Caltech Open-Sources FNO: A Deep Learning Method For Solving PDEs (Partial differential equations)

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Caltech’s Dolcit group recently open-sourced FNO, Fourier Neural Operator, a deep-learning method for Solving the PDEs (Partial differential equations). FNO being three times faster than traditional solvers outperforms the existing deep-learning techniques for solving PDEs. FNO is used to speed up the calculations and weather predictions.

The model and experiments were described in a paper published on arXiv. For the family of PDEs that are invariant of mesh resolution, the neural network can represent solutions By learning a mapping from one function to another. 

By applying a Fourier transformation, the model efficiently calculates a global convolution. The error rates achieved by FNO are (comparatively) 30% lower on Navier-Stokes equations and 60% lower on the Darcy flow. 

PDEs find use in many engineering and physics areas as they describe a wide variety of phenomena, including fluid dynamics, heat transfer, and quantum mechanics. The solution of the PDE is a function, often of time and space. However, when we talk practically, there is no closed-form PDE solution, which forces the scientists and engineers to resort to numerical approximations, using finite difference methods (FDM)or finite element methods (FEM). A fine-grained mesh of discrete points of interest is created in the above Techniques, and the PDE’s behavior in a small neighborhood is analyzed around each mesh point for a short span of time. 

These methods have some disadvantages, which are as follow:

  1. The process is time-consuming.
  2. The process must be rerun if any changes are made to the parameters of the problem or the grid definition.

Deep-learning and Neural networks have shown excellent results in speeding up scientific simulations. The PDEs are often solved by producing a model that can quickly generate sample data for statistical analysis. The above is applied in the inverse problem where the final observations are given, and we try to determine a system’s initial conditions. The two previous deep-learning approaches are:

  1. Finite-dimensional operators: The models are not mesh-independent. They use convolutional neural networks (CNN) and produce a parameterized approximation of a solution.
  2. Neural FEM: They are mesh-independent but represent the solution for only a specific instance of a PDE. FEM has to be trained again in case the parameters are changed. 
https://arxiv.org/pdf/2010.08895.pdf

The team’s approach is to build a neural network that can learn the mapping between a PDE and its solution. The FNO work is built on the previous graph kernel network (GKN) paper presented by the team at the recent NeurIPS conference.

https://zongyi-li.github.io/blog/2020/fourier-pde/

Paper: https://arxiv.org/abs/2010.08895

Github: https://github.com/zongyi-li/fourier_neural_operator

Saksham Goyal
Consultant Intern: He is Currently pursuing his Third year of B.Tech in Mechanical field from Indian Institute of Technology(IIT), Goa. He is motivated by his vision to bring remarkable changes in the society by his knowledge and experience. Being a ML enthusiast with keen interest in Robotics, he tries to be up to date with the latest advancements in Artificial Intelligence and deep learning.

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