Physics-informed　neural　networks　(PINNs)　based　machine　learning　is　an　emerging　framework　for　solving　nonlinear　differential　equations.　However,　due　to　the　implicit　regularity　of　neural　network　structure,　PINNs　can　only　find　the　flattest　solution　in　most　cases　by　minimizing　the　loss　functions.　In　this　paper,　we　combine　PINNs　with　the　homotopy　continuation　method,　a　classical　numerical　method　to　compute　isolated　roots　of　polynomial　systems,　and　propose　a　new　deep　learning　framework,　named　homotopy　physics-informed　neural　networks　(HomPINNs),　for　solving　multiple　solutions　of　nonlinear　elliptic　differential　equations.　The　implementation　of　an　HomPINN　is　a　homotopy　process　that　is　composed　of　the　training　of　a　fully　connected　neural　network,　named　the　starting　neural　network,　and　training　processes　of　several　PINNs　with　different　tracking　parameters.　The　starting　neural　network　is　to　approximate　a　starting　function　constructed　by　the　trivial　solutions,　while　other　PINNs　are　to　minimize　the　loss　functions　defined　by　boundary　condition　and　homotopy　functions,　varying　with　different　tracking　parameters.　These　training　processes　are　regraded　as　different　steps　of　a　homotopy　process,　and　a　PINN　is　initialized　by　the　well-trained　neural　network　of　the　previous　step,　while　the　first　starting　neural　network　is　initialized　using　the　default　initialization　method.　Several　numerical　examples　are　presented　to　show　the　efficiency　of　our　proposed　HomPINNs,　including　reaction-diffusion　equations　with　a　heart-shaped　domain.　©　2022　Elsevier　Ltd