Neural Networks vs Traditional Numerical Methods

In summary, the conversation discusses the performance of neural networks as 'universal approximators' for nonlinear functions in comparison to traditional numerical methods for solving nonlinear PDEs. The speaker mentions finding papers on applications of neural networks to Navier Stokes equations and other problems, but is unsure of their potential usefulness. They question whether neural networks can offer comparable accuracy with less computational intensity compared to current numerical methods for modeling NS equations. The speaker also suggests that this topic may be better suited for discussion in the Computer Science forum.
  • #1
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As neural networks are 'universal approximators' for nonlinear functions, in general how do they perform in comparison to traditional numerical methods for nonlinear PDEs? Just googling, I can find papers on applications to Navier Stokes and other problems, but I don't really have the background to judge how potentially useful they are. For example, can NNs perform better (i.e. comparable accuracy but less computationally intensive) than current numerical methods for modelling the NS equations?

(this may be better in the Computer Science forum)

https://en.wikipedia.org/wiki/Universal_approximation_theorem
 
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  • #2
The question is a little confusing to me. I am not an expert in neural networks. Neural networks does have the ability to fit a model through data points and in the examples I have seen they did a better job than other curve fitting or statistical regression algorithms. That agrees with how I would interpret the term "universal estimator". I'm not sure that finding the solution of a PDE is the same thing. But maybe I am missing something.

PS. It only took a quick Google search to find articles on solving PDEs using neural networks, so I will leave this to others with more expertise.
 

1. What is the difference between neural networks and traditional numerical methods?

Neural networks are a type of supervised learning algorithm that uses interconnected layers of nodes to learn patterns and make predictions. Traditional numerical methods, on the other hand, rely on mathematical equations and algorithms to solve problems.

2. Which approach is better for solving complex problems?

Both neural networks and traditional numerical methods have their own strengths and weaknesses. Neural networks are better suited for handling complex and nonlinear relationships, while traditional numerical methods are more efficient for solving well-defined problems with known equations.

3. Can neural networks replace traditional numerical methods?

It depends on the specific problem at hand. In some cases, neural networks may outperform traditional numerical methods, but in others, traditional methods may still be the preferred approach. Both techniques have their own advantages and applications, so it is unlikely that one will completely replace the other.

4. Are neural networks more accurate than traditional numerical methods?

In general, neural networks have the potential to be more accurate than traditional numerical methods because they can learn from data and adapt to complex patterns. However, the accuracy of a neural network depends on the quality and quantity of data it is trained on, and the specific problem being solved.

5. Which approach is more computationally intensive?

Neural networks tend to be more computationally intensive because they require large amounts of data and training time to learn and make accurate predictions. Traditional numerical methods rely on mathematical equations and algorithms, which can be solved more efficiently with less data and computational power.

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