On universal approximation and error bounds for Fourier Neural Operators

By Nikola B. Kovachki, Samuel Lanthaler, Siddhartha Mishra. Journal of machine learning research, 2021

383 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Geophysics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d15a11a7be64ccc52b709b44b9fac1e4d6302062

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Anima Anandkumar
2. G. Em Karniadakis
3. Zong-Yi Li
4. Kamyar Azizzadenesheli
5. Nikola B. Kovachki
6. Lu Lu
7. Xuhui Meng
8. Burigede Liu

In Geophysics

1. G. Em Karniadakis
2. Andrew M. Stuart

In Statistical and Nonlinear Physics

1. Anima Anandkumar
2. Zong-Yi Li
3. G. Em Karniadakis
4. Lu Lu
5. Nikola B. Kovachki
6. Zhiping Mao
7. Pengzhan Jin
8. Shuai Meng

Citations in the corpus

• Factorized Fourier Neural Operators (2021, background): “More generally, Kovachki et al. (2021a) prove that the FNO can approximate any continuous operator.”
• Neural Operator: Learning Maps Between Function Spaces (2021, background): “We note that the theory of Section 4 is designed for general kernels and does not apply to the FNO formulation; however, similar universal approximation results were developed for it in (Kovachki et al., 2021) when the input and output spaces are Hilbert space.”
• A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data (2021, methodology): “In addition to computational tests, we also perform a theoretical comparison of DeepONet versus FNO, following the published work on the theory of DeepONet in [2, 24, 25, 26], and on the more recent theory of FNO in [4, 27].”
• MIONet: Learning multiple-input operators via tensor product (2022, background): “The current theoretical analysis of neural operators focuses on the approximation capability such as the universal approximation theorems for DeepONet [3, 27] and FNO [17].”
• Fourier Neural Operator with Learned Deformations for PDEs on General Geometries (2022, background): “We also leave this direction as future work.”