Over-PINNs: Enhancing Physics-Informed Neural Networks via Higher-Order Partial Derivative Overdetermination of PDEs

• URL: http://arxiv.org/abs/2506.05918v1
• Date: Fri, 06 Jun 2025 09:38:13 GMT
• Title: Over-PINNs: Enhancing Physics-Informed Neural Networks via Higher-Order Partial Derivative Overdetermination of PDEs
• Authors: Wenxuan Huo, Qiang He, Gang Zhu, Weifeng Huang,
• Abstract summary: We introduce the Over-PINNs framework to generate auxiliary equations that impose additional physical constraints.<n>It achieves a significant improvement in solution accuracy without incurring substantial additional computational costs.
• Score: 6.968551465693266
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Partial differential equations (PDEs) serve as the cornerstone of mathematical physics. In recent years, Physics-Informed Neural Networks (PINNs) have significantly reduced the dependence on large datasets by embedding physical laws directly into the training of neural networks. However, when dealing with complex problems, the accuracy of PINNs still has room for improvement. To address this issue, we introduce the Over-PINNs framework, which leverages automatic differentiation (AD) to generate higher-order auxiliary equations that impose additional physical constraints. These equations are incorporated as extra loss terms in the training process, effectively enhancing the model's ability to capture physical information through an "overdetermined" approach. Numerical results illustrate that this method exhibits strong versatility in solving various types of PDEs. It achieves a significant improvement in solution accuracy without incurring substantial additional computational costs.

Related papers

• High precision PINNs in unbounded domains: application to singularity formulation in PDEs [83.50980325611066]
  We study the choices of neural network ansatz, sampling strategy, and optimization algorithm.<n>For 1D Burgers equation, our framework can lead to a solution with very high precision.<n>For the 2D Boussinesq equation, we obtain a solution whose loss is $4$ digits smaller than that obtained in citewang2023asymptotic with fewer training steps.
  arXiv  Detail & Related papers  (2025-06-24T02:01:44Z)
• Stiff Transfer Learning for Physics-Informed Neural Networks [1.5361702135159845]
  We propose a novel approach, stiff transfer learning for physics-informed neural networks (STL-PINNs) to tackle stiff ordinary differential equations (ODEs) and partial differential equations (PDEs)<n>Our methodology involves training a Multi-Head-PINN in a low-stiff regime, and obtaining the final solution in a high stiff regime by transfer learning.<n>This addresses the failure modes related to stiffness in PINNs while maintaining computational efficiency by computing "one-shot" solutions.
  arXiv  Detail & Related papers  (2025-01-28T20:27:38Z)
• Advancing Generalization in PINNs through Latent-Space Representations [71.86401914779019]
  Physics-informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs)<n>We propose PIDO, a novel physics-informed neural PDE solver designed to generalize effectively across diverse PDE configurations.<n>We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier-Stokes equations.
  arXiv  Detail & Related papers  (2024-11-28T13:16:20Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• Physics-aware deep learning framework for linear elasticity [0.0]
  The paper presents an efficient and robust data-driven deep learning (DL) computational framework for linear continuum elasticity problems. For an accurate representation of the field variables, a multi-objective loss function is proposed. Several benchmark problems including the Airimaty solution to elasticity and the Kirchhoff-Love plate problem are solved.
  arXiv  Detail & Related papers  (2023-02-19T20:33:32Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Physics-informed attention-based neural network for solving non-linear partial differential equations [6.103365780339364]
  Physics-Informed Neural Networks (PINNs) have enabled significant improvements in modelling physical processes. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Here, we address the question of which network architectures are best suited to learn the complex behavior of non-linear PDEs.
  arXiv  Detail & Related papers  (2021-05-17T14:29:08Z)
• Physics informed deep learning for computational elastodynamics without labeled data [13.084113582897965]
  We present a physics-informed neural network (PINN) with mixed-variable output to model elastodynamics problems without resort to labeled data. Results show the promise of PINN in the context of computational mechanics applications.
  arXiv  Detail & Related papers  (2020-06-10T19:05:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.