The accessibility of spatially distributed data, enabled by affordable sensors, field, and numerical experiments, has facilitated the development of data-driven solutions for scientific problems, including climate change, weather prediction, and urban planning. Neural Partial Differential Equations (Neural PDEs), which combine deep learning (DL) techniques with domain expertise (e.g., governing equations) for parameterization, have proven to be effective in capturing valuable correlations within spatiotemporal datasets. However, sparse and noisy measurements coupled with modeling approximation introduce aleatoric and epistemic uncertainties. Therefore, quantifying uncertainties propagated from model inputs to outputs remains a challenge and an essential goal for establishing the trustworthiness of Neural PDEs. This work evaluates various Uncertainty Quantification (UQ) approaches for both Forward and Inverse Problems in scientific applications. Specifically, we investigate the effectiveness of Bayesian methods, such as Hamiltonian Monte Carlo (HMC) and Monte-Carlo Dropout (MCD), and a more conventional approach, Deep Ensembles (DE). To illustrate their performance, we take two canonical PDEs: Burger's equation and the Navier-Stokes equation. Our results indicate that Neural PDEs can effectively reconstruct flow systems and predict the associated unknown parameters. However, it is noteworthy that the results derived from Bayesian methods, based on our observations, tend to display a higher degree of certainty in their predictions as compared to those obtained using the DE. This elevated certainty in predictions suggests that Bayesian techniques might underestimate the true underlying uncertainty, thereby appearing more confident in their predictions than the DE approach.

翻译：由廉价传感器、实地实验及数值实验所支撑的空间分布数据可获取性，推动了气候变化、天气预报及城市规划等科学问题数据驱动求解方案的发展。神经偏微分方程（Neural PDEs）通过将深度学习技术与领域专业知识（如控制方程）相结合进行参数化，已被证明能有效捕捉时空数据中的有价值关联。然而，稀疏且含噪声的测量数据与建模近似共同引入了偶然不确定性和认知不确定性。因此，量化从模型输入到输出所传递的不确定性仍是建立神经偏微分方程可信度的挑战与核心目标。本研究系统评估了面向科学应用中正问题与逆问题的多种不确定性量化（UQ）方法，具体探究了贝叶斯方法（如哈密顿蒙特卡洛HMC与蒙特卡洛丢弃法MCD）及传统集成学习（DE）的有效性。为展示性能差异，我们选取Burger方程与Navier-Stokes方程两类经典偏微分方程进行验证。结果表明，神经偏微分方程能有效重构流动系统并预测相关未知参数。值得注意的是，基于观测数据的贝叶斯方法所得结果相较于DE方法，其预测结果呈现更高的置信度。这种预测置信度的提升暗示贝叶斯方法可能低估了真实潜在不确定性，从而表现出比DE方法更强的预测可靠性假象。