Data-driven discovery of PDEs in complex datasets

By Jens Berg, Kaj Nyström. 2018

158 citations recorded by the source corpus.

Abstract Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities of interest. A different approach is to measure the quantities of interest and use deep learning to reverse engineer the PDEs which are describing the physical process. In this paper we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. We include examples of data from a known model problem, and real data from weather station measurements. We show how necessary transformations of the input data amounts to coordinate transformations in the discovered PDE, and we elaborate on feature and model selection. It is shown that the dynamics of a non-linear, second order PDE can be accurately described by an ordinary differential equation which is automatically discovered by our deep learning algorithm. Even more interestingly, we show that similar results apply in the context of more complex simulations of the Swedish temperature distribution.

Subfields: Artificial Intelligence, Atmospheric Science, Statistical and Nonlinear Physics

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Citations in the corpus

A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate (2021, background): “Berg and Nystr ¨o m [46] proposed a uniﬁed deep neural network approach to approximate solutions to PDEs and then used deep learning to discover PDEs hidden in complex data sets from measurement data [47].”
PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2020, methodology): “This idea has also been extended to utilize multi-fidelity datasets [34] and solve system identification problems [35–37].”
Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks (2019, background): “These types of approaches are emerging with increasing interest [54, 55, 56, 57, 58], and it is easy to see how such approaches naturally lend themselves to the DNN frameworks.”
Adaptive activation functions accelerate convergence in deep and physics-informed neural networks (2019, methodology): “proposed a sparse regression which is based on library of candidate terms and sparse model selection to select the important terms involved in the governing equation and recently by Berg and Nystr¨om [7]. One of the ecient machine learning based approach for solving an inverse problem given by Raiss”
B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data (2020, background): “The state-of-the-art in data-driven modeling has advanced signiﬁcantly recently in applications across diﬀerent ﬁelds [1, 2, 3, 4, 5], due to the rapid development of machine learning and explosive growth of available data collected from diﬀerent sensors (e.g., satellites, cameras, etc.).”
Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks (2020, background): “In the past few years, there has been intensive research on understanding how deep neural networks can be adopted to solve and discover differential equations [1–3,7,8,15, 17, 20–22, 24, 28].”