Understanding uncertainty is critical, especially when data are sparse and variations are large. Bayesian neural networks offer a powerful strategy to build predictable models from sparse data, and inherently quantify both, aleatoric uncertainties of the data and epistemic uncertainties of the model. Yet, classical Bayesian neural networks ignore the fundamental laws of physics, they are non-interpretable, and their parameters have no physical meaning. Here we integrate concepts of Bayesian learning and constitutive neural networks to discover interpretable models, parameters, and uncertainties that best explain soft matter systems. Instead of training an individual constitutive neural network and learning point values of the network weights, we train an ensemble of networks and learn probability distributions of the weights, along with their means, standard deviations, and credible intervals. We use variational Bayesian inference and adopt an efficient backpropagation-compatible algorithm that approximates the true probability distributions by simpler distributions and minimizes their divergence through variational learning. When trained on synthetic data, our Bayesian constitutive neural network successfully rediscovers the initial model, even in the presence of noise, and robustly discovers uncertainties, even from incomplete data. When trained on real data from healthy and aneurysmal human arteries, our network discovers a model with more stretch stiffening, more anisotropy, and more uncertainty for diseased than for healthy arteries. Our results demonstrate that Bayesian constitutive neural networks can successfully discriminate between healthy and diseased arteries, robustly discover interpretable models and parameters for both, and efficiently quantify uncertainties in model discovery. We anticipate our approach to generalize to other soft biomedical systems for which real-world data are rare and inter-personal variations are large. Ultimately, our calculated uncertainties will help enhance model robustness, promote personalized predictions, enable informed decision-making, and build confidence in automated model discovery and simulation. Our source code, data, and examples are available at https://github.com/LivingMatterLab/CANN.

中文翻译：

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发现不确定性：贝叶斯本构人工神经网络


了解不确定性至关重要，尤其是在数据稀疏且变化较大的情况下。贝叶斯神经网络提供了一种强大的策略，可以从稀疏数据中构建可预测的模型，并在本质上量化数据的随机不确定性和模型的认识不确定性。然而，经典贝叶斯神经网络忽略了物理学的基本定律，它们是不可解释的，并且它们的参数没有物理意义。在这里，我们整合了贝叶斯学习和本构神经网络的概念，以发现最能解释软物质系统的可解释模型、参数和不确定性。我们不是训练单个本构神经网络并学习网络权重的点值，而是训练一组网络并学习权重的概率分布，以及它们的平均值、标准差和可信区间。我们使用变分贝叶斯推理，并采用一种高效的反向传播兼容算法，该算法通过更简单的分布来近似真实概率分布，并通过变分学习最小化它们的发散。当在合成数据上进行训练时，我们的贝叶斯本构神经网络成功地重新发现了初始模型，即使在存在噪声的情况下，也能稳健地发现不确定性，即使来自不完整的数据也是如此。当使用来自健康动脉瘤和动脉瘤性人类动脉的真实数据进行训练时，我们的网络发现了一个模型，与健康动脉相比，患病动脉具有更多的拉伸刚度、更多的各向异性和更多的不确定性。 我们的结果表明，贝叶斯本构神经网络可以成功区分健康动脉和患病动脉，稳健地发现两者的可解释模型和参数，并有效地量化模型发现中的不确定性。我们预计我们的方法将推广到其他软生物医学系统，这些系统的真实世界数据很少且人际差异很大。最终，我们计算的不确定性将有助于增强模型的稳健性，促进个性化预测，实现明智的决策，并建立对自动模型发现和仿真的信心。我们的源代码、数据和示例可在 https://github.com/LivingMatterLab/CANN 上获得。