用排斥机制提升物理信息神经网络的不确定性估计精度
Repulsive Ensembles for Bayesian Inference in Physics-informed Neural Networks
- 在损失函数中加入排斥项，防止模型坍缩
- 相比传统集成，不确定性估计更准确，样本多样性更高
- 适合需要可信度评估的科学计算场景
物理信息神经网络（PINNs）在求解微分方程方面表现出色，尤其适用于非标准或不适定问题。当从数据中推断方程的解和参数时，不确定性估计比点估计更有价值，能反映解的可靠性。本文研究逆问题，提出排斥型PINN集成（RE-PINN），通过在损失函数中引入特定排斥项，使集成预测在无限成员数下趋近真实贝叶斯后验。在可能情况下，将集成结果与蒙特卡洛基线对比。标准集成易坍缩至最大后验解，而排斥集成则显著提升不确定性估计准确性，并展现更高样本多样性。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) have proven an effective tool for solving differential equations, in particular when considering non-standard or ill-posed settings. When inferring solutions and parameters of the differential equation from data, uncertainty estimates are preferable to point estimates, as they give an idea about the accuracy of the solution. In this work, we consider the inverse problem and employ repulsive ensembles of PINNs (RE-PINN) for obtaining such estimates. The repulsion is implemented by adding a particular repulsive term to the loss function, which has the property that the ensemble predictions correspond to the true Bayesian posterior in the limit of infinite ensemble members. Where possible, we compare the ensemble predictions to Monte Carlo baselines. Whereas the standard ensemble tends to collapse to maximum-a-posteriori solutions, the repulsive ensemble produces significantly more accurate uncertainty estimates and exhibits higher sample diversity.
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