Levenberg-Marquardt　(LM)　algorithm　is　commonly　used　for　training　radial　basis　function　neural　networks　(RBFNN)　due　to　its　fast　convergence.　However,　its　successful　application　is　limited　by　the　vanishing　gradient　problem　in　the　learning　process.　To　solve　this　problem,　an　accelerated　LM　(ALM)　algorithm　is　designed　to　improve　the　learning　performance　of　RBFNN　in　this　paper.　First,　an　error　information　filtering　mechanism　(EIFM)　is　developed　to　discard　training　samples　information　that　the　RBFNN　has　already　learned,　then　the　same　amount　of　computation　can　obtain　the　more　error　gradient　information.　Second,　a　width　adjustment　mechanism　(WAM)　is　designed　to　help　training　samples　escape　from　the　saturated　region　of　the　activation　function　to　increment　gradient　value.　Finally,　some　benchmark　and　real　problems　are　used　to　exhibition　the　effectiveness　of　proposed　ALM　algorithm.　The　results　demonstrate　the　proposed　ALM-RBFNN　can　effectively　mitigate　the　vanishing　gradient　and　has　good　learning　performance.