Gradient-based　algorithms　are　commonly　used　for　training　radial　basis　function　neural　network　(RBFNN).　However,　it　is　still　difficult　to　avoid　vanishing　gradient　to　improve　the　learning　performance　in　the　training　process.　For　this　reason,　in　this　paper,　an　accelerated　second-order　learning　(ASOL)　algorithm　is　developed　to　train　RBFNN.　First,　an　adaptive　expansion　and　pruning　mechanism　(AEPM)　of　gradient　space,　based　on　the　integrity　and　orthogonality　of　hidden　neurons,　is　designed.　Then,　the　effective　gradient　information　is　constantly　added　to　gradient　space　and　the　redundant　gradient　information　is　eliminated　from　gradient　space.　Second,　with　AEPM,　the　neurons　are　generated　or　pruned　accordingly.　In　this　way,　a　self-organizing　RBFNN　(SORBFNN)　which　reduces　the　structure　complexity　and　improves　the　generalization　ability　is　obtained.　Then,　the　structure　and　parameters　in　the　learning　process　can　be　optimized　by　the　proposed　ASOL-based　SORBFNN　(ASOL-SORBFNN).　Third,　some　theoretical　analyses　including　the　efficiency　of　the　proposed　AEPM　on　avoiding　the　vanishing　gradient　and　the　stability　of　SORBFNN　in　the　process　of　structural　adjustment　are　given,　then　the　successful　application　of　the　proposed　ASOL-SORBFNN　is　guaranteed.　Finally,　to　illustrate　the　advantages　of　the　proposed　ASOL-SORBFNN,　several　experimental　studies　are　examined.　By　comparing　with　other　existing　approaches,　the　results　show　that　ASOL-SORBFNN　performs　well　in　terms　of　both　learning　speed　and　prediction　accuracy.　©　2021　Elsevier　B.V.