When the proposed method is compared to state-of-the-art variants of PINN, it is established that the method is superior to the current methods in a variety of high-dimensional PDEs with very small error magnitudes, even in the 20D case.
Abstract
This paper proposes a modified physics-informed neural network (PINN), known as Adaptive Weighted Loss Gradient-Enhanced PINNs (AWL-gPINNs) to the numerical approximation of high-dimensional nonlinear sine-Gordon equations (SGEs). The proposed algorithm is an extension of the typical PINN formulation, i.e. it adds gradient-based residual constraints and an adaptive weighting strategy in order to balance the importance of PDE residual terms, initial conditions, and boundary conditions in training. The existing governing SGE is reduced to a coupled first-order form, which allows the automatic differentiation to be easily integrated to evaluate higher-order derivatives. The resulting composite loss functional consists of the residual and gradient-regularization terms that have trainable weights, which reduce the issue of stiffness and imbalance in multi-objective optimization. Benchmark problems of 3D to 20D damped and undamped SGEs in both long and short time domains are extensively numerically experimented with. The findings show that AWL-gPINNs perform much better than standard PINNs and a variety of existing algorithms and obtain orders of error reduction between 1e-3 -1e-2 and 1e-5-1e-4. The technique also demonstrates rapid convergence, increased training robustness, and stability across different collocation densities, noise perturbations, and initialization conditions. Moreover, when the proposed method is compared to state-of-the-art variants of PINN, it is established that the method is superior to the current methods in a variety of high-dimensional PDEs with very small error magnitudes, even in the 20D case. The efficacy, robustness, and practical efficiency of the suggested AWL-gPINN framework for high-dimensional nonlinear SGE are further validated by ablation studies and multi-seed stability assessments. The results support that the proposed AWL-gPINNs is a scalable and successful technique for high-dimensional nonlinear PDE solutions.
To address the issues of insufficient accuracy and susceptibility to numerical oscillations in traditional physics-informed neural networks (PINNs) when solving large-gradient problems of hyperbolic conservation laws, this paper proposes an adaptive regularized hybrid PINN (ARH-PINN) to accurately capture large-gradient fields such as shock waves and suppress numerical oscillations. The ARH-PINNs method introduces a novel neural network architecture that integrates a Fourier embedding layer, a multi-layer perceptron, and a radial basis function layer to effectively capture both the global features and local structures of solutions to hyperbolic conservation laws. To improve the solution accuracy of large-gradient fields such as shock waves and suppress numerical oscillations, we propose an adaptive regularization strategy based on a compression indicator and a sharpness indicator and further conduct a sensitivity analysis of the associated parameters. To balance convergence speed and numerical stability, we incorporate input symmetric normalization and learning rate annealing into the training pipeline. Verified through extensive numerical examples on classical one-dimensional (1D) and two-dimensional (2D) conservation laws, the proposed ARH-PINNs can accurately resolve large-gradient fields such as shock waves, while effectively suppressing non-physical oscillations and retaining low numerical dissipation.
Tingjie Li, Supei Zheng, Feng Hu et al.· The Physics of Fluids· 0 citations
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.
Hailong Liu, S. Hedayatrasa, Yunpeng Zhu et al.· e-Journal of Nondestructive...· 0 citations
Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-frequency features; and representation-coefficient coupling, the entanglement of representation learning and coefficient fitting within a single nonconvex optimization objective. In this work, we propose the Fourier-enhanced alternating Levenberg--Marquardt PINN (FALM-PINN), an optimization framework that decouples representation learning from coefficient fitting. The upper-level problem learns a Fourier-enhanced basis that enriches the latent space with high-frequency components, while the lower-level problem resolves the coupling by fitting the projection coefficients on this basis, solving a nonlinear least-squares problem with the Levenberg--Marquardt algorithm. The framework applies to general nonlinear and coupled PDE systems, and reduces to a single-step convex optimization problem for linear PDEs. We prove global convergence of the alternating training scheme in both cases. Numerical examples on multiple challenging high-frequency and nonlinear PDEs show that FALM-PINN achieves relative $L^2$ errors up to two orders of magnitude lower than state-of-the-art baselines.
Yulun Wu, Matthieu Barreau, Miguel Aguiar et al.· 0 citations
Physics-Informed Neural Networks (PINNs) solve differential equations by minimizing the residual of a nonlinear operator over a neural parameterization of the solution. However, monolithic PINNs often suffer from ill-conditioning, spectral bias, and optimization instability. We introduce a variational boosting framework in which solutions are constructed additively in function space. Each stage trains a weak learner whose converged correction satisfies a local orthogonality condition, equivalent to a projected functional gradient descent step onto the tangent space of the network's function manifold. Because each correction network is deliberately small, the restricted minimization admits full Newton or conjugate gradient updates, which are typically infeasible in large PINNs. The resulting method separates global nonlinear refinement into a sequence of well-conditioned subproblems while preserving the full variational structure of the operator. This framework provides a geometric interpretation of multi-stage PINNs as projected functional gradient descent and enables stable second-order optimization for nonlinear differential equations.
Traditional Physics-Informed Neural Networks (PINNs) incorporate partial differential equations (PDEs), boundary conditions, and initial conditions into model training. However, their application to multidimensional problems may involve high computational costs associated with high-order automatic differentiation and increasingly demanding sampling requirements as dimensionality grows, potentially limiting the accuracy of solution approximation. To address these issues, we propose a Functional Constraint-based Variational Physics-Informed Neural Network (FC-VPINN) for solving PDEs admitting self-adjoint or weighted self-adjoint representations. It reformulates strong-form PDE constraints as corresponding variational constraints, improving mathematical interpretability while reducing reliance on high-order automatic differentiation. The stationarity condition of the constructed quadratic variational functional is mathematically equivalent to the governing equation under the prescribed conditions. For the second-order PDEs considered in this work, the resulting variational objective involves at most first-order derivatives of the network output, thereby reducing the derivative order required during training. Boundary and initial conditions are imposed through corresponding loss terms, without requiring a predefined set of test functions. Experiments on two-dimensional advection-diffusion, three-dimensional diffusion, and three-dimensional Poisson equations show that FC-VPINN achieves approximately one-order-of-magnitude lower prediction errors than the traditional PINN and reduces memory usage to 40% of that required by the baseline, demonstrating improved accuracy and computational efficiency in multidimensional problems.
Wenjie Zhang, Yubo Li, Weidong Cui et al.· Chinese Physics B· 0 citations
Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving forward and inverse partial differential equations (PDEs), but conventional real-valued PINNs (RV-PINNs) often suffer from spectral bias, limited expressivity, and reduced accuracy for high-frequency, oscillatory, and phase-dependent dynamics. In this work, we propose a generalized split complex-valued physics-informed neural network (SCV-PINN), in which network parameters and latent representations are defined in the complex domain. The framework employs split complex-valued activation functions by independently applying standard real-valued activations to the real and imaginary components, providing numerical stability, computational efficiency, and improved approximation capability. This formulation enables simultaneous learning of amplitude and phase information, enhancing the representation of nonlinear and oscillatory systems. Extensive ablation studies evaluate different split activation functions and collocation sampling strategies. The proposed framework is validated on forward and inverse PDE benchmarks including Burgers, Allen-Cahn, Korteweg-de Vries, nonlinear Schrodinger, Helmholtz, Poisson, Kovasznay flow (Re = 20), lid-driven cavity flow (Re = 100), the Lorenz system, inverse Burgers, inverse Navier-Stokes (Re = 100), and a three-dimensional Navier-Stokes Beltrami flow. For the Beltrami benchmark, SCV-PINN achieves a relative L2 error of 4.07 x 10^-5. Numerical results consistently demonstrate lower relative L2 errors and more accurate parameter identification than RV-PINNs and several existing PINN variants. 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