[论文解读] Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanics

Jan N. Fuhg, Reese E. Jones+1 more|arXiv (Cornell University)|Oct 5, 2023

Model Reduction and Neural NetworksPhysics and Astronomy被引用 3

一句话总结

本文提出使用平滑L⁰-正则化对物理增强神经网络进行极端稀疏化，以在固体力学中发现可解释、可信的本构模型。通过在联合训练中同时优化数据拟合与参数稀疏性，并施加热力学一致性约束，该方法可获得紧凑、人类可读的超弹性、屈服函数和硬化定律的函数形式——在合成数据和实验数据上均取得高精度与强外推能力。

ABSTRACT

Data-driven constitutive modeling with neural networks has received increased interest in recent years due to its ability to easily incorporate physical and mechanistic constraints and to overcome the challenging and time-consuming task of formulating phenomenological constitutive laws that can accurately capture the observed material response. However, even though neural network-based constitutive laws have been shown to generalize proficiently, the generated representations are not easily interpretable due to their high number of trainable parameters. Sparse regression approaches exist that allow to obtaining interpretable expressions, but the user is tasked with creating a library of model forms which by construction limits their expressiveness to the functional forms provided in the libraries. In this work, we propose to train regularized physics-augmented neural network-based constitutive models utilizing a smoothed version of $L^{0}$-regularization. This aims to maintain the trustworthiness inherited by the physical constraints, but also enables interpretability which has not been possible thus far on any type of machine learning-based constitutive model where model forms were not assumed a-priory but were actually discovered. During the training process, the network simultaneously fits the training data and penalizes the number of active parameters, while also ensuring constitutive constraints such as thermodynamic consistency. We show that the method can reliably obtain interpretable and trustworthy constitutive models for compressible and incompressible hyperelasticity, yield functions, and hardening models for elastoplasticity, for synthetic and experimental data.

研究动机与目标

• 为克服尽管表达能力强且泛化性能优异，但数据驱动神经网络本构模型缺乏可解释性的问题。
• 消除稀疏回归方法中对预定义模型形式库的依赖，从而限制函数表达能力。
• 通过正则化结合物理约束与参数稀疏性，实现可信的、可外推的本构建模。
• 从有限的实验或合成数据中自动发现可解释、物理解释一致的本构定律。
• 弥合高容量神经网络与固体力学建模中可解释、人类可读数学表达式之间的差距。

提出的方法

• 该方法采用L⁰-正则化的平滑近似，通过惩罚训练过程中活跃参数的数量来促进极端稀疏性。
• 物理增强神经网络在训练中同时拟合训练数据并最小化非零参数数量，确保模型简洁性。
• 在神经网络架构与损失函数设计中，强制实施热力学一致性、客观性及材料对称性。
• 每个神经元仅使用一个非线性激活函数，通过平滑L⁰惩罚引导剪枝，仅保留关键参数。
• 最终模型以稀疏、显式的函数形式提取，具备可解释性，可直接用于有限元求解器。
• 该方法在合成与实验数据上应用于超弹性、屈服函数及各向同性硬化模型。

实验结果

研究问题

• RQ1 能否通过对物理增强神经网络进行极端稀疏化，获得无需预先假设函数形式的可解释本构模型？
• RQ2 该方法在低数据量情形下，泛化与外推能力如何，尤其在训练数据范围之外？
• RQ3 在稀疏、数据驱动模型中，施加物理约束在多大程度上能提升泛化能力并减少过拟合？
• RQ4 该方法能否发现复杂材料行为（如弹塑性硬化）的紧凑、人类可读的函数形式？
• RQ5 与传统剪枝或稀疏回归相比，平滑L⁰-正则化在可解释性与准确性方面表现如何？

主要发现

• 该方法在合成数据上成功发现了可解释的可压缩与不可压缩超弹性本构模型，精度极高。
• 在实验SS316L不锈钢数据上，模型拟合出形式为R(r) = 0.023 + 1.662/(1+1.071e^(-190.683r)) + 0.362/(1+1.071e^(-2200.640r))的硬化函数，且在r=0时无硬化行为。
• 在40Cr3MoV贝氏体钢数据上，模型生成了包含三个活跃项的复杂但可解释的硬化函数，实现了精确的插值与外推。
• 损失函数与活跃参数数量在训练过程中单调下降，表明收敛稳定且稀疏性诱导有效。
• 由于物理约束的强制实施，即使在数据有限的情况下，该方法仍能实现可靠的外推，超越训练应变范围。
• 最终的稀疏模型结构紧凑且可解析表达，可直接集成至现有有限元框架中。