Physics-informed Neural Networks Research Articles

Chebyshev spectral approximation-based physics-informed neural network for solving higher-order nonlinear differential equations

Chebyshev spectral approximation-based physics-informed neural network for solving higher-order nonlinear differential equations

A brief review of reduced order models using intrusive and non‐intrusive techniques

AbstractReduced Order Models (ROMs) have gained a great attention by the scientific community in the last years thanks to their capabilities of significantly reducing the computational cost of the numerical simulations, which is a crucial objective in applications like real time control and shape optimization. This contribution aims to provide a brief overview about such a topic. We discuss both a classic intrusive framework based on a Galerkin projection technique and hybrid/non‐intrusive approaches, including Physics Informed Neural Networks (PINN), purely Data‐Driven Neural Networks (NN), Radial Basis Functions (RBF), Dynamic Mode Decomposition (DMD) and Gaussian Process Regression (GPR). We also briefly mention geometrical parametrization and dimensionality reduction methods like Active Subspaces (ASs). Then we test the performance of such approaches in terms of efficiency and accuracy against three academic test cases, the lid driven cavity, the flow past a cylinder and the geometrically parametrized Stanford Bunny. Moreover, we also briefly present some preliminary results related to a more complex case involving an industrial application.

Riemann-hilbert problem and physics-informed neural networks method for the nonlocal Sasa-Satsuma equation

Riemann-hilbert problem and physics-informed neural networks method for the nonlocal Sasa-Satsuma equation

A Physics Informed Neural Network for Solving the Inverse Heat Transfer Problem in Gas Turbine Rotating Cavities

Abstract Recently, Physics-Informed Neural Networks have displayed great potential in delivering swift and accurate solutions for inverse problems. Here, a Physics-Informed Neural Network (PINN) was used to solve the inverse heat conduction problem in the rotating cavities of a aeroengine high-pressure compressor internal air system. The neural network was designed to receive experimentally captured radially distributed temperature profiles as inputs and predict the associated surface heat fluxes. The correctness of these predicted heat fluxes are assessed by numerically solving the direct heat conduction equation using a 2D Finite-Element model, thereby recovering the original temperature profiles. A comparative analysis is conducted between the predicted temperature profiles and the initial inputs. The physics informed neural network is trained using noise free synthetic data, created from a range of radial temperature curve fit coefficients, and subsequently tested on noisy experimental data at engine representative conditions. The predicted temperature values exhibit some good agreement with their respective actual counterparts. Furthermore, the sensitivity of model hyperparameters are explored to showcase the capability of the proposed approach. The results show that Physics- Informed Neural Networks exhibit reduced susceptibility to experimental uncertainties when addressing inverse problems, in contrast to inverse solution methods, and offer a possible new approach for analysis of experimental data if trained on a sufficiently large dataset.

Computationally Efficient Inference via Time-Aware Modular Control Systems

Control in multi-agent decision-making systems is an important issue with a wide variety of existing approaches. In this work, we offer a new comprehensive framework for distributed control. The main contributions of this paper are summarized as follows. First, we propose PHIMEC (physics-informed meta control)—an architecture for learning optimal control by employing a physics-informed neural network when the state space is too large for reward-based learning. Second, we offer a way to leverage impulse response as a tool for system modeling and control. We propose IMPULSTM, a novel approach for incorporating time awareness into recurrent neural networks designed to accommodate irregular sampling rates in the signal. Third, we propose DIMAS, a modular approach to increasing computational efficiency in distributed control systems via domain-knowledge integration. We analyze the performance of the first two contributions on a set of corresponding benchmarks and then showcase their combined performance as a domain-informed distributed control system. The proposed approaches show satisfactory performance both individually in their respective applications and as a connected system.

Research on temperature prediction model of molten steel of tundish in continuous casting

To achieve the desired superheat of molten steel during the continuous casting process, optimization of process parameters such as molten steel temperature in ladle furnace, casting speed, and baking temperature is necessary. Therefore, obtaining the superheat corresponding to these process parameters in advance is particularly important. To address this issue, a model for predicting the temperature of molten steel in the tundish during continuous casting is designed. The model adopts a combined modeling approach of mechanistic model and data model. To address the issue of the mechanism model’s inability to capture the variation of the lining’s thermal parameters, this article improves the traditional physics-informed neural network (PINN) algorithm. It combines the constraints from both the forward and inverse problems, allowing for obtaining solutions to the equations while capturing the variation of equation parameters. Actual data from multiple casting sequences at a steel plant are collected to validate the accuracy and interpretability of the model. The results show that the error of the model is about 2.1k which has better accuracy compared to pure mechanistic model and pure data model. Additionally, it can capture the variation patterns of tundish lining thermal parameters under different operating conditions. Therefore, the model designed in this article can provide both profound physical interpretation ability and more practical predictions of molten steel temperature.

Physics-Informed Neural Networks-Based Online Excavation Trajectory Planning for Unmanned Excavator

As a large-scale mining excavator, the electric shovel (ES) has been extensively employed in open-pit mines for overburden removal and mineral loading. In the development of unmanned operations for ES, dynamic excavation trajectory planning is essential, as it directly influences operational efficiency and energy consumption by guiding the dipper during excavation. However, conventional optimization-based methods for excavation trajectory planning typically start from scratch, resulting in a time-consuming process that fails to meet real-time requirements. To address this challenge, we propose an innovative online trajectory planning framework based on physics-informed neural networks (PINNOTP) that utilizes advanced data-driven techniques. The input to PINNOTP consists of on-site working conditions, including the initial state of the ES and the material surface being excavated. The output is a smooth, polynomial-based curve that serves as the reference trajectory for the dipper. To ensure smooth execution of the generated trajectory, prior domain knowledge—such as physics-based target-oriented constraints, essential system dynamics, and mechanical constraints—is explicitly incorporated into the loss function during training. A case study is presented to validate the proposed method, demonstrating that PINNOTP effectively addresses the challenges of online excavation trajectory planning.

MRF-PINN: a multi-receptive-field convolutional physics-informed neural network for solving partial differential equations

MRF-PINN: a multi-receptive-field convolutional physics-informed neural network for solving partial differential equations

Learning Gravity Fields of Small Bodies: Self-adaptive Physics-informed Neural Networks

The reconstruction of the gravity field within the surface region of small bodies is crucial for the surface proximity operations of a probe. However, the irregular shape, uneven mass distribution, and sparse gravitational data of small bodies pose challenges in the reconstruction. We propose a self-adaptive physics-informed neural network (PINN) for the reconstruction of the gravity field within the surface region of irregular and heterogeneous small bodies. First, we introduce an auxiliary-point-based data augmentation strategy to reduce the model’s dependency on the quantity of data. Second, we incorporate a residual-based adaptive sampling strategy to enhance the prediction accuracy of the model in regions with significant variations in small-body density. Finally, we introduce an adaptive weight module based on gradient ascent to mitigate the balancing issue of loss terms in the PINN. Experiments indicate that our algorithm achieves improved accuracy for reconstructing the gravity field within the surface region of small bodies. This work is expected to contribute to the enhancement of safety in surface proximity operations around the surfaces of small bodies.

Approximating families of sharp solutions to Fisher's equation with physics-informed neural networks

This paper employs physics-informed neural networks (PINNs) to solve Fisher's equation, a fundamental reaction-diffusion system with both simplicity and significance. The focus is on investigating Fisher's equation under conditions of large reaction rate coefficients, where solutions exhibit steep traveling waves that often present challenges for traditional numerical methods. To address these challenges, a residual weighting scheme is introduced in the network training to mitigate the difficulties associated with standard PINN approaches. Additionally, a specialized network architecture designed to capture traveling wave solutions is explored. The paper also assesses the ability of PINNs to approximate a family of solutions by generalizing across multiple reaction rate coefficients. The proposed method demonstrates high effectiveness in solving Fisher's equation with large reaction rate coefficients and shows promise for meshfree solutions of generalized reaction-diffusion systems.

Parameter identification of shallow water waves using the generalized equal width equation and physics-informed neural networks: a conservative approximation scheme

Parameter identification of shallow water waves using the generalized equal width equation and physics-informed neural networks: a conservative approximation scheme

Diffusion-derived intravoxel-incoherent motion anisotropy relates to CSF and blood flow.

Multi-b-value diffusion MRI was acquired in 32 gradient directions for 11 healthy volunteers, and in six directions for 29 patients with cSVD and 14 controls at 3 T. A physics-informed neural network was used to estimate intravoxel incoherent motion (IVIM)-DTI model parameters, including the parenchymal slow diffusion (D-)tensor and the pseudo-diffusion (D*)-tensor, from which the fractional anisotropy (FA), mean diffusivity (MD), axial diffusivity (AD), and radial diffusivity (RD) were derived. Comparisons of D*-tensor metrics were made between lateral, third, and fourth ventricles and between the middle cerebral arteries and superior sagittal sinus. Group differences in D*-tensor metrics in normal-appearing white matter were analyzed using multivariable linear regression, correcting for age and sex. D*-anisotropy aligned well with CSF flow and arterial blood flow. FA(D*), MD(D*), AD(D*), and RD(D*) were highest in the third, moderate in the fourth, and lowest in the lateral ventricles. The arteries showed higher MD(D*), AD(D*), and RD(D*) than the sagittal sinus. Higher FA(D*) in the normal-appearing white matter was related to cSVD diagnosis and older age, suggesting microvascular architecture alterations. Multi-b-value, multi-directional diffusion analysis using the IVIM-DTI model enables assessment of the cerebral microstructure, fluid flow, and microvascular architecture, providing information on neurodegeneration, glymphatic waste clearance, and the vasculature in one measurement.

M-IPISincNet: An explainable multi-source physics-informed neural network based on improved SincNet for rolling bearings fault diagnosis

Timely and accurate diagnosis of bearing faults can effectively reduce the chance of accidents in equipment. However, deep learning methods are mostly completely dependent on data and lack interpretability. It is difficult to deal with the differences between real-time data and training data under changing working conditions and noisy environments. In this study, we proposed M-IPISincNet, an explainability multi-source physics-informed convolutional network based on improved SincNet. Rolling bearing fault diagnosis is realized by extracting fault features from vibration and current signals. Firstly, a physics-informed convolutional layer is designed based on inverse Fourier transform and bandpass filters. Fault features are extracted by multi-scale convolution and multi-layer nonlinear mapping. A DBN network is applied extract unsupervised hidden fusion features in the vibration and current signals. The proposed method is validated under the datasets of Paderborn University (PU) and Case Western Reserve University (CWRU), which proves that the proposed method has explainability, robustness and great accuracy under multiple working conditions and noises.

A Comprehensive Review of Remaining Useful Life Estimation Approaches for Rotating Machinery

This review paper comprehensively analyzes the prognosis of rotating machines (RMs), focusing on mechanical-flaw and remaining-useful-life (RUL) estimation in industrial and renewable energy applications. It introduces common mechanical faults in rotating machinery, their causes, and their potential impacts on RM performance and longevity, particularly in wind, wave, and tidal energy systems, where reliability is crucial. The study outlines the primary procedures for RUL estimation, including data acquisition, health indicator (HI) construction, failure threshold (FT) determination, RUL estimation approaches, and evaluation metrics, through a detailed review of published work from the past six years. A detailed investigation of HI design using mechanical-signal-based, model-based, and artificial intelligence (AI)-based techniques is presented, emphasizing their relevance to condition monitoring and fault detection in offshore and hybrid renewable energy systems. The paper thoroughly explores the use of physics-based, data-driven, and hybrid models for prognosis. Additionally, the review delves into the application of advanced methods such as transfer learning and physics-informed neural networks for RUL estimation. The advantages and disadvantages of each method are discussed in detail, providing a foundation for optimizing condition-monitoring strategies. Finally, the paper identifies open challenges in prognostics of RMs and concludes with critical suggestions for future research to enhance the reliability of these technologies.

Surface Profile Recovery from Electromagnetic Fields with Physics-Informed Neural Networks

Physics-informed neural networks (PINN) have shown their potential in solving both direct and inverse problems of partial differential equations. In this paper, we introduce a PINN-based deep learning approach to reconstruct one-dimensional rough surfaces from field data illuminated by an electromagnetic incident wave. In the proposed algorithm, the rough surface is approximated by a neural network, with which the spatial derivatives of surface function can be obtained via automatic differentiation, and then the scattered field can be calculated using the method of moments. The neural network is trained by minimizing the loss between the calculated and the observed field data. Furthermore, the proposed method is an unsupervised approach, independent of any surface data, where only the field data are used. Both transverse electric (TE) field (Dirichlet boundary condition) and transverse magnetic (TM) field (Neumann boundary condition) are considered. Two types of field data are used here: full-scattered field data and phaseless total field data. The performance of the method is verified by testing with Gaussian-correlated random rough surfaces. Numerical results demonstrate that the PINN-based method can recover rough surfaces with great accuracy and is robust with respect to a wide range of problem regimes.

Exact and Data-Driven Lump Wave Solutions for the (3+1)-Dimensional Hirota–Satsuma–Ito-like Equation

In this paper, the lump wave solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like (HSIl) equation are constructed by employing the Hirota bilinear method and quadratic function approach, and the corresponding propagation behaviors and nonlinear dynamical properties are also investigated. At the same time, the physics informed neural network (PINN) deep learning technique is employed to study the data-driven solutions for the HSIl equation from the derived lump wave solutions. The machine learning results show high effectiveness and accuracy, providing new techniques for discussing more nonlinear dynamics of lump waves and discovering new lump wave solutions.

Point neuron learning: a new physics-informed neural network architecture

Machine learning and neural networks have advanced numerous research domains, but challenges such as large training data requirements and inconsistent model performance hinder their application in certain scientific problems. To overcome these challenges, researchers have investigated integrating physics principles into machine learning models, mainly through (i) physics-guided loss functions, generally termed as physics-informed neural networks, and (ii) physics-guided architectural design. While both approaches have demonstrated success across multiple scientific disciplines, they have limitations including being trapped to a local minimum, poor interpretability, and restricted generalizability beyond sampled data range. This paper proposes a new physics-informed neural network (PINN) architecture that combines the strengths of both approaches by embedding the fundamental solution of the wave equation into the network architecture, enabling the learned model to strictly satisfy the wave equation. The proposed point neuron learning method can model an arbitrary sound field based on microphone observations without any dataset. Compared to other PINN methods, our approach directly processes complex numbers, offers better interpretability, and can be generalized to out-of-sample scenarios. We evaluate the versatility of the proposed architecture by a sound field reconstruction problem in a reverberant environment. Results indicate that the point neuron method outperforms two competing methods and can efficiently handle noisy environments with sparse microphone observations.

Prediction of microstructural evolution of multicomponent polymers by Physics-Informed neural networks

As emerging concepts, leveraging physical information with machine learning to construct hybrid models enables a scientific data-driven paradigm for the investigation of complex kinetic issues in the field of polymer science. Herein, the model of physics-informed neural networks (PINNs), integrating advantages of neural networks with physics-based knowledge, is extended to predict the spatiotemporal patterns of phase-separated microstructures of multicomponent polymers. It is demonstrated that the PINN-based data-driven model can achieve the forward, backward and bidirectional predictions of spatiotemporally phase-separated patterns of homopolymer blends. All predicted patterns reveal a high accuracy and robustness of PINN-based data-driven model by leveraging a small amount of training data. Particularly, the PINN-based method can be harnessed to infer the latent physical law about the growth kinetics of phase-separated microstructures. In addition, PINN-based data-driven model can also be generalized to forecast the spatiotemporal patterns of microphase-separated nanostructures of block copolymers and infer their growth kinetics. Our study opens the possibility of learning non-equilibrium phenomena of complex polymer systems from only a few snapshots of microstructural evolution.

Adaptive fractional physics-informed neural networks for solving forward and inverse problems of anomalous heat conduction in functionally graded materials

In this paper, adaptive fractional physics-informed neural networks (PINNs) are employed to solve the forward and inverse problems of anomalous heat conduction in three-dimensional functionally graded materials. In adaptive fractional PINNs, the finite difference L1 scheme is employed to discretize the time fractional derivatives in anomalous heat conduction problems. By combining the finite difference L1 scheme with automatic differentiation technique, adaptive fractional PINNs minimize a loss function constructed based on the governing equation, initial and boundary conditions to obtain solutions to heat conduction problems. To avoid competition among various loss terms during the training process, an adaptive loss balancing algorithm is adopted to balance the interactions among different terms of the loss functions. In addition, polynomial basis functions are used to expand unknown functions characterizing material parameters or heat sources, aiming to enhance the performance of adaptive fractional PINNs in resolving inverse problems. Five numerical examples involving forward, inverse, and nonlinear problems related to anomalous heat conduction are carried out to demonstrate the feasibility and effectiveness of adaptive fractional PINNs.

Physics-informed neural networks and higher-order high-resolution methods for resolving discontinuities and shocks: A comprehensive study

Addressing discontinuities in fluid flow problems is inherently difficult, especially when shocks arise due to the nonlinear nature of the flow. While handling discontinuities is a well-established practice in computational fluid dynamics (CFD), it remains a major challenge when applying physics-informed neural networks (PINNs). In this study, we compare the shock-resolving capabilities of traditional CFD methods with those of PINNs, highlighting the advantages of the latter. Our findings show that PINNs exhibit less dissipative behavior compared to conventional techniques. We evaluated the performance of both PINNs and traditional methods on linear and nonlinear test cases, demonstrating that PINNs offer superior shock-resolving properties. Notably, PINNs can accurately resolve inviscid shocks with just three grid points, whereas traditional methods require at least seven points. This suggests that PINNs are more effective at resolving shocks and discontinuities when using the same grid for both PINN and CFD simulations. However, it’s important to note that PINNs, in this context, are computationally more expensive than traditional methods on a given grid.