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We develop an improved variant of $U(1)$-symmetric infinite projectedentangled-pair state (iPEPS) ansatz to investigate the ground state phasediagram of the spin-$1/2$ square $J_{1}-J_{2}$ Heisenberg model. In order toimprove the accuracy of the ansatz, we discuss a simple strategy to selectautomatically relevant symmetric sectors and also introduce an optimizationmethod to treat second-neighbor interactions more efficiently. We show thatvariational ground-state energies of the model obtained by the $U(1)$-symmetriciPEPS ansatz (for a fixed bond dimension $D$) set a better upper bound,improving previous tensor-network-based results. By studying the finite-$D$scaling of the magnetically order parameter, we find a N\'{e}el phase for$J_2/J_1<0.53$. For $0.53<J_2/J_1<0.61$, a non-magnetic columnar valence bondsolid (VBS) state is established as observed by the pattern of local bondenergy. The divergent behavior of correlation length $\xi \sim D^{1.2}$ andvanishing order parameters are consistent with a deconfined N\'{e}el-to-VBStransition at $J^{c_1}_{2}/J_1=0.530(5)$, where estimated critical anomalousexponents are $\eta_{s} \sim 0.6$ and $\eta_{d} \sim 1.9$ for spin and dimercorrelations respectively. We show that the associated VBS order parametermonotonically increases with $J_2/J_1$ and finally a first-order quantum phasetransition takes place at $J^{c_2}_{2}/J_1=0.610(2)$ to the conventional Stripephase. We compare our results with earlier DMRG and PEPS studies and suggestfuture directions for resolving remaining issues.

$U(1)$-symmetric infinite projected entangled-pair state study of the spin-$1/2$ square $J_{1}-J_{2}$ Heisenberg model