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About this paper
Samuel Lanthaler, Andrew M. Stuart
This work by Lanthaler and Stuart at ETH Zürich and Caltech formally establishes an "infinite-dimensional curse of parametric complexity" for general neural operators, demonstrating that achieving high accuracy requires an exponential number of parameters. The paper then presents HJ-Net, an architecture that leverages the method of characteristics to learn the solution operator for Hamilton-Jacobi equations with provable algebraic complexity, significantly improving efficiency for this specific problem class.
Key insights
- Generic neural operators suffer from an "infinite-dimensional curse of parametric complexity," meaning the number of parameters required to achieve a desired accuracy scales exponentially with the inverse of the error.
- Overcoming this curse for specific operator classes necessitates the design of "structure-aware" neural operators that incorporate inherent mathematical properties of the problem.
- By decomposing the operator learning task into learning a finite-dimensional flow map and using classical reconstruction methods, algebraic complexity can be achieved for certain PDEs like the Hamilton-Jacobi equation.
Summary by alphaXiv.
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