Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equations
arxiv(2022)
摘要
Partial differential equations (PDEs) are widely used for the description of
physical and engineering phenomena. Some key parameters involved in PDEs, which
represent certain physical properties with important scientific
interpretations, are difficult or even impossible to measure directly.
Estimating these parameters from noisy and sparse experimental data of related
physical quantities is an important task. Many methods for PDE parameter
inference involve a large number of evaluations for numerical solutions to PDE
through algorithms such as the finite element method, which can be
time-consuming, especially for nonlinear PDEs. In this paper, we propose a
novel method for the inference of unknown parameters in PDEs, called the
PDE-Informed Gaussian Process (PIGP) based parameter inference method. Through
modeling the PDE solution as a Gaussian process (GP), we derive the manifold
constraints induced by the (linear) PDE structure such that, under the
constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an
augmentation method that transforms the nonlinear PDE into an equivalent PDE
system linear in all derivatives, which our PIGP-based method can handle. The
proposed method can be applied to a broad spectrum of nonlinear PDEs. The
PIGP-based method can be applied to multi-dimensional PDE systems and PDE
systems with unobserved components. Like conventional Bayesian approaches, the
method can provide uncertainty quantification for both the unknown parameters
and the PDE solution. The PIGP-based method also completely bypasses the
numerical solver for PDEs. The proposed method is demonstrated through several
application examples from different areas.
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