NSF-AoF:A Bayesian Paradigm for Physics-Informed Machine Learning

机器学习算法已被证明是现代工业和社会不可或缺的。然而，传统的机器学习从观测数据中提取信息，而忽略了编入自然科学定律的大量信息。这项研究涉及基于物理的机器学习，这是一个新兴领域，通过将科学定律直接编码到机器学习算法中，有望对科学和工程产生深远而持久的影响。这大大降低了这些算法对数据大小的要求，甚至允许它们外推到没有数据的领域。该项目将为物理信息机器学习开发贝叶斯范式，其中将包括具有量化不确定性的新概率方法、新的计算和分析方法以及新的无监督算法。研究结果将有利于石油工程、航空航天工程、材料科学和天文学等领域的应用。本研究将为物理信息神经网络（pinn）和物理信息高斯过程（PIGPs）开发贝叶斯范式。研究人员将开发非线性偏微分方程的概率解算器，利用最新的概率解算器方法结合PINN和PIGP模型来解决物理信息机器学习问题。为了研究贝叶斯神经网络的性能，将发展多部分损失函数神经网络的训练动态分析方法。研究人员将研究基于传统多重初始化、粒子群和变分推理的PINN集成的贝叶斯模型平均。此外，将开发结合PINN和PIGP算法的新的无监督方法，以改善整个物理领域传播信息的问题，这是物理信息机器学习算法的常见故障模式。这项工作将产生贝叶斯物理知识的机器学习工具，用于油藏模拟、计算流体动力学、微观结构信息学中的相场建模和超新星大气中的辐射传输等问题，以及由研究人员及其合作者在最近成立的德克萨斯a&&m数据科学研究所（TAMIDS）的科学机器学习实验室进行的其他多学科研究项目。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Machine learning algorithms have proved to be indispensable to modern industry and society. However, traditional machine learning extracts information from observational data, while ignoring the tremendous amount of information encoded into scientific laws of nature. This research concerns physics-informed machine learning, an emerging area that promises to have a profound and lasting impact in science and engineering, by coding scientific laws directly into machine learning algorithms. This dramatically reduces the data size requirement of these algorithms, and even allows them to extrapolate to domains where there is no data. This project will develop a Bayesian paradigm for physics-informed machine learning, which will include new probabilistic methods with quantified uncertainty, new computation and analysis methods, and new unsupervised algorithms. The results of this research will benefit applications in petroleum engineering, aerospace engineering, materials science, and astronomy being developed by the investigators and their collaborators. This research will develop a Bayesian paradigm for physics-informed neural networks (PINNs) and physics-informed Gaussian processes (PIGPs). The investigators will develop probabilistic solvers for nonlinear partial differential equations that leverage recent probabilistic solver methods in combination with PINN and PIGP models to solve physics-informed machine learning problems. Training dynamic analysis methods for neural networks with multi-part loss functions will be developed in order to investigate the performance of Bayesian PINNs. The investigators will study Bayesian model averaging for PINN ensembles based on traditional multiple initialization, particle swarms, and variational inference. In addition, new unsupervised methods to combine PINN and PIGP algorithms will be developed to ameliorate the issue of propagating information throughout the physical domain, which is a common failure mode of physics-informed machine learning algorithms. This work will result in Bayesian physics-informed machine learning tools for problems in oil reservoir simulation, computational fluid dynamics, phase-field modeling in microstructure informatics, and radiative transfer in supernova atmospheres, among other multidisciplinary research projects conducted by the investigators and their collaborators at the recently-established Scientific Machine Learning Laboratory of the Texas A&M Institute of Data Science (TAMIDS).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.