发表机构
Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出CL-PINN，将不同参数的PDE实例视为相关任务依次学习，结合贝叶斯主动参数选择等技术，提升参数化PDE解的精度与泛化能力，优于相关基线方法。
AI 中文摘要
物理信息神经网络（PINNs）将控制方程融入神经网络训练，无需大量观测数据集即可近似偏微分方程（PDE）的解。参数化PINNs（ParamPINNs）进一步将物理参数作为输入，使单个模型能够表示参数域内一系列PDE的解。然而，现有的ParamPINNs仍存在训练效率低、不同参数间精度不均、对有限采样参数任务过拟合等问题，这会损害其对未采样参数的泛化能力。为解决这些问题，我们提出了持续学习物理信息神经网络（CL-PINN），它将不同参数值下的PDE实例视为相关任务并依次学习。CL-PINN结合了基于贝叶斯优化的主动参数选择、任务级动态损失加权、稀疏物理约束重放以及可选的参数子网络，以在有限的主动任务容量下优化任务分配并保留知识。它无需观测数据，旨在计算资源有限的情况下求解相对宽参数域内的参数化PDE。对五个基准（包括一个连续函数和四个参数化PDE）的多种子评估显示，贝叶斯选择相比网格贪婪搜索大幅减少了目标损失查询次数，而稀疏重放缓解了早期任务的遗忘。在规定的案例内资源协议下，CL-PINN通常比固定采样和网格贪婪基线提供更高且更均衡的解精度，为学习能跨物理参数泛化的PDE解提供了实用途径，并有潜力为大规模工程参数研究提供可复用的物理信息代理。
英文摘要
Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets. Parameterized PINNs (ParamPINNs) further take physical parameters as inputs, allowing a single model to represent a family of PDE solutions over a parameter domain. Existing ParamPINNs, however, still face inefficient training, uneven accuracy across parameters, and overfitting to a limited set of sampled parameter tasks, which can impair generalization to unsampled parameters. To address these issues, we propose a continual-learning physics-informed neural network (CL-PINN), which treats PDE instances at different parameter values as related tasks and learns them sequentially. CL-PINN combines Bayesian-optimization-based active parameter selection, task-wise dynamic loss weighting, sparse physics-constrained replay, and an optional parameter subnetwork to improve task allocation and knowledge retention under bounded active-task capacity. It requires no observational data and is designed to solve parameterized PDEs over relatively broad parameter domains under limited computational resources. Multi-seed evaluations on five benchmarks, including one continuous function and four parameterized PDEs, show that Bayesian selection substantially reduces objective-loss queries relative to grid-greedy search, while sparse replay mitigates forgetting of earlier tasks. Under the prescribed within-case resource protocols, CL-PINN generally provides higher and more balanced solution accuracy than fixed-sampling and grid-greedy baselines. CL-PINN offers a practical route toward learning PDE solutions that generalize across physical parameters and has the potential to support reusable physics-informed surrogates for large-scale engineering parameter studies.
Comments124 pages in total, including the main text (63 pages, 25 figures, and 16 tables) and supplementary material (61 pages, 30 figures, and 17 tables)
