Physics-Informed Error Field Learning: A Post-Training Optimization Framework for Physics-Informed Neural Networks

• URL: http://arxiv.org/abs/2608.24970v1
• Date: Tue, 25 Aug 2026 09:34:46 GMT
• Title: Physics-Informed Error Field Learning: A Post-Training Optimization Framework for Physics-Informed Neural Networks
• Abstract summary: This paper proposes a Physics-Informed Error Field Learning framework for PINNs.<n>By deriving error control equations under physical constraints, PIEFL learns the discrepancy between the current approximation and the exact solution.<n> Numerical experiments on representative PDEs demonstrate that PIEFL achieves higher solution accuracy under the same computational budget.
• Score: 5.123157239082347
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Physics-Informed Broad Learning System: An Efficient Backpropagation-Free Framework for Solving Partial Differential Equations [9.497148303350697]
  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.<n>We propose a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs.<n>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.
  arXiv  Detail & Related papers  (2026-07-28T11:36:34Z)
• Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System [47.73467147921344]
  Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems.<n>Recent Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but frequently suffer from slow convergence and optimization instability.<n>This article proposes the Physics-Informed Broad Learning System (PIBLS), a novel backpropagation-free framework that reformulates PDE solving as a direct least-squares optimization.
  arXiv  Detail & Related papers  (2026-06-18T03:34:06Z)
• Building Trust in PINNs: Error Estimation through Finite Difference Methods [52.92887996882041]
  Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs)<n>Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality.<n>We propose a lightweight post-hoc method that addresses this gap by producing pointwise error estimates for PINN predictions.
  arXiv  Detail & Related papers  (2026-03-16T16:51:42Z)
• Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective [14.65008276932511]
  Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs)<n>PINNs are typically optimized using a finite set of points, which poses significant challenges in guaranteeing their convergence and accuracy.<n>We propose a new weighting scheme that will adaptively change the weights to the loss functions from isolated points to their continuous neighborhood regions.
  arXiv  Detail & Related papers  (2025-06-24T17:13:51Z)
• DiffGrad for Physics-Informed Neural Networks [0.0]
  Burgers' equation, a fundamental equation in fluid dynamics that is extensively used in PINNs, provides flexible results with the Adamprop. This paper introduces a novel strategy for solving Burgers' equation by incorporating DiffGrad with PINNs.
  arXiv  Detail & Related papers  (2024-09-05T04:39:35Z)
• RoPINN: Region Optimized Physics-Informed Neural Networks [66.38369833561039]
  Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) This paper proposes and theoretically studies a new training paradigm as region optimization. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm.
  arXiv  Detail & Related papers  (2024-05-23T09:45:57Z)
• Failure-informed adaptive sampling for PINNs [5.723850818203907]
  Physics-informed neural networks (PINNs) have emerged as an effective technique for solving PDEs in a wide range of domains. Recent research has demonstrated, however, that the performance of PINNs can vary dramatically with different sampling procedures. We present an adaptive approach termed failure-informed PINNs, which is inspired by the viewpoint of reliability analysis.
  arXiv  Detail & Related papers  (2022-10-01T13:34:41Z)
• Adaptive Anomaly Detection for Internet of Things in Hierarchical Edge Computing: A Contextual-Bandit Approach [81.5261621619557]
  We propose an adaptive anomaly detection scheme with hierarchical edge computing (HEC) We first construct multiple anomaly detection DNN models with increasing complexity, and associate each of them to a corresponding HEC layer. Then, we design an adaptive model selection scheme that is formulated as a contextual-bandit problem and solved by using a reinforcement learning policy network.
  arXiv  Detail & Related papers  (2021-08-09T08:45:47Z)
• Optimal Transport Based Refinement of Physics-Informed Neural Networks [0.0]
  We propose a refinement strategy to the well-known Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs) based on the concept of Optimal Transport (OT) PINNs solvers have been found to suffer from a host of issues: spectral bias in fully-connected pathologies, unstable gradient, and difficulties with convergence and accuracy. We present a novel training strategy for solving the Fokker-Planck-Kolmogorov Equation (FPKE) using OT-based sampling to supplement the existing PINNs framework.
  arXiv  Detail & Related papers  (2021-05-26T02:51:20Z)
• A Meta-Learning Approach to the Optimal Power Flow Problem Under Topology Reconfigurations [69.73803123972297]
  We propose a DNN-based OPF predictor that is trained using a meta-learning (MTL) approach. The developed OPF-predictor is validated through simulations using benchmark IEEE bus systems.
  arXiv  Detail & Related papers  (2020-12-21T17:39:51Z)
• Self-Directed Online Machine Learning for Topology Optimization [58.920693413667216]
  Self-directed Online Learning Optimization integrates Deep Neural Network (DNN) with Finite Element Method (FEM) calculations. Our algorithm was tested by four types of problems including compliance minimization, fluid-structure optimization, heat transfer enhancement and truss optimization. It reduced the computational time by 2 5 orders of magnitude compared with directly using methods, and outperformed all state-of-the-art algorithms tested in our experiments.
  arXiv  Detail & Related papers  (2020-02-04T20:00:28Z)

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.