The　article　presents,　a　reliable　numerical　framework　supported　by　feed-forward　Artificial　Neural　Network　(ANN)　optimized　with　hybrid　swarm　intelligence　technique　for　non-linear　system　of　Flierl–Petviashivili　(FP)　problem.　The　universal　approximation　capabilities　of　ANN　are　exploited　for　mathematical　approximation　of　the　system　in　an　unsupervised　way　based　upon　various　performance　metrics　like　fitness　value,　absolute　error　and　execution　time.　The　optimization　of　the　cost　function　is　subject　to　finding　the　appropriate　weights　which　are　highly　stochastic　in　nature　for　the　problem　as　well　as　its　initial　and　boundary　conditions.　Therefore,　hybrid　approach　based　on　Particle　Swarm　Optimization　(PSO)　and　Interior　Point　Algorithm　(IPA)　is　exploited　for　tuning　of　the　adaptive　weights　in　such　a　way　that　exploration　is　performed　by　PSO　while　the　exploitation　is　done　using　IPA　algorithm.　The　designed　scheme　is　evaluated　for　standard　FP　problem　along　with　its　variants　supported　on　various　scenarios.　The　reliability,　accuracy　and　robustness　of　the　solvers　are　validated　through　a　statistical　analysis　applied　on　two　hundred　independent　runs.　©　2020,　Springer　Nature　Switzerland　AG.