Inverse Design of Ultra-Wideband Frequency Selective Surface Using a Graph based Conditional Variational Autoencoder (G-CVAE) integrated with a Physics Informed Neural Network (PINN)

In recent years great interest has risen towards surrogate reservoir models based on data-driven methodologies with the purpose of speeding up reservoir management decisions. In this work, a Physics Informed Neural Network (PINN) based on a Capacitance Resistance Model (CRM) has been developed and tested on a synthetic and on a real dataset to predict the production of oil reservoirs under waterflooding conditions. CRMs are simple models based on material balance that estimate the liquid production as a function of injected water and bottom hole pressure. PINNs are Artificial Neural Networks (ANNs) that incorporate prior physical knowledge of the system under study to regularize the network. A PINN based on a CRM is obtained by including the residual of the CRM differential equations in the loss function designed to train the neural network on the historical data. During training, weights and biases of the network and parameters of the physical equations, such as connectivity factors between wells, are updated with the backpropagation algorithm. To investigate the effectiveness of the novel methodology on waterflooded scenarios, two test cases are presented: a small synthetic one and a real mature reservoir. Results obtained with PINN are compared with respect to CRM and ANN alone. In the synthetic case CRM and PINN give slightly better quality history matches and predictions than ANN. The connectivity factors estimated by CRM and PINN are very similar and correctly represent the underlying geology. In the real case PINN gives better quality history matches and predictions than ANN, and both significantly outperform CRM. Even though the CRM formulation is too simple to predict the complex behavior of a real reservoir, the CRM based regularization contributes to improving the PINN predictions quality compared to the purely data-driven ANN model. The connectivity factors estimated by CRM and PINN are not in agreement. However, the latter method provided results closer to our understanding of the flooding process after many years of operations and data analysis. All considered, PINN outperformed both CRM and ANN in terms of predictivity and interpretability, effectively combining strengths from both methodologies. The presented approach does not require the construction of a 3D model since it learns directly from production data, while preserving physical consistency. Moreover, it represents a computationally inexpensive alternative to traditional full-physics reservoir simulations which could have vast applications for problems requiring many forward evaluations, like the optimization of water allocation for mature reservoirs.

Physics-informed Neural Implicit Flow neural network for parametric PDEs.

We propose a new approach to the solution of the wave propagation and full waveform inversions (FWIs) based on a recent advance in deep learning called physics‐informed neural networks (PINNs). In this study, we present an algorithm for PINNs applied to the acoustic wave equation and test the method with both forward models and FWI case studies. These synthetic case studies are designed to explore the ability of PINNs to handle varying degrees of structural complexity using both teleseismic plane waves and seismic point sources. PINNs' meshless formalism allows for a flexible implementation of the wave equation and different types of boundary conditions. For instance, our models demonstrate that PINN automatically satisfies absorbing boundary conditions, a serious computational challenge for common wave propagation solvers. Furthermore, a priori knowledge of the subsurface structure can be seamlessly encoded in PINNs' formulation. We find that the current state‐of‐the‐art PINNs provide good results for the forward model, even though spectral element or finite difference methods are more efficient and accurate. More importantly, our results demonstrate that PINNs yield excellent results for inversions on all cases considered and with limited computational complexity. We discuss the current limitations of the method with complex velocity models as well as strategies to overcome these challenges. Using PINNs as a geophysical inversion solver offers exciting perspectives, not only for the full waveform seismic inversions, but also when dealing with other geophysical datasets (e.g., MT, gravity) as well as joint inversions because of its robust framework and simple implementation.

Investigating steady unconfined groundwater flow using Physics Informed Neural Networks

In the field of structural engineering and materials science, particularly in aerospace engineering, optimizing structural design for strength while minimizing material usage presents a complex challenge, especially under variable loading conditions. The current research for the first time introduces an approach to strength evaluation through the innovative application of Physics Informed Neural Networks (PINNs) in shakedown analysis. This paper presents a novel methodology that combines the principles of physics-informed machine learning with shakedown analysis’s rigorous demands. Shakedown analysis offers a sophisticated framework for determining the safe load-bearing capacity of structures beyond the conventional elastic limit but within the plastic threshold, without the need to consider the history of loading conditions. This methodology enables engineers to design lighter, more material-efficient structures by safely harnessing the structure’s capacity to withstand loads without reaching failure. Our approach leverages the concept of Physics Informed Neural Networks (PINNs), which integrates differential equations governing physical laws directly into the learning process of deep neural networks. PINNs are further advanced by incorporating self-equilibrating stress field relations, essential for shakedown analysis. This integration enables to accurately predict the shakedown limit strength, crucial for determining a structure’s ability to endure repeated loading without failure. By adding these relations to the mechanical equilibrium equation and constitutive equations within the neural network architecture, it is now offer a comprehensive modeling capability, extending their application to more complex scenarios in solid mechanics, including accurate shakedown limit predictions. The proposed methodology introduces key innovations by extending PINNs to nonlinear problems for complex elastoplastic behavior through shakedown analysis and proposing a multi-network PINN model for more accurate structural response representation. To validate our approach, we employ synthetic data derived from analytical and numerical reference solutions, focusing on convergence behavior and accuracy. Our research highlights the robustness of PINNs in handling sparse data and extrapolating across a wide range of parameters, a critical aspect in the context of shakedown analysis where the design space is vast and complex. This ability to predict accurately under previously unseen conditions not only underscores the potential of PINNs in surrogate modeling but also in the further sensitivity analysis, providing a powerful tool for engineers to explore and optimize structural designs efficiently. Furthermore, the technique is used to determine the shakedown strength for a manned airtight module.

Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networks from additional information obtained by employing the physical laws and evaluating them at random points in the space–time domain. Such PIML integrates multimodality and multifidelity data with mathematical models, and implements them using neural networks or graph networks. Here, we review some of the prevailing trends in embedding physics into machine learning, using physics-informed neural networks (PINNs) based primarily on feed-forward neural networks and automatic differentiation. For more complex systems or systems of systems and unstructured data, graph neural networks (GNNs) present some distinct advantages, and here we review how physics-informed learning can be accomplished with GNNs based on graph exterior calculus to construct differential operators; we refer to these architectures as physics-informed graph networks (PIGNs). We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.

Physics Informed Neural Networks (PINNs) are shown to be a promising method for the approximation of partial differential equations (PDEs). PINNs approximate the PDE solution by minimizing physics-based loss functions over a given domain. Despite substantial progress in the application of PINNs to a range of problem classes, investigation of error estimation and convergence properties of PINNs, which is important for establishing the rationale behind their good empirical performance, has been lacking. This paper presents convergence analysis and error estimates of PINNs for a multi-physics problem of thermally coupled incompressible Navier-Stokes equations. Through a model problem of Beltrami flow it is shown that a small training error implies a small generalization error. Posteriori convergence rates of total error with respect to the training residual and collocation points are presented. This is of practical significance in determining appropriate number of training parameters and training residual thresholds to get good PINNs prediction of thermally coupled steady state laminar flows. These convergence rates are then generalized to different spatial geometries as well as to different flow parameters that lie in the laminar regime. A pressure stabilization term in the form of pressure Poisson equation is added to the PDE residuals for PINNs. This physics informed augmentation is shown to improve accuracy of the pressure field by an order of magnitude as compared to the case without augmentation. Results from PINNs are compared to the ones obtained from stabilized finite element method and good properties of PINNs are highlighted.

This paper uses Physics-Informed Neural Network (PINN) to design Frequency Selective Surface (FSS). PINN integrates physical information into the loss function, so training PINN does not require a dataset, which will be faster than traditional neural networks for inverse design. The specific implementation process of this paper is to construct a PINN using field solutions of mode matching method, and given the design goal, the PINN can train the shape of the diaphragm. The single frequency FSS that meets the design goal was designed using the inverse design method proposed in this paper without a dataset, verifying the rationality of using PINN to design metasurfaces. Using PINN for inverse design is not limited to single frequency FSS, but can also be used for more complex metasurface.

Frequency selective surface based on metallic mesh can realize the physical properties of both high infrared transmittance and millimeter-wave band-pass filter. In order to improve the optical transmittance, reduce surface resistance and suppress the effect of high order diffraction energy on the imaging quality of the optical system, a new design of frequency selective surface based on hybrid period metallic mesh is obtained. In this paper, the diffraction intensity distribution and surface current of frequency selective surface are analyzed based on metallic mesh. Simulation and experimental results show that frequency selective surface based on hybrid period metallic mesh realizes a stable millimeter-wave band-pass filter property, at the same time, it obtains 5% increase of infrared transmittance and 4 Ω reduce of surface resistance. New design of frequency selective surface based on hybrid period metallic mesh effectively suppresses the effect of high order diffraction energy on the imaging quality of the optical system.

Salinity in estuarine environments has been traditionally simulated using process-based models. More recently, data-driven models including artificial neural networks (ANNs) have been developed for simulating salinity. Compared to process-based models, ANNs yield faster salinity simulations with comparable accuracy. However, ANNs are often purely data-driven and not constrained by physical laws, making it difficult to interpret the causality between input and output data. Physics-informed neural networks (PINNs) are emerging machine-learning models to integrate the benefits of both process-based models and data-driven ANNs. PINNs can embed the knowledge of physical laws in terms of the partial differential equations (PDE) that govern the dynamics of salinity transport into the training of the neural networks. This study explores the application of PINNs in salinity modeling by incorporating the one-dimensional advection–dispersion salinity transport equation into the neural networks. Two PINN models are explored in this study, namely PINNs and FoNets. PINNs are multilayer perceptrons (MLPs) that incorporate the advection–dispersion equation, while FoNets are an extension of PINNs with an additional encoding layer. The exploration is exemplified at four study locations in the Sacramento–San Joaquin Delta of California: Pittsburg, Chipps Island, Port Chicago, and Martinez. Both PINN models and benchmark ANNs are trained and tested using simulated daily salinity from 1991 to 2015 at study locations. Results indicate that PINNs and FoNets outperform the benchmark ANNs in simulating salinity at the study locations. Specifically, PINNs and FoNets have lower absolute biases and higher correlation coefficients and Nash–Sutcliffe efficiency values than ANNs. In addition, PINN models overcome some limitations of purely data-driven ANNs (e.g., neuron saturation) and generate more realistic salinity simulations. Overall, this study demonstrates the potential of PINNs to supplement existing process-based and ANN models in providing accurate and timely salinity estimation.

Summary In recent years, Physics Informed Neural Networks (PINNs) have generated considerable interest in the scientific computing community as an alternative, or potential contender, to traditional numerical discretization methods such as finite- difference, volume, and element methods, for solving partial differential equations (PDE). In PINNs, the governing equations are incorporated into a loss function as a regularization term to guide the neural network so that the solution respects the underlying physics, hence the name ‘physics informed’. In this work, we investigate the performance of PINNs in solving the highly anisotropic diffusion equation that models fluid flow in subsurface porous media. Several levels of permeability anisotropy are tested. The results show that PINNs have excellent performance when the solution is smooth regardless of the strength of permeability anisotropy. However, PINNs struggle to give adequate results when the solution has large gradients, for example, when the solution is induced by a concentrated source term. The problem is exacerbated by higher levels of permeability anisotropy. Our results highlight a few limitations in the current implementation of physics-informed neural networks for fluid flow in porous media and show that we still have ways to go before it can compete with traditional numerical methods.

To solve the difficulty of inverse design of metal-loaded electromagnetic devices, we propose a physics-informed neural network (PINN) framework. With the emergence of PINNs, some scholars within the field of electromagnetism have utilized them to design dielectric-loaded electromagnetic devices. However, using PINN to design metal-loaded electromagnetic devices is still uncommon due to the difficulty of binarization of metal shapes. In light of this, we propose a PINN framework to address the inverse physical design of metal-loaded electromagnetic devices. In PINN, we embed mathematical-physics models into the network, which endows the training parameters of PINN with practical physical significance. The main innovation of this paper is to solve the binarization problem in the inverse design of metal-loaded devices within the PINN. To verify the feasibility of our method, we provide an example of designing a dual-layer metal diaphragm-loaded parallel plate waveguide device. The PINN presented with metal binarization in this paper offers a novel approach to the inverse physical design of metal-loaded electromagnetic devices.

Physics-Informed Neural Networks (PINNs) integrate physics principles with machine learning, offering innovative solutions for complex modeling challenges. Laminated composites, characterized by their anisotropic behavior, multi-layered structures, and intricate interlayer interactions, pose significant challenges for traditional computational methods. PINNs address these issues by embedding governing physical laws directly into neural network architectures, enabling efficient and accurate modeling. This review provides a comprehensive overview of PINNs applied to laminated composites, highlighting advanced methodologies such as hybrid PINNs, k-space PINNs, Theory-Constrained PINNs, optimal PINNs, and disjointed PINNs. Key applications, including structural health monitoring (SHM), structural analysis, stress-strain and failure analysis, and multi-scale modeling, are explored to illustrate how PINNs optimize material configurations and enhance structural reliability. Additionally, this review examines the challenges associated with deploying PINNs and identifies future directions to further advance their capabilities. By bridging the gap between classical physics-based models and data-driven techniques, this review advances the understanding of PINN methodologies for laminated composites and underscores their transformative role in addressing modeling complexities and solving real-world problems.

Physics-Informed Neural Networks (PINNs) have emerged as a powerful paradigm at the intersection of deep learning and scientific computing, offering a novel approach to solving forward and inverse problems governed by partial differential equations (PDEs). By embedding physical laws into the training of neural networks through constrained loss formulations, PINNs unify data-driven learning with mechanistic modeling. This survey provides an in-depth review of the theoretical foundations, algorithmic advances, and practical applications of PINNs across diverse scientific and engineering disciplines. We begin by exploring the motivation and historical context for physics-informed learning, contrasting it with traditional numerical methods and conventional machine learning models. The core methodologies underlying PINNs—including loss construction, network architecture design, optimization techniques, and extensions to stochastic and multi-physics systems—are discussed in detail. A taxonomy of recent variants, such as variational PINNs, probabilistic PINNs, domain-decomposed PINNs, and operator-learning PINNs, is presented to highlight the rapid diversification of the field. Representative applications in fluid dynamics, materials science, geophysics, biomedical engineering, and climate modeling are examined to demonstrate the real-world impact of PINNs. Despite their promise, PINNs face several challenges, including optimization stiffness, high computational cost, generalization limitations, sensitivity to noise, and lack of theoretical guarantees. We provide a critical analysis of these issues and survey emerging solutions. Looking ahead, we identify future research directions such as foundation models for physics, self-supervised learning, symbolic and probabilistic hybrid frameworks, scalable training strategies, and autonomous scientific discovery. By systematically bridging the gap between data and physical laws, PINNs represent a foundational step toward interpretable, generalizable, and trustworthy scientific machine learning. This review aims to serve as a comprehensive reference for researchers and practitioners seeking to understand and advance the frontiers of physics-informed neural computation.

Correcting model misspecification in physics-informed neural networks (PINNs)