用贝叶斯后验方差替代集成，稳定物理信息神经网络训练
B-PL-PINN: Stabilizing PINN Training with Bayesian Pseudo Labeling
- 以贝叶斯后验方差代替集成共识，动态扩展有效训练区域
- 在多个基准问题上优于传统集成方法，性能接近最优混合优化器
- 适合需稳定训练的PINN应用，如复杂偏微分方程求解
针对前向问题中物理信息神经网络（PINN）训练易发收敛困难的问题，本文提出B-PL-PINN方法。该方法将原基于集成共识与伪标签点邻近性的策略，替换为基于贝叶斯后验方差的评估机制，从而更数学严谨地扩展各PINN的有效训练域。实验表明，该方法在一系列基准问题上优于现有集成方案，且性能与结合Adam和LBFGS的混合优化器训练的集成相当，显著提升了信息从初始条件向计算域内部传播的能力。
原文摘要 · Abstract (English)
Training physics-informed neural networks (PINNs) for forward problems often suffers from severe convergence issues, hindering the propagation of information from regions where the desired solution is well-defined. Haitsiukevich and Ilin (2023) proposed an ensemble approach that extends the active training domain of each PINN based on i) ensemble consensus and ii) vicinity to (pseudo-)labeled points, thus ensuring that the information from the initial condition successfully propagates to the interior of the computational domain. In this work, we suggest replacing the ensemble by a Bayesian PINN, and consensus by an evaluation of the PINN's posterior variance. Our experiments show that this mathematically principled approach outperforms the ensemble on a set of benchmark problems and is competitive with PINN ensembles trained with combinations of Adam and LBFGS.
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