Communications in Mathematical Sciences

Volume 20 (2022)

Number 5

Universal approximation of symmetric and anti-symmetric functions

Pages: 1397 – 1408



Jiequn Han (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.; and Center for Computational Mathematics, Flatiron Institute, New York, N.Y., USA)

Yingzhou Li (School of Mathematical Sciences, Fudan University, Shanghai, China)

Lin Lin (Department of Mathematics, University of California, Berkeley, Cal., U.S.A.; and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A.)

Jianfeng Lu (Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, North Carolina, U.S.A.)

Jiefu Zhang (Department of Mathematics, University of California, Berkeley, Cal., U.S.A.)

Linfeng Zhang (Program in Applied and Computational Mathematics, Princeton University, Princeton, NJew Jersey, U.S.A.)


We consider universal approximations of symmetric and anti-symmetric functions, which are important for applications in quantum physics, as well as other scientific and engineering computations. We give constructive approximations with explicit bounds on the number of parameters with respect to the dimension and the target accuracy $\epsilon$. While the approximation still suffers from the curse of dimensionality, to the best of our knowledge, these are the first results in the literature with explicit error bounds for functions with symmetry or anti-symmetry constraints.


universal approximation, symmetric function, anti-symmetric function, neural network, Vandermonde determinant, quantum many-body problem

2010 Mathematics Subject Classification

41A25, 41A29, 41A63

The work of Lin Lin and Jiefu Zhang was supported in part by the Department of Energy under grants DE-SC0017867 and DE-AC02-05CH11231.

The work of Yingzhou Li and Jianfeng Lu was also supported in part by the National Science Foundation via grants DMS-1454939 and ACI-1450280.

Received 3 August 2021

Received revised 2 December 2021

Accepted 2 December 2021

Published 26 May 2022