用证据深度学习提升物理信息神经网络的不确定性估计能力
Evidential Physics-Informed Neural Networks for Scientific Discovery
- 引入证据深度学习损失函数，实现输出不确定性的量化
- 在1D泊松方程和2DFisher-KPP方程上覆盖概率校准优于贝叶斯PINN与深度集成
- 适用于需要可靠置信度的科学发现场景，如糖尿病生理建模
我们提出了证据物理信息神经网络（E-PINN）的基础理论与实现规范——一种新型的不确定性感知物理信息神经网络。它利用证据深度学习的边缘分布损失函数来估计输出不确定性，并通过学习到的后验分布推断偏微分方程中的未知参数。在两个典型案例研究中验证：一维泊松方程带高斯源项，二维Fisher-KPP方程。结果表明，E-PINN生成的经验覆盖概率显著优于贝叶斯PINN与深度集成方法。为展示实际应用价值，还简要展示了E-PINN在分析糖尿病病理生理学研究中常用的临床葡萄糖-胰岛素数据集的应用。
原文摘要 · Abstract (English)
We present the fundamental theory and implementation guidelines underlying Evidential Physics-Informed Neural Network (E-PINN) -- a novel class of uncertainty-aware PINN. It leverages the marginal distribution loss function of evidential deep learning for estimating uncertainty of outputs, and infers unknown parameters of the PDE via a learned posterior distribution. Validating our model on two illustrative case studies -- the 1D Poisson equation with a Gaussian source and the 2D Fisher-KPP equation, we found that E-PINN generated empirical coverage probabilities that were calibrated significantly better than Bayesian PINN and Deep Ensemble methods. To demonstrate real-world applicability, we also present a brief case study on applying E-PINN to analyze clinical glucose-insulin datasets that have featured in medical research on diabetes pathophysiology.
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