The　fixed/preassigned　time　(FT/PT)　synchronization　issue　of　multi-layer　networks　via　a　discontinuous　control　strategy　is　addressed　in　this　paper.　Firstly,　the　inventiveness　FT　theorem　is　established,　and　a　precise　estimation　of　setting　time　is　given　by　employing　mathematical　induction,　proof　by　contradiction,　and　maximum　value　theory.　Subsequently,　based　on　the　FT　stability　results,　several　FT　synchronization　criteria　for　multi-layer　networks　are　proposed　by　designing　a　periodically　switching　protocol.　Additionally,　the　topological　connectivity　constraints　are　eliminated　through　the　implementation　of　synchronized　states　as　virtual　nodes.　Correspondingly,　the　PT　synchronization　issue　is　investigated　by　developing　several　extraordinary　control　strategies　in　which　the　control　gains　are　finite.　Ultimately,　the　feasibility　of　the　developed　control　design　and　the　established　criteria　is　illustrated　by　two　numerical　examples.　©　2024　Elsevier　Ltd