frameworks for integrating data and physical laws for learning. In this study, we apply physics-informed neural networks (PINNs) for super-resolution of flow-field data both in time and space from a limited set of noisy measurements without having any high-resolution reference data. Our objective is to obtain a continuous solution of the problem, providing a physically-consistent prediction at any point in the solution domain. We demonstrate the applicability of PINNs for the super-resolution of flow-field data in time and space through three canonical cases: Burgers’ equation, two-dimensional vortex shedding behind a circular cylinder and the minimal turbulent channel flow. The robustness of the models is also investigated by adding synthetic Gaussian noise. Furthermore, we show the capabilities of PINNs to improve the resolution and reduce the noise in a real experimental dataset consisting of hot-wire-anemometry measurements. Our results show the adequate capabilities of PINNs in the context of data augmentation for experiments in fluid mechanics.