2022/02/24 by Da Long, Long, Da, Zheng Wang +11 · 5 citations
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Computer science #FOS: Computer and information sciences #Function (biology) #Gaussian #Gaussian Processes and Bayesian Inference #Gaussian function #Gaussian process #Generative Adversarial Networks and Image Synthesis #Kernel (algebra) #Kriging #Latent variable #Machine Learning (cs.LG) #Machine learning #Mathematical optimization #Mathematics #Model Reduction and Neural Networks #Nonlinear system #Physics #Stochastic differential equation #cs.LG
paper · pdf · doi:10.48550/arxiv.2202.12316
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2022/02/24 · openalex created_date 2022/04/03 · arxiv created 2022/07/20 · arxiv updated 2022/07/22 · openalex updated_date 2026/08/08
Physical modeling is critical for many modern science and engineering applications. From a data science or machine learning perspective, where more domain-agnostic, data-driven models are pervasive, physical knowledge -- often expressed as differential equations -- is valuable in that it is complementary to data, and it can potentially help overcome issues such as data sparsity, noise, and inaccuracy. In this work, we propose a simple, yet powerful and general framework -- AutoIP, for Automatically Incorporating Physics -- that can integrate all kinds of differential equations into Gaussian Processes (GPs) to enhance prediction accuracy and uncertainty quantification. These equations can be linear or nonlinear, spatial, temporal, or spatio-temporal, complete or incomplete with unknown source terms, and so on. Based on kernel differentiation, we construct a GP prior to sample the values of the target function, equation-related derivatives, and latent source functions, which are all jointly from a multivariate Gaussian distribution. The sampled values are fed to two likelihoods: one to fit the observations, and the other to conform to the equation. We use the whitening method to evade the strong dependency between the sampled function values and kernel parameters, and we develop a stochastic variational learning algorithm. AutoIP shows improvement upon vanilla GPs in both simulation and several real-world applications, even using rough, incomplete equations.