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Kriging (or Gaussian process regression) is a popular machine learning method for its flexibility and closed-form prediction expressions. However, one of the key challenges in applying kriging to engineering systems is that the available measurement data is scarce due to the measurement limitations and high sensing costs. On the other hand, physical knowledge of the engineering system is often available and represented in the form of partial differential equations (PDEs). We present in this work a PDE Informed Kriging model (PIK), which introduces PDE information via a set of PDE points and conducts posterior prediction similar to the standard kriging method. The proposed PIK model can incorporate physical knowledge from both linear and nonlinear PDEs. To further improve learning performance, we propose an Active PIK framework (APIK) that designs PDE points to leverage the PDE information based on the PIK model and measurement data. The selected PDE points not only explore the whole input space but also exploit the locations where the PDE information is critical in reducing predictive uncertainty. Finally, an expectation-maximization algorithm is developed for parameter estimation. We demonstrate the effectiveness of APIK in two synthetic examples, a shock wave case study, and a laser heating case study.
Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the deep learn
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high nonlinearit
Most simulation schemes for partial differential equations (PDEs) focus on minimizing a simple error norm of a discretized version of a field. This paper takes a fundamentally different approach; the discretized field is interpreted as data providing
Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system discovery that has been shown to successfully recover governing dynamical systems from data (Brunton et al., PNAS, 16; Rudy et al., Sci. Adv. 17). Recently, several groups have
Deep learning-based surrogate modeling is becoming a promising approach for learning and simulating dynamical systems. Deep-learning methods, however, find very challenging learning stiff dynamics. In this paper, we develop DAE-PINN, the first effect