新模型融合物理规律与不确定性，提升科学计算精度与可信度。
Evidential Physics-Informed Neural Networks
- 用证据深度学习构建可量化不确定性的物理神经网络
- 在噪声数据下保持边界条件，覆盖率更接近理论值
- 适合需要可信预测的科学逆问题研究
我们提出一种基于证据深度学习原理的新型物理信息神经网络，通过学习高阶分布参数实现不确定性量化。将偏微分方程残差损失和数据拟合损失中的依赖变量重新表述为证据先验分布超参数的函数。模型配备基于信息论的正则项，包含两个逆伽马分布之间的KL散度以刻画预测不确定性。相比贝叶斯物理信息神经网络，本框架对数据噪声更敏感，能更准确保持边界条件，且经验覆盖概率更接近名义水平。为验证其在科学发现数据挖掘中的适用性，我们展示了该模型在求解一维和二维非线性微分方程逆问题中的应用。
原文摘要 · Abstract (English)
We present a novel class of Physics-Informed Neural Networks that is formulated based on the principles of Evidential Deep Learning, where the model incorporates uncertainty quantification by learning parameters of a higher-order distribution. The dependent and trainable variables of the PDE residual loss and data-fitting loss terms are recast as functions of the hyperparameters of an evidential prior distribution. Our model is equipped with an information-theoretic regularizer that contains the Kullback-Leibler divergence between two inverse-gamma distributions characterizing predictive uncertainty. Relative to Bayesian-Physics-Informed-Neural-Networks, our framework appeared to exhibit higher sensitivity to data noise, preserve boundary conditions more faithfully and yield empirical coverage probabilities closer to nominal ones. Toward examining its relevance for data mining in scientific discoveries, we demonstrate how to apply our model to inverse problems involving 1D and 2D nonlinear differential equations.
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