AutoIP: A United Framework to Integrate Physics into Gaussian Processes
This work addresses the challenge of incorporating physical knowledge into data-driven models for science and engineering applications, though it is incremental as it builds on existing kernel methods.
The authors tackled the problem of integrating diverse differential equations into Gaussian processes to improve prediction accuracy and uncertainty quantification, achieving improvements over vanilla GPs in simulations and real-world applications, even with rough or incomplete equations.
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.