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On universal approximation and error bounds for Fourier Neural Operators
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On universal approximation and error bounds for Fourier Neural Operators
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Fourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinite-dimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any continuous operator to desired accuracy. Moreover, we suggest a mechanism by which FNOs can approximate operators associated with PDEs efficiently. Explicit error bounds are derived to show that the size of the FNO, approximating operators associated with a Darcy type elliptic PDE and with the incompressible Navier-Stokes equations of fluid dynamics, only increases sub (log)-linearly in terms of the reciprocal of the error. Thus, FNOs are shown to efficiently approximate operators arising in a large class of PDEs.
Forward citations
Cited by 1 Pith paper
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Stability and Discretization Error of State Space Model Neural Operators
Derives discretization error bounds and input-to-state stability guarantees for SS-NOs and FNOs, with empirical validation on 1D and 2D PDE benchmarks.
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