Uncertainty quantification (UQ) in scientific machine learning (SciML) becomes increasingly critical as neural networks (NNs) are being widely adopted in addressing complex problems across various scientific disciplines. Representative SciML models are physics-informed neural networks (PINNs) and neural operators (NOs). While UQ in SciML has been increasingly investigated in recent years, very few works have focused on addressing the uncertainty caused by the noisy inputs, such as spatial–temporal coordinates in PINNs and input functions in NOs. The presence of noise in the inputs of the models can pose significantly more challenges compared to noise in the outputs of the models, primarily due to the inherent nonlinearity of most SciML algorithms. As a result, UQ for noisy inputs becomes a crucial factor for reliable and trustworthy deployment of these models in applications involving physical knowledge. To this end, we introduce a Bayesian approach to quantify uncertainty arising from noisy inputs–outputs in PINNs and NOs. We show that this approach can be seamlessly integrated into PINNs and NOs, when they are employed to encode the physical information. PINNs incorporate physics by including physics-informed terms via automatic differentiation, either in the loss function or the likelihood, and often take as input the spatial–temporal coordinate. Therefore, the present method equips PINNs with the capability to address problems where the observed coordinate is subject to noise. On the other hand, pretrained NOs are also commonly employed as equation-free surrogates in solving differential equations and Bayesian inverse problems, in which they take functions as inputs. The proposed approach enables them to handle noisy measurements for both input and output functions with UQ. We present a series of numerical examples to demonstrate the consequences of ignoring the noise in the inputs and the effectiveness of our approach in addressing noisy inputs–outputs with UQ when PINNs and pretrained NOs are employed for physics-informed learning.

中文翻译：

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物理信息神经网络和神经运算符中噪声输入-输出的不确定性量化


随着神经网络 （NN） 被广泛用于解决各个科学学科的复杂问题，科学机器学习 （SciML） 中的不确定性量化 （UQ） 变得越来越重要。具有代表性的 SciML 模型是物理信息神经网络 （PINN） 和神经运算符 （NO）。虽然近年来越来越多地研究 SciML 中的 UQ，但很少有工作专注于解决由噪声输入引起的不确定性，例如 PINN 中的时空坐标和 NO 中的输入函数。与模型输出中的噪声相比，模型输入中存在的噪声可能会带来更多的挑战，这主要是由于大多数 SciML 算法固有的非线性。因此，用于噪声输入的 UQ 成为在涉及物理知识的应用中可靠和可信地部署这些模型的关键因素。为此，我们引入了一种贝叶斯方法来量化 PINN 和 NO 中噪声输入-输出引起的不确定性。我们表明，当 PINN 和 NO 用于编码物理信息时，这种方法可以无缝集成到 PINN 和 NO 中。PINN 通过自动微分在损失函数或似然中包括物理学依据的术语来整合物理学，并且通常将时空坐标作为输入。因此，该方法使 PINN 能够解决观察到的坐标受噪声影响的问题。另一方面，预训练的 NO 也通常用作求解微分方程和贝叶斯逆问题时的无方程代理，其中它们将函数作为输入。 所提出的方法使他们能够使用 UQ 处理输入和输出函数的噪声测量。我们提供了一系列数值示例，以证明当 PINN 和预训练的 NO 用于物理知情学习时，忽略输入中噪声的后果以及我们的方法在使用 UQ 解决噪声输入-输出方面的有效性。