pinn implementation

Existing PINN surveys suffer from incomplete coverage and insufficient depth—either focusing narrowly on specific applications or neglecting architectural evolution and empirical evaluation. To address this, we present the first systematic and in-depth comprehensive survey of Physics-Informed Neural Networks (PINNs). Our work establishes a unified analytical framework encompassing architectural variants, classes of differential equations (including PDEs and ODEs), canonical application domains (e.g., fluid dynamics, heat transfer), and real-world case studies. By integrating loss function design principles, numerical stability analysis, interpretability assessment, and empirical computational efficiency benchmarks, we quantitatively characterize the applicability boundaries of mainstream PINN variants for the first time. We identify three fundamental challenges: weak convergence guarantees, poor generalization under distributional shift, and prohibitive computational overhead. Based on this diagnosis, we propose a verifiable, roadmap-driven agenda for future research, prioritizing theoretical foundations, scalable architectures, and robust training paradigms.

This work proposes NewPINNs, a novel framework that addresses the well-known optimization failures of traditional physics-informed neural networks (PINNs) when solving partial differential equations (PDEs), which often stem from ill-conditioned residual loss formulations, sensitivity to loss weighting, and challenges posed by stiffness or strong nonlinearity. NewPINNs uniquely integrates classical numerical solvers—such as finite volume, finite element, and spectral methods—directly into the neural network training process. Instead of relying on explicit PDE residual and boundary condition losses, the method enforces consistency between the neural network predictions and the states evolved by the embedded numerical solver through a pull-push interaction mechanism. This approach effectively circumvents common failure modes of PINNs, significantly enhancing solution stability and accuracy across a range of forward and inverse PDE problems, particularly in stiff and highly nonlinear regimes.

This work addresses the well-known challenges of traditional physics-informed neural networks (PINNs)—including difficult training dynamics, slow convergence, and limited interpretability—by introducing domain-aware Fourier features (DaFFs). DaFFs uniquely embed geometric and boundary information directly into the input encoding, thereby eliminating the need for explicit boundary loss terms and complex loss balancing strategies. Furthermore, the authors integrate a Layer-wise Relevance Propagation (LRP) framework tailored for PINNs to enable physically consistent feature attribution. Experimental results demonstrate that the proposed approach reduces prediction errors by several orders of magnitude compared to conventional PINNs and methods based on random Fourier features, while also achieving significantly faster convergence and producing explanations that align more closely with underlying physical principles.

Physics-informed neural networks (PINNs) suffer from slow convergence and spectral bias when solving partial differential equations (PDEs) with rapid oscillations, boundary layers, or strong nonlinearity. Method: This work systematically investigates the representational disparity between learnable activation functions and learnable basis functions. We propose two PINN architectures: a multi-layer perceptron (MLP) with learnable activations and a Kolmogorov–Arnold network (KAN) with learnable basis functions. For the first time, we quantitatively compare their performance in high-frequency approximation, convergence speed, and spectral bias mitigation across diverse PDEs—including oscillatory solutions, nonlinear waves, multiphysics couplings, and fluid dynamics. Results: Learnable basis functions significantly improve approximation accuracy for boundary layers and sharp gradients while accelerating convergence. The empirical findings reveal problem-dependent structural requirements for activations versus bases, leading to principled design guidelines for PDE-solving neural architectures. Code and pretrained models are publicly released.

This work addresses the lack of a unified understanding of the design principles, applicability, and performance differences between Physics-Informed Neural Networks (PINNs) and Neural Operators (NOs), which hinders the development of reliable data-driven PDE solvers. It proposes the first unified analytical framework that systematically characterizes the design space of both approaches along three dimensions: learning objectives, mechanisms for embedding physical structure, and strategies for computational load distribution. By elucidating the intrinsic connections and fundamental distinctions between these methods, the study not only clarifies the positioning and performance origins of existing techniques but also provides theoretical guidance and novel pathways for designing efficient and robust PDE solvers that effectively integrate physical priors with data-driven learning.

To address encoder redundancy and spectral bias in Physics-Informed Neural Networks (PINNs), this paper proposes Spectral PINNSformer (S-Pformer). Methodologically, S-Pformer discards the conventional encoder-decoder architecture in favor of a lightweight, encoder-free decoder-only Transformer; incorporates Fourier feature embeddings to explicitly encode multi-scale frequency-domain information, thereby mitigating low-frequency bias; and leverages self-attention to efficiently capture long-range spatiotemporal dependencies. These design choices jointly reduce model parameter count while enhancing multi-scale representation fidelity for complex PDE solutions. Experimental results demonstrate that S-Pformer consistently outperforms existing PINNSformer variants across multiple benchmark PDE tasks and achieves accuracy comparable to—or exceeding—that of standard MLP-based baselines. Overall, S-Pformer achieves a balanced optimization of predictive accuracy and computational efficiency.

This work addresses the limitations of existing parameter-efficient fine-tuning (PEFT) methods when adapting physics-informed neural networks under out-of-distribution conditions, where such approaches often disrupt the underlying physical manifold structure and introduce redundancy. Conventional SVD-based techniques further suffer from subspace locking and truncation of high-frequency spectral modes. To overcome these challenges, the authors propose MODE, a lightweight micro-architecture that decomposes physical evolution into complementary mechanisms via dense mixing of dominant spectral components within an orthogonal basis, a residual spectral reactivation mechanism, and affine Galilean decoupling. This design effectively activates high-frequency modes while preserving the manifold structure, using only a minimal number of trainable parameters. Evaluated on tasks including 1D convection–diffusion–reaction equations and 2D Helmholtz equations, MODE substantially outperforms existing PEFT methods and demonstrates superior out-of-distribution generalization.

This work addresses the significant limitations in accuracy and convergence of Physics-Informed Neural Networks (PINNs) when training data are inconsistent with the governing partial differential equation (PDE). The authors introduce the concept of a “consistency barrier,” which reveals an inherent lower bound on error arising from the tension between data fidelity and PDE residual enforcement. Through controlled experiments on the 1D viscous Burgers equation—leveraging analytical solutions, multi-fidelity numerical data, and residual-driven training—they demonstrate that PINN errors under low-fidelity data are fundamentally constrained by this consistency barrier. In contrast, when high-fidelity data are employed, the barrier is effectively eliminated, yielding PINN solutions statistically indistinguishable from the analytical solution. This study provides both theoretical insight and empirical evidence to advance the understanding and robustness of PINNs in practical applications.

Traditional physics-informed neural networks (PINNs) with fixed domain decomposition lack adaptivity in solving multiscale partial differential equations (PDEs), failing to align with the intrinsic multiscale structure of solutions. Method: We propose Adaptive Basis PINN (AB-PINN), which introduces a residual-driven dynamic domain decomposition mechanism inspired by numerical mesh refinement: subdomains are generated and adjusted in real time during training based on local PDE residuals, enabling co-evolution of subdomain distribution and solution features. A parameter-sharing and collaborative optimization framework across subdomains enhances representational capacity and mitigates local minima. Contribution/Results: AB-PINN achieves superior accuracy and convergence stability on multiple benchmark problems involving strongly nonlinear, multiscale PDEs—requiring less hyperparameter tuning than baseline methods—demonstrating enhanced robustness and generalization capability.

This study addresses key challenges in power system modeling—namely data scarcity, poor interpretability, and the imperative to strictly adhere to physical laws—by systematically reviewing and comparing hybrid architectures that integrate physical priors with machine learning. These include physics-informed neural networks (PINNs), DeepONets, Fourier neural operators, and their enhanced variants incorporating graph structures, domain decomposition, and extreme learning machines. By embedding fundamental physical laws such as Maxwell’s equations directly into neural network training, this paradigm substantially improves prediction accuracy and generalization under sparse, noisy data conditions. It achieves simulation speeds orders of magnitude faster than finite element methods and outperforms purely data-driven approaches in dynamic modeling, parameter sensitivity analysis, and real-time digital twin calibration, thereby advancing power system intelligence from opaque black-box models toward transparent, physically interpretable frameworks.