Gaussian Process Assisted Active Learning of Physical Laws
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In many areas of science and engineering, discovering the governing differential equations from the noisy experimental data is an essential challenge. It is also a critical step in understanding the physical phenomena and prediction of the future behaviors of the systems. However, in many cases, it is expensive or time-consuming to collect experimental data. This article provides an active learning approach to estimate the unknown differential equations accurately with reduced experimental data size. We propose an adaptive design criterion combining the D-optimality and the maximin space-filling criterion. The D-optimality involves the unknown solution of the differential equations and derivatives of the solution. Gaussian process models are estimated using the available experimental data and used as surrogates of these unknown solution functions. The derivatives of the estimated Gaussian process models are derived and used to substitute the derivatives of the solution. Variable-selection-based regression methods are used to learn the differential equations from the experimental data. The proposed active learning approach is entirely data-driven and requires no tuning parameters. Through three case studies, we demonstrate the proposed approach outperforms the standard randomized design in terms of model accuracy and data economy.
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