As　extension　of　traditional　echo　state　networks　(ESNs),　the　polynomial　echo　state　networks　(PESNs)　have　been　proposed　in　our　previous　work　(Yang　et　al.,　208)　by　employing　the　polynomial　function　of　complete　input　variable　as　output　weight　matrix.　In　practice,　the　generalization　performance　and　computational　burden　of　PESNs　are　perturbed　by　redundant　or　irrelevant　inputs.　To　construct　output　weights　with　a　suitable　subset　of　input　variables,　the　forward　selection　based　PESN　(FS-PESN)　and　backward　selection　based　PESN　(BS-PESN)　are　proposed.　Firstly,　the　forward　selection　method　is　used　in　FS-PESN　to　choose　the　input　variable　which　incurs　the　maximum　reduction　on　objective　function,　and　the　backward　selection　shame　is　introduced　in　BS-PESN　to　remove　the　input　variable　which　leads　to　the　smallest　increment　on　objective　function.　Then,　the　iterative　updating　strategies　are　designed　to　avoid　repetitive　computations　in　FS-PESN　and　BS-PESN.　Specially,　an　accelerating　scheme　is　introduced　into　BS-PESN　to　simplify　training　process.　Finally,　numerical　simulations　are　carried　out　to　illustrate　effectiveness　of　the　proposed　techniques　in　terms　of　generalization　ability　and　testing　time.　(C)　2020　Elsevier　B.V.　All　rights　reserved.