Deep learning algorithms for solving differential equations: a survey

ABSTRACT

Differential equations (DEs) are widely employed in the mathematical modelling of a wide range of scientific and engineering problems. The analytical solution of these DEs is typically unknown for a variety of practical problems of relevance. Several numerical methods have been developed over time to find the solution to such DEs and numerous new approaches are still being proposed daily. In recent years, deep learning has emerged as a promising method for solving high-dimensional DEs. Due to the universal approximation capability of neural networks, there is no doubt that studies in this field will continue to grow in the near future. However, there is a need to understand the best-performing neural network architectures and algorithms that demonstrated their effectiveness and ability over traditional algorithms for solving various types of high-dimensional DEs. In this survey, we provide a review of deep learning algorithms classified as artificial neural networks (ANNs) and deep neural networks (DNNs) for solutions of DEs, that have been published in the last decade (between 2011 and 2022). The key purpose of this study is to explore the research papers published in the area of numerical solutions of DEs in order to get a deeper understanding of the current situation.

KEYWORDS:

• Artificial neural networks
• deep neural networks
• differential equations
• Gradient descent
• adam optimiser

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Disclosure statement

No potential conflict of interest was reported by the author(s).