Although deep learning has achieved remarkable success in various scientific machine learning applications, its black-box nature poses concerns regarding interpretability and generalization capabilities beyond the training data. Interpretability is crucial and often desired in modeling physical systems. Moreover, acquiring extensive datasets that encompass the entire range of input features is challenging in many physics-based learning tasks, leading to increased errors when encountering out-of-distribution (OOD) data. In this work, motivated by the field of functional data analysis (FDA), we propose generalized functional linear models as an interpretable surrogate for a trained deep learning model. We demonstrate that our model could be trained either based on a trained neural network (post-hoc interpretation) or directly from training data (interpretable operator learning). A library of generalized functional linear models with different kernel functions is considered and sparse regression is used to discover an interpretable surrogate model that could be analytically presented. We present test cases in solid mechanics, fluid mechanics, and transport. Our results demonstrate that our model can achieve comparable accuracy to deep learning and can improve OOD generalization while providing more transparency and interpretability. Our study underscores the significance of interpretability in scientific machine learning and showcases the potential of functional linear models as a tool for interpreting and generalizing deep learning.

翻译：尽管深度学习在各种科学机器学习应用中取得了显著成功，但其黑箱特性引发了关于可解释性以及泛化能力（超出训练数据范围）的担忧。在物理系统建模中，可解释性至关重要且常被期望。此外，在许多基于物理的学习任务中，获取覆盖全部输入特征范围的大规模数据集具有挑战性，这导致在遇到分布外（OOD）数据时误差增加。受功能数据分析（FDA）领域启发，本研究提出将广义功能线性模型作为已训练深度学习模型的可解释替代方案。我们证明该模型既可基于已训练的神经网络进行训练（事后解释），也可直接从训练数据中学习（可解释算子学习）。研究中考虑了具有不同核函数的广义功能线性模型库，并采用稀疏回归来发现可解析表达的可解释替代模型。我们展示了固体力学、流体力学和输运领域的测试案例。结果表明，我们的模型能够达到与深度学习相当的精度，在提供更高透明度和可解释性的同时，还能提升OOD泛化性能。本研究强调了科学机器学习中可解释性的重要性，并展示了功能线性模型作为解释和泛化深度学习工具的潜力。