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Anima Anandkumar highlights that neural operators, despite simple benchmarks, have achieved massive speedups (10,000–million times) in hard real-world problems like high-resolution AI weather modeling (FourCastNet) and nuclear fusion turbulence, referencing a new paper showing learned solvers become more cost-effective as PDE tasks get harder.
This paper establishes quantitative Sobolev approximation bounds for neural operators, proving that operators can be uniformly approximated with explicit complexity-error relations. It validates these theoretical bounds using Fourier Neural Operators on the Burgers' equation, demonstrating that Sobolev-space approximation theory accurately predicts scaling behavior.