An Analysis of Physics-Informed Neural Networks
- URL: http://arxiv.org/abs/2303.02890v1
- Date: Mon, 6 Mar 2023 04:45:53 GMT
- Title: An Analysis of Physics-Informed Neural Networks
- Authors: Edward Small
- Abstract summary: We present a new approach to approximating the solution to physical systems - physics-informed neural networks.
The concept of artificial neural networks is introduced, the objective function is defined, and optimisation strategies are discussed.
The partial differential equation is then included as a constraint in the loss function for the problem, giving the network access to knowledge of the dynamics of the physical system it is modelling.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Whilst the partial differential equations that govern the dynamics of our
world have been studied in great depth for centuries, solving them for complex,
high-dimensional conditions and domains still presents an incredibly large
mathematical and computational challenge. Analytical methods can be cumbersome
to utilise, and numerical methods can lead to errors and inaccuracies. On top
of this, sometimes we lack the information or knowledge to pose the problem
well enough to apply these kinds of methods. Here, we present a new approach to
approximating the solution to physical systems - physics-informed neural
networks. The concept of artificial neural networks is introduced, the
objective function is defined, and optimisation strategies are discussed. The
partial differential equation is then included as a constraint in the loss
function for the optimisation problem, giving the network access to knowledge
of the dynamics of the physical system it is modelling. Some intuitive examples
are displayed, and more complex applications are considered to showcase the
power of physics informed neural networks, such as in seismic imaging. Solution
error is analysed, and suggestions are made to improve convergence and/or
solution precision. Problems and limitations are also touched upon in the
conclusions, as well as some thoughts as to where physics informed neural
networks are most useful, and where they could go next.
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