发表机构
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出PIEFL框架作为PINNs的训练后优化策略，通过引入辅助误差网络修正预测误差，在相同计算预算下提升了PINNs的解精度。
AI 中文摘要
物理信息神经网络（Physics-Informed Neural Networks, PINNs）已成为求解偏微分方程（Partial Differential Equations, PDEs）的重要数值方法。然而，在优化后期，进一步的参数更新往往仅能带来精度的小幅提升，却会增加计算成本。为解决该问题，本文提出一种针对PINNs的物理信息误差场学习（Physics-Informed Error Field Learning, PIEFL）框架。与传统方法持续用单个网络近似解场不同，PIEFL在主网络达到满意精度后引入辅助误差网络，并将学习目标从解场转移至误差场。通过推导物理约束下的误差控制方程，误差网络学习当前近似解与精确解的差异，将学习到的误差修正项与主网络预测结合以提升解的精度。该框架避免了对整个解空间的持续优化，将计算资源集中于修正现有预测误差，且无需修改主网络架构，可兼容现有PINN模型，作为通用的训练后优化策略使用。对代表性PDEs的数值实验表明，在相同计算预算下，PIEFL可获得更高的解精度，验证了其提升PINNs性能的有效性。
英文摘要
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.
