physics-informed neural networks

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 instability arising when enforcing Neumann and Robin boundary conditions on piecewise $C^1$ (globally only $C^0$) boundaries in Physics-Informed Neural Operators (PINOs). We propose a novel strong-constraint method based on orthogonal projection, extending the Sukumar & Srivastava framework to relax the conventional requirement of globally $C^1$ boundaries. By constructing admissible trial functions and applying orthogonal projection onto the constrained function space, our approach ensures exact and numerically stable satisfaction of boundary conditions. The method unifies weak, semi-weak, and strong formulations for boundary treatment. We validate it on scalar Darcy flow and steady-state Navier–Stokes equations: results demonstrate significantly improved training stability and numerical accuracy, enhanced robustness on complex geometries, faster convergence, and lower approximation errors compared to standard approaches.

Traditional numerical solvers for parametric partial differential equations (PDEs) incur prohibitive computational costs in multi-query scenarios. Method: This paper proposes a unified framework to systematically compare physics-informed neural networks (PINNs) and neural operators—including DeepONet and the Fourier neural operator—in their ability to learn solution mappings over infinite-dimensional function spaces. It innovatively integrates soft-constraint physics embedding with operator approximation theory to characterize fundamental differences in their generalization mechanisms. Contribution/Results: The analysis yields theoretically grounded guidelines for method selection. Experiments on canonical fluid and solid mechanics tasks demonstrate that the proposed approach achieves real-time inference and inverse problem solving up to 10³–10⁵× faster than conventional solvers, while maintaining comparable accuracy—thereby significantly enhancing parameter-space exploration efficiency.

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.

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 study addresses the numerical instability of neural spectral architectures (NeuSA) when solving stiff differential equations by proposing a novel framework that integrates neural spectral methods with exponential time differencing (ETD). The approach leverages spectral representations and high-order exponential integrators to precisely handle linear stiff terms, while employing physics-informed neural networks to model nonlinear residuals, thereby achieving efficient decoupling of stiff–nonstiff dynamics. Experimental results demonstrate that the proposed framework attains both high stability and accuracy on stiff PDE benchmarks. Furthermore, it supports the identification of unknown physical parameters through inverse problem learning, offering a reliable new paradigm for modeling stiff systems.

This work proposes a novel paradigm for physics-informed neural networks (PINNs) by systematically integrating the finite difference method (FDM) to replace automatic differentiation in constructing partial differential equation (PDE) loss terms. Traditional PINNs rely on automatic differentiation, which entails complex implementation and substantial computational overhead. In contrast, the proposed FDM-PINN framework significantly simplifies model implementation and enhances training efficiency. Demonstrated on canonical PDEs such as the Laplace and Burgers equations, FDM-PINN not only outperforms purely data-driven deep learning models lacking physical constraints but also achieves accuracy comparable to conventional automatic differentiation-based PINNs while substantially reducing computational cost. This approach establishes a new, efficient, and lightweight pathway for physics-driven modeling.

This work proposes implicit Finite Operator Learning (iFOL), a physics-informed operator learning framework for multiphysics problems governed by coupled partial differential equations on arbitrary domains, which operates without requiring ground-truth labeled data. Built upon the finite element weighted residual formulation, iFOL establishes a resolution-independent mapping from input parameters to the solution space and provides a unified treatment of complex geometries and multiphysics coupling. Implemented within the Folax system on the JAX platform, the approach integrates FNO, DeepONet, and iFOL, leveraging finite element residuals to construct physics-constrained loss functions. Experiments demonstrate that iFOL achieves high efficiency on complex geometries in two- and three-dimensional nonlinear thermo-mechanical coupling and industrial casting scenarios, while FNO excels in accuracy on regular domains; furthermore, a single-network end-to-end training strategy significantly outperforms baseline methods.

This study addresses the ill-conditioning and insufficient accuracy of loss functions induced by differential operators during the training of physics-informed neural operators. To overcome this limitation, this work proposes a preconditioned residual loss that integrates geometric and algebraic multigrid techniques to achieve mesh-independent condition number control. Notably, this strategy is architecture-agnostic and incurs zero overhead during inference. By effectively resolving these optimization challenges, the proposed approach breaks through the bottlenecks of conventional unsupervised physics-informed learning. Experimental results demonstrate that the method attains supervised-level accuracy on benchmark equations such as the Poisson equation, yielding a four- to twenty-five-fold improvement in accuracy over existing state-of-the-art approaches.

This work addresses the challenge that existing neural operators struggle to simultaneously maintain physical fidelity, generalization capability, and inference efficiency under varying PDE parameters and boundary conditions. The authors propose a generalized neural operator framework that explicitly embeds PDE parameters and boundary conditions to ensure well-posedness and enable efficient cross-domain solutions. Key innovations include a parameter-gated kernel mixture mechanism, a generalized boundary transfer operator, and a unified Dirichlet latent-space representation. A physics-constrained training objective is designed to eliminate instance-specific optimization. Experiments demonstrate that the proposed method significantly outperforms current neural operators across diverse heterogeneous physical scenarios, achieving high accuracy while matching the inference speed of conventional numerical solvers, thereby offering a balanced combination of universality, efficiency, and physical consistency.