Physics-informed Neural Networks Research Articles

Anti-derivatives approximator for enhancing physics-informed neural networks

This study presents a novel strategy for constructing an approximator for arbitrary univariate functions. The proposed approximation utilizes the anti-derivatives of a Fourier series expansion for the presumed piecewise function, resulting in a remarkable feature that enables the simultaneous approximation of an arbitrary function and its anti-derivatives. These anti-derivatives can be employed to discover solution curves for systems of ordinary differential equations based on an optimization scheme, even in the presence of chaotic dynamics. Additionally, the anti-derivatives approximator is extended as an adaptive activation function for physics-informed neural networks, leveraging the high-order differentiability of the anti-derivatives. Systematic experiments have demonstrated the outstanding merits of the proposed anti-derivatives-based approximator, including its ability to construct regression models for volatile data and their anti-derivatives, solve differential equations, and enhance the capabilities of physics-informed neural networks.

Efficiently Training Physics-Informed Neural Networks via Anomaly-Aware Optimization

Efficiently Training Physics-Informed Neural Networks via Anomaly-Aware Optimization

Splitting Physics-Informed Neural Networks for Inferring the Dynamics of Integer- and Fractional-Order Neuron Models

Splitting Physics-Informed Neural Networks for Inferring the Dynamics of Integer- and Fractional-Order Neuron Models

A novel method for response probability density of nonlinear stochastic dynamic systems

AbstractThis paper presents a novel method for analyzing high-dimensional nonlinear stochastic dynamic systems. In particular, we attempt to obtain the solution of the Fokker–Planck–Kolmogorov (FPK) equation governing the response probability density of the system without using the FPK equation directly. The method consists of several important components including the radial basis function neural networks (RBFNN), Feynman–Kac formula and the short-time Gaussian property of the response process. In the area of solving partial differential equations (PDEs) using neural networks, known as physics-informed neural network (PINN), the proposed method presents an effective alternative for obtaining solutions of PDEs without directly dealing with the equation, thus avoids evaluating the derivatives of the equation. This approach has a potential to make the neural network-based solution more efficient and accurate. Several highly challenging examples of nonlinear stochastic systems are presented in the paper to illustrate the effectiveness of the proposed method in comparison to the equation-based RBFNN approach.

Newtonian Physics Informed Neural Network (NwPiNN) for Spatio-Temporal Forecast of Visual Data

AbstractMachine intelligence is at great height these days and has been evident with its effective provenance in almost all domains of science and technology. This work will focus on one handy and profound application of machine intelligence-time series forecast, and that too on visual data points, i.e., our objective is to predict future visual data points, given a subtle lag to work on. For the same, we would propose a deep learner, Newtonian physics informed neural network (NwPiNN) with the critical modelling capabilities of the physics informed neural networks, modelled on the laws of Newtonian physics. For computational efficacy, we would work on the gray-scale values of pixels. Since the variation in data pixel values is not only provoked by the pixel gray values but also by the velocity component of each pixel, the final prediction of the model would be a weighted average of the gray value forecast and the kinematics of each pixel, as modelled by the PINN. Besides its’ proposal, NwPiNN is subjected to benchmark visual dataset, and compared with existing models for visual time series forecast, like ConvLSTM, and CNN-LSTM, and in most of the occasions, NwPiNN is found to outperform its preliminaries.

A multifidelity approach to continual learning for physical systems

We introduce a novel continual learning method based on multifidelity deep neural networks. This method learns the correlation between the output of previously trained models and the desired output of the model on the current training dataset, limiting catastrophic forgetting. On its own the multifidelity continual learning method shows robust results that limit forgetting across several datasets. Additionally, we show that the multifidelity method can be combined with existing continual learning methods, including replay and memory aware synapses, to further limit catastrophic forgetting. The proposed continual learning method is especially suited for physical problems where the data satisfy the same physical laws on each domain, or for physics-informed neural networks, because in these cases we expect there to be a strong correlation between the output of the previous model and the model on the current training domain.

Data-assisted training of a physics-informed neural network to predict the separated Reynolds-averaged turbulent flow field around an airfoil under variable angles of attack

AbstractPhysics-informed neural networks are a promising method to yield surrogate models of flow fields. We present a metamodeling technique for variable geometries based on physics-informed neural networks. The method was applied to the DU99W350 airfoil at a Reynolds number of $$1\times 10^{5}$$ 1 × 10 5 . Using our technique, the angle of attack was introduced as an additional input parameter of the network and the model was trained to predict the Reynolds-averaged velocity and pressure fields around the airfoil for arbitrary angles of attack between 10.0 and 17.5$$^{\circ }$$ ∘ . Furthermore, we present an effective method to generate the training points for the parameterized geometry. The model was trained with data from simulations for a limited set of angles of attack. Additionally, satisfaction of the a priori known boundary conditions as well as the Reynolds-averaged Navier–Stokes equations was attained. A sensitivity analysis concerning the Reynolds number, the amount and distribution of training data, and the turbulence model was conducted showing the superiority of the pseudo-Reynolds stress method and the demand for labeled training data in the domain. The trained network was capable of predicting the flow separation progressing with angle of attack on the suction surface and exhibited excellent agreement with numerically simulated results, even in the proximity of the wall for interpolations as well as extrapolations from the labeled data set. Our study demonstrates that physics-informed neural networks can be used to obtain accurate flow field surrogate models of variable geometries.

Soliton Solutions of Some Ocean Waves Supported by Physics Informed Neural Network Method

In this study, we aim to obtain numerical results of the modified Benjamin-Bona-Mahony equation, Ostrovsky-Benjamin-Bona-Mahony equation and Mikhailov-Novikov-Wang equation via the physics-informed neural networks (PINN) method. The equations are modeled for shallow and long water waves, as well as fundamental and phenomenonal models in ocean engineering. According to the implementation, we obtained the PINN solutions of kink, bright, multisoliton (two-soliton) and mixed dark-bright soliton solutions. According to the inference from the obtained results, we achieved good results in some cases compared to other approximate solution methods in the literature. However, it was also observed that the best possible results could not be obtained in cases where the soliton type was intricate and layered. While the results were obtained, the number of hidden layers and the number of neural networks in the layers also varied. These results are shown in tables. Since it is known that the aforementioned models are not solved by the PINN method, we anticipate that the study will lead to other studies in the field of ocean engineering.

TDOR-MPINNs: Multi-output physics-informed neural networks based on time differential order reduction for solving coupled Klein–Gordon–Zakharov systems

With the continuous development in the field of deep learning, in recent years, it has also been widely used in the field of solving partial differential equations, especially the physics-informed neural networks (PINNs) method. However, the PINNs method has some limitations in solving coupled Klein–Gordon–Zakharov (KGZ) systems. To this end, in this article, inspired by the PINNs method and combined with the characteristics of the coupled KGZ systems, we design a neural network model, named multi-output physics-informed neural networks based on time differential order reduction (TDOR-MPINNs), to solve the coupled KGZ systems. Compared with the PINNs, the TDOR-MPINNs first reduces the time derivatives, and thus can increase supervised learning tasks. And through comparing the numerical results obtained by using TDOR-MPINNs and PINNs for solving the one-dimensional (1-D) and two-dimensional (2-D) coupled KGZ systems, we further validate the effectiveness, accuracy and reliability of the TDOR-MPINNs.

Research on Modeling Method of Autonomous Underwater Vehicle Based on a Physics-Informed Neural Network

Research on Modeling Method of Autonomous Underwater Vehicle Based on a Physics-Informed Neural Network

Physics-informed neural networks for high-resolution weather reconstruction from sparse weather stations

Background The accurate provision of weather information holds immense significance to many disciplines. One example corresponds to the field of air traffic management, in which one basis for weather detection is set upon recordings from sparse weather stations on ground. The scarcity of data and their lack of precision poses significant challenges to achieve a detailed description of the atmosphere state at a certain moment in time. Methods In this article, we foster the use of physics-informed neural networks (PINNs), a type of machine learning (ML) architecture which embeds mathematically accurate physics models, to generate high-quality weather information subject to the regularization provided by the Navier-Stokes equations. Results The application of PINNs is oriented to the reconstruction of dense and precise wind and pressure fields in areas where only a few local measurements provided by weather stations are available. Our model does not only disclose and regularize such data, which are potentially corrupted by noise, but is also able to precisely compute wind and pressure in target areas. Conclusions The effect of time and spatial resolution over the capability of the PINN to accurately reconstruct fluid phenomena is thoroughly discussed through a parametric study, concluding that a proper tuning of the neural network’s loss function during training is of utmost importance.

Predicting transformer temperature field based on physics‐informed neural networks

AbstractThe safe operation of oil‐immersed transformers is critical to the safety and stability of the power grid. As the operating time increases, the failure rate of oil‐immersed transformers shows an increasing trend, posing serious challenges to safe operation. It is necessary to investigate the internal state of the oil‐immersed transformer to improve the digital degree and achieve digitalisation and intelligent operation and maintenance. A physics‐informed neural network (PINN) for oil‐immersed transformers was introduced to reconstruct the temperature distribution inside the transformer. According to the approach, the loss function of the network would be optimised by incorporating physical constraint loss terms including heat transfer equations, initial conditions and boundary conditions. The results show that the method proposed can be used to reconstruct and predict the temperature field of transformers in a few seconds with satisfactory accuracy. In conclusion, the PINN proposed outperforms deep neural networks in terms of accuracy, reliability and interpretability, especially in data‐poor cases.

Physics-informed neural networks for an optimal counterdiabatic quantum computation

A novel methodology that leverages physics-informed neural networks to optimize quantum circuits in systems with NQ qubits by addressing the counterdiabatic (CD) 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 CD 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.

Swirling flow field reconstruction and cooling performance analysis based on experimental observations using physics-informed neural networks

The design of thermal protection modules (such as film cooling) for combustion chambers requires a high-fidelity swirling flow field. Although numerical methods provide insights into three-dimensional mechanisms of swirling flow, their predictions of key features such as recirculation zones and swirling jets are often unsatisfactory due to inherent anisotropy and the isotropic nature of the Boussinesq hypothesis. Experimental methods, such as hot wire, laser Doppler velocimetry, and planar particle image velocimetry (PIV), offer more accurate reference data but are limited by sparse or planar observations. In this study, considering the outperformed capability of solving inverse problems, physics-informed neural network (PINN) was adopted to reconstruct the mean swirling flow field based on limited experimental observations from two-dimensional and two-component (2D2C) results. It was found that adding partial information characterizing the swirling flow, such as the swirling jet, could significantly improve the reconstruction of flow field. In addition, film cooling effectiveness was the key variable to evaluate the film cooling performance, which was relatively measurable in the scalar field. To further improve the accuracy of the reconstruction, the multi-source strategy was adopted into the neural network, where the film cooling effectiveness (FCE) of the effusion plate was imported as the scalar source. It was found that the prediction of the flow field near the target plate was improved, where the highest error reduction could reach 76.5%. Finally, through the reconstructed three-dimensional vortex distribution, it was found that swirling flow vortex structures near the swirler exit had a significant impact on cooling effectiveness, causing a non-uniform cooling distribution. This study aims to diagnose the three-dimensional swirling flow field with deep learning by leveraging limited experimental data and deepen the understanding of effusion cooling under swirling flow condition so that obtains a more accurate reference in the design of thermal protection modules.

Weak-formulated physics-informed modeling and optimization for heterogeneous digital materials

Abstract Numerical solutions to partial differential equations (PDEs) are instrumental for material structural design where extensive data screening is needed. However, traditional numerical methods demand significant computational resources, highlighting the need for innovative optimization algorithms to streamline design exploration. Direct gradient-based optimization algorithms, while effective, rely on design initialization and require complex, problem-specific sensitivity derivations. The advent of machine learning offers a promising alternative to handling large parameter spaces. To further mitigate data dependency, researchers have developed physics-informed neural networks (PINNs) to learn directly from PDEs. However, the intrinsic continuity requirement of PINNs restricts their application in structural mechanics problems, especially for composite materials. Our work addresses this discontinuity issue by substituting the PDE residual with a weak formulation in the physics-informed training process. The proposed approach is exemplified in modeling digital materials, which are mathematical representations of complex composites that possess extreme structural discontinuity. This paper also introduces an interactive process that integrates physics-informed loss with design objectives, eliminating the need for pre-trained surrogate models or analytical sensitivity derivations. The results demonstrate that our approach can preserve the physical accuracy in data-free material surrogate modeling but also accelerates the direct optimization process without model pre-training.

Hypersonic inlet flow field reconstruction dominated by shock wave and boundary layer based on small sample physics-informed neural networks

High Mach number inlets face complex challenges such as shock wave/boundary layer interference, adversely impacting the aerodynamic characteristics and stable operating boundaries. Traditional computational fluid dynamics (CFD) simulations for performance and flow field analysis are slow, and data-driven deep learning technologies struggle with predicting flow fields in complex aerodynamic phenomena like shock wave/boundary layer interactions, boundary layer separation, and Mach disks. This study introduces a rapid, robust method for reconstructing hypersonic viscous inlet flow fields using inlet geometry design parameters. This method is called shock wave and boundary layer dominated two-dimensional multiphysical field reconstruction model based on multi-scale sensory field fusion residual physical information neural network (PIMSRF_ResNet), ensuring high-precision training data. Using the optimal Pareto solution set from multi-objective optimization with a small sample intelligence method, 150 sets of multi-physics fields under Ma10 conditions with varying geometric design parameters are calculated. Compared to traditional pure data-driven models, PIMSRF_ResNet's structural similarity in the test set reaches 0.9773, with a peak signal-to-noise ratio close to 30 dB and a correlation coefficient exceeding 98%. These experimental results demonstrated that the proposed method is a promising tool for real-time prediction of complex internal flow fields dominated by high Mach number shock waves and boundary layers.

A peridynamic-informed deep learning model for brittle damage prediction

Physics-informed Neural Network (PINN) has been introduced recently to predict and understand complex physical phenomena by directly incorporating feedback from governing equations. While PINNs offer remarkable capabilities, challenges arise when addressing discontinuities and heterogeneity, potentially compromising accuracy and reliability in prediction outcomes. In this study, a novel approach that combines the principles of peridynamic (PD) theory with PINN is presented to predict quasi-static damage and crack propagation in brittle materials. To achieve high prediction accuracy and convergence rate, the linearized PD governing equation is enforced in the PINN’s residual-based loss function. The proposed PD-INN is able to learn and capture complicated displacement patterns associated with different geometrical parameters, such as pre-crack position and length. The computational method presented in this study can be implemented to any PD constitutive model like Bond-Based PD or State-Based PD as well as any horizon factor. Several enhancements like cyclical annealing schedule and deformation gradient aware optimization technique are proposed to ensure the model would not get stuck in its trivial solution. The model’s performance assessment is conducted by monitoring the behavior of loss function throughout the training process. The results obtained through PD-INN demonstrate a satisfactory level of agreement in 2D and 3D when compared with the ground truth deformation from high-fidelity techniques like PD direct numerical method and Extended finite element method (X-FEM). Our results show the ability of the nonlocal PD-INN to predict damage and crack propagation accurately and efficiently.

Physics-informed generator-encoder adversarial networks with latent space matching for stochastic differential equations

We propose a new class of physics-informed neural networks, called Physics-Informed Generator-Encoder Adversarial Networks, to effectively address the challenges posed by forward, inverse, and mixed problems in stochastic differential equations (SDEs). In these scenarios, while governing equations are known, the available data consist of only a limited set of snapshots for system parameters. Our model consists of two key components: the generator and the encoder, both updated alternately by gradient descent. In contrast to previous approaches that directly match approximated solutions with real snapshots, we employ an indirect matching operating within the lower-dimensional latent feature space. This method circumvents challenges associated with high-dimensional inputs and complex data distributions in solving SDEs. Numerical experiments indicate that, compared to existing deep learning solvers, our proposed approach not only demonstrates superior accuracy but also exhibits advantages in both computational efficiency and model complexity.

Predicting positon solutions of a family of nonlinear Schrödinger equations through deep learning algorithm

We consider a hierarchy of nonlinear Schrödinger equations (NLSEs) and forecast the evolution of positon solutions using a deep learning approach called Physics Informed Neural Networks (PINN). Notably, the PINN algorithm accurately predicts positon solutions not only in the standard NLSE but also in other higher order versions, including cubic, quartic and quintic NLSEs. The PINN approach also effectively handles two coupled NLSEs and two coupled Hirota equations. In addition to the above, we report exact second-order positon solutions of the sextic NLSE and coupled generalized NLSE. These solutions are not available in the existing literature and we construct them through generalized Darboux transformation method. Further, we utilize PINNs to forecast their behaviour as well. To validate PINN's accuracy, we compare the predicted solutions with exact solutions obtained from analytical methods. The results show high fidelity and low mean squared error in the predictions generated by our PINN model.

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.