kovachki2024neural: Neural Operator: Learning Maps Between Function Spaces
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tags :
- source
- http://arxiv.org/abs/2108.08481
- authors
- Kovachki, N., Li, Z., Liu, B., Azizzadenesheli, K., Bhattacharya, K., Stuart, A., & Anandkumar, A.
- year
- 2024
Neural operators are designed to learn general solutions for maps between two function spaces and to be discretization-invariant. Current data-driven solutions using neural networks are not discretization invariant and often require new networks/datasets/training for different levels of discretization. Neural operators describe a process to estimate these maps with the following properties:
- acts on any discretization of the input function, i.e. accepts any set of points in the input domain,
- can be evaluated at any point of the output domain
- converges toa continuum operator as the discretization is refined. (Converging to a continuum operator means that as the discretization is refined, the function more closely estimates the true continuous function).
Generic Parametric PDEs
This paper considers the generic form of PDEs as
\begin{align} (L_a u)(x) &= f(x), &x \in D \\ u(x) &= 0, &x \in \partial D \end{align}
Where \(u(x)\) is the solution function, \(f(x)\) is the function we get sampled data from, \(L_a\) is the mapping from the banach space of \(u\) to its dual space.