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1. Integrating Physics-Informed Neural Networks and Power Flow for Scalable Online Transient Analysis

   https://doi.org/10.1109/SmartGridComm65349.2025.11204635  

   Gu, Zhangrong; Zhao, Yue; Yue, Meng; Zhao, Tianqiao; Li, Jiaming (September 2025, IEEE)

   Transient dynamic analysis is crucial for power system stability, particularly with increasing renewable integration, yet traditional numerical integration methods are computation- ally expensive for large-scale online applications. We propose a novel framework integrating Physics-Informed Neural Networks (PINNs) with an AC Power Flow (AC-PF) solver to efficiently simulate dynamic trajectories in power systems in a scalable fashion. A separate PINN is trained for each generator to accurately capture the differential equations of the generator dynamic model. The trained PINNs and an AC-PF solver iteratively update the dynamic and algebraic variables to simulate dynamic trajectories for the entire system. The PINN training process consists of an unsupervised stage to enforce physical laws, followed by a supervised stage leveraging simulation data for enhanced accuracy. As the predictor training is performed separately for individual generators, the method’s computational complexity scales linearly with the number of generators in both training and testing. Experiments on a 3-generator 9-bus system demonstrate the very high accuracy of the developed method in simulating full system dynamic trajectories. 

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2. Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks

   https://doi.org/10.3390/e26050396  

   Xing, Siyuan; Charalampidis, Efstathios G (May 2024, Entropy)

   In this paper, we apply a machine-learning approach to learn 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 Separable Gaussian Neural Networks (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs that treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into the so-called co-traveling wave frame. This reformulation 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 investigations, and explore not only peakon solutions in the ab-family but also compacton solutions in (2+1)-dimensional, Rosenau-Hyman family of PDEs. A comparative analysis with multi-layer perceptron (MLP) reveals that SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader application in solving complex nonlinear PDEs. 

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3. Gradient-Driven Physics Informed Neural Networks for Conduction Heat Transfer and Incompressible Laminar Flow

   https://doi.org/10.1115/1.4070545  

   Lu, Tingying; Shahadat, M_R B; Liu, Qilin; He, Runlin; Jiang, Xiaoyu; Li, Zheng (April 2026, Journal of Computational and Nonlinear Dynamics)

   Abstract Physics-Informed Neural Networks (PINNs) have opened new possibilities for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. However, despite their flexibility, traditional PINNs often struggle to capture sharp gradients and intricate solution features, which limits their effectiveness in many practical problems. In this work, we have introduced Gradient-Driven Physics-Informed Neural Networks (GDPINNs) that improve the ability of traditional PINNs to resolve sharp gradients. By incorporating gradient information directly into the loss function, GDPINNs better target regions where traditional PINNs typically fail. We validated the method on steady-state and transient heat conduction problems, including a central heating source and a sinusoidal boundary condition, and found strong agreement with reference solutions. To further understand the framework's capability, we applied it to a high-gradient steady-state and transient heat conduction problem, where GDPINNs show clear advantages over traditional PINNs and align closely with reference results. We also extended GDPINNs to incompressible laminar flow in a lid-driven cavity, demonstrating its broader applicability. In these cases, GDPINNs consistently provide higher accuracy and better capture critical solution features, highlighting their potential to improve PINNs-based approaches for complex physical problems with sharp gradients. 

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4. Minimizing nature's cost: Exploring data-free physics-informed neural network solvers for fluid mechanics applications

   https://doi.org/10.1063/5.0250022  

   Elmaradny, Abdelrahman; Atallah, Ahmed; Taha, Haithem (February 2025, Physics of Fluids)

   In this paper, we present a novel approach for fluid dynamic simulations by leveraging the capabilities of Physics-Informed Neural Networks (PINNs) guided by the newly unveiled Principle of Minimum Pressure Gradient (PMPG). In a PINN formulation, the physics problem is converted into a minimization problem (typically least squares). The PMPG asserts that for incompressible flows, the total magnitude of the pressure gradient over the domain must be minimum at every time instant, turning fluid mechanics into minimization problems, making it an excellent choice for PINNs formulation. Following the PMPG, the proposed PINN formulation seeks to construct a neural network for the flow field that minimizes Nature's cost function for incompressible flows in contrast to traditional PINNs that minimize the residuals of the Navier–Stokes equations. This technique eliminates the need to train a separate pressure model, thereby reducing training time and computational costs. We demonstrate the effectiveness of this approach through a case study of inviscid flow around a cylinder. The proposed approach outperforms the traditional PINNs approach in terms of training time, convergence rate, and compliance with physical metrics. While demonstrated on a simple geometry, the methodology is extensible to more complex flow fields (e.g., three-dimensional, unsteady, and viscous flows) within the incompressible realm, which is the region of applicability of the PMPG. 

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5. Temporal consistency loss for physics-informed neural networks

   https://doi.org/10.1063/5.0211398  

   Thakur, Sukirt; Raissi, Maziar; Mitra, Harsa; Ardekani, Arezoo M (July 2024, Physics of Fluids)

   Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations (PDEs) in a forward and inverse manner using neural networks. However, balancing individual loss terms can be challenging, mainly when training these networks for stiff PDEs and scenarios requiring enforcement of numerous constraints. Even though statistical methods can be applied to assign relative weights to the regression loss for data, assigning relative weights to equation-based loss terms remains a formidable task. This paper proposes a method for assigning relative weights to the mean squared loss terms in the objective function used to train PINNs. Due to the presence of temporal gradients in the governing equation, the physics-informed loss can be recast using numerical integration through backward Euler discretization. The physics-uninformed and physics-informed networks should yield identical predictions when assessed at corresponding spatiotemporal positions. We refer to this consistency as “temporal consistency.” This approach introduces a unique method for training physics-informed neural networks (PINNs), redefining the loss function to allow for assigning relative weights with statistical properties of the observed data. In this work, we consider the two- and three-dimensional Navier–Stokes equations and determine the kinematic viscosity using the spatiotemporal data on the velocity and pressure fields. We consider numerical datasets to test our method. We test the sensitivity of our method to the timestep size, the number of timesteps, noise in the data, and spatial resolution. Finally, we use the velocity field obtained using particle image velocimetry experiments to generate a reference pressure field and test our framework using the velocity and pressure fields. 

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