Aug 2026· e-Journal of Nondestructive Testing· 0 citations
TL;DR
The results demonstrate that PINN achieves more accurate and stable full-field vibration reconstructions than conventional PINNs, particularly under conditions involving high-frequency modes, and highlights the potential of hybrid data-physics neural frameworks as an efficient and reliable approach for solving complex PDE-governed dynamical systems.
Abstract
Physics-informed neural networks (PINNs) have emerged as a powerful paradigm in scientific machine learning by embedding governing physical laws into neural network training through loss functions. They have demonstrated remarkable success in solving various forward and inverse problems governed by partial differential equations (PDEs). However, in practical applications, purely physics-constrained PINNs that rely solely on PDE residuals often suffer from slow or non-convergence and limited prediction accuracy, particularly when modeling high-order dynamical systems (e.g., second-order and above). Moreover, conventional PINNs struggle to effectively capture high-frequency components in complex physical fields, which further limits their generalization and representational capability.
To address these challenges, this study proposes a data-guided physics-informed neural network with Fourier feature enhancement. In the proposed framework, a small amount of high-fidelity measurement or simulation data is incorporated to guide the training process, providing explicit guidance that complement the physics-based constraints. Meanwhile, Fourier feature embeddings are introduced into the input layer of the network to enhance its ability to represent high-frequency variations and multi-scale solution structures. This synergistic integration of data guidance and Fourier-enhanced representations accelerates convergence, improves robustness, and enhances the accuracy of PDE solutions.
The effectiveness of the proposed PINN model is validated through numerical and simulation studies on Euler-Bernoulli beam vibration problems, which serve as representative examples of high-order mechanical systems. The results demonstrate that PINN achieves more accurate and stable full-field vibration reconstructions than conventional PINNs, particularly under conditions involving high-frequency modes. These findings highlight the potential of hybrid data-physics neural frameworks as an efficient and reliable approach for solving complex PDE-governed dynamical systems.
Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding governing physical laws into deep neural networks. However, their reliance on computationally expensive gradient-based optimization and deep architectures often results in slow training, high computational cost, and limited scalability. In this work, we propose a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs. The proposed formulation embeds the governing differential operator and the associated initial and boundary constraints directly into a linear output-layer optimization problem, thereby replacing nonlinear gradient-based training with a deterministic least-squares solution obtained via the pseudoinverse. Consequently, the entire learning process is reduced to a single linear optimization stage while preserving the underlying physical constraints. As a result, PI-BLS offers an efficient learning paradigm for a physics-informed learning framework for solving PDEs that eliminates iterative backpropagation while preserving the underlying physical constraints. Experimental results on representative forward PDE benchmarks demonstrate that PI-BLS achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.
Pinki Khatun, M. Sajid, Abhinav Jha et al.· 0 citations
Physics-Informed Neural Networks (PINNs) have emerged as an important class of numerical methods for solving partial differential equations (PDEs). However, during the late-stage optimization process, further parameter updates often yield diminishing accuracy improvements while increasing computational costs. To address this issue, this paper proposes a Physics-Informed Error Field Learning (PIEFL) framework for PINNs. Unlike conventional approaches that continuously approximate the solution field using a single network, PIEFL introduces an auxiliary error network after the primary network achieves satisfactory accuracy and shifts the learning objective from the solution field to the error field. By deriving error control equations under physical constraints, the error network learns the discrepancy between the current approximation and the exact solution, and the learned error correction is combined with the primary prediction to improve solution accuracy. The proposed framework avoids continuous optimization of the entire solution space and focuses computational resources on correcting existing prediction errors. Moreover, PIEFL requires no modification to the primary network architecture, making it compatible with existing PINN models and applicable as a general post-training optimization strategy. Numerical experiments on representative PDEs demonstrate that PIEFL achieves higher solution accuracy under the same computational budget, validating its effectiveness in improving the performance of PINNs.
This review synthesizes recent advances in loss function designs for Physics-Informed Neural Networks (PINNs), a transformative approach to solving partial differential equations (PDEs) by embedding physical laws into deep learning frameworks. We begin by exploring the foundational role of loss functions in deep neural networks and their adaptation in PINNs to enforce PDE residuals, boundary conditions, and data constraints. The article systematically categorizes core loss components and early optimization strategies, including collocation point selection, which underpin effective PINN training. A comprehensive survey examines advanced loss function designs, including adaptive and self-adaptive weighting strategies, alternative formulations such as weak forms, energy-based, and gradient-enhanced methods, as well as uncertainty quantification and training enhancements like meta-learning, transfer learning, loss landscape engineering, regularization techniques, and optimization algorithms. We evaluated their impacts on convergence speed, predictive accuracy, and computational efficiency in applications such as fluid dynamics, solid mechanics, wave propagation, and geotechnical modeling. Challenges such as gradient imbalances, multi-scale dynamics, and high-dimensional scaling are critically analyzed, with emerging solutions like automated loss balancing, physics-aware regularization, and innovative architectures proposed to enhance robustness. A comparative analysis highlights performance trade-offs, while future directions emphasize dynamic optimization, hierarchical multi-fidelity losses, and scalable uncertainty quantification. This structured synthesis, drawing on developments up to 2025, equips researchers with insights to refine PINN methodologies, bridging data-driven and physics-based paradigms for transformative scientific computing.
M. Esmaeilbeigi, Daniela Annunziata, Salvatore Cuomo et al.· Journal of Scientific Comput...· 0 citations
The proposed SCV-PINN provides a robust and generalized extension of standard PINNs for complex-valued, multiscale, oscillatory, high-dimensional, and real-valued nonlinear PDEs.
Biswanath Barman, Rajendra K. Ray, Debdeep Chatterjee· 0 citations
This work embeds feature interaction modules derived from factorization machines (FMs) into physics-informed neural networks (PINNs) and neural operator learning, to enhance model expressiveness for solution manifolds of parameterized partial differential equations (PDEs). Motivated by the second-order Taylor expansion of multivariate functions to characterize variable couplings, we first propose FM-PINN. It explicitly captures spatio-temporal variable interactions and improves the approximation accuracy for smooth high-order PDEs. We further group spatial coordinates, time, physical parameters, and initial and boundary conditions into independent feature sets and model their cross-group interactions. Based on this strategy, we develop FM-Operator and FM-DeepONet, which are particularly effective for nonlinear conservation laws and problems with sharp gradients or discontinuities, while offering no consistent advantage on smooth operator learning benchmarks. Numerical tests demonstrate that the proposed mechanism delivers substantial accuracy gains on challenging shock-dominated equations, indicating a promising direction for physics-consistent modeling of parameterized PDEs with strong cross-field dependencies.
Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets. Parameterized PINNs (ParamPINNs) further take physical parameters as inputs, allowing a single model to represent a family of PDE solutions over a parameter domain. Existing ParamPINNs, however, still face inefficient training, uneven accuracy across parameters, and overfitting to a limited set of sampled parameter tasks, which can impair generalization to unsampled parameters. To address these issues, we propose a continual-learning physics-informed neural network (CL-PINN), which treats PDE instances at different parameter values as related tasks and learns them sequentially. CL-PINN combines Bayesian-optimization-based active parameter selection, task-wise dynamic loss weighting, sparse physics-constrained replay, and an optional parameter subnetwork to improve task allocation and knowledge retention under bounded active-task capacity. It requires no observational data and is designed to solve parameterized PDEs over relatively broad parameter domains under limited computational resources. Multi-seed evaluations on five benchmarks, including one continuous function and four parameterized PDEs, show that Bayesian selection substantially reduces objective-loss queries relative to grid-greedy search, while sparse replay mitigates forgetting of earlier tasks. Under the prescribed within-case resource protocols, CL-PINN generally provides higher and more balanced solution accuracy than fixed-sampling and grid-greedy baselines. CL-PINN offers a practical route toward learning PDE solutions that generalize across physical parameters and has the potential to support reusable physics-informed surrogates for large-scale engineering parameter studies.
Xujia Chen, Xinyu Hu, Letian Chen et al.· 0 citations