When characterizing materials, it can be important to not only predict their mechanical properties, but also to estimate the probability distribution of these properties across a set of samples. Constitutive neural networks allow for the automated discovery of constitutive models that exactly satisfy physical laws given experimental testing data, but are only capable of predicting the mean stress response. Stochastic methods treat each weight as a random variable and are capable of learning their probability distributions. Bayesian constitutive neural networks combine both methods, but their weights lack physical interpretability and we must sample each weight from a probability distribution to train or evaluate the model. Here we introduce a more interpretable network with fewer parameters, simpler training, and the potential to discover correlated weights: Gaussian constitutive neural networks. We demonstrate the performance of our new Gaussian network on biaxial testing data, and discover a sparse and interpretable four-term model with correlated weights. Importantly, the discovered distributions of material parameters across a set of samples can serve as priors to discover better constitutive models for new samples with limited data. We anticipate that Gaussian constitutive neural networks are a natural first step towards generative constitutive models informed by physical laws and parameter uncertainty.

翻译：在材料表征中，不仅预测其力学性能，还需估计这些性能在样本集上的概率分布至关重要。本构神经网络能够基于实验测试数据自动发现严格满足物理定律的本构模型，但仅能预测平均应力响应。随机方法将每个权重视为随机变量，能够学习其概率分布。贝叶斯本构神经网络结合了两种方法，但其权重缺乏物理解释性，且训练或评估模型时必须从概率分布中对每个权重进行采样。本文提出一种更具解释性、参数更少、训练更简单且能发现相关权重的网络：高斯本构神经网络。我们在双轴测试数据上验证了新高斯网络的性能，并发现了一个具有相关权重的稀疏且可解释的四项模型。重要的是，所发现的材料参数在样本集上的分布可作为先验知识，为数据有限的新样本发现更优的本构模型。我们预期高斯本构神经网络是迈向基于物理定律和参数不确定性的生成式本构模型的自然第一步。