Thermal　deformation　is　the　main　factor　affecting　the　machining　accuracy　of　grinding　machines，which　severely　limits　further　improvements　in　the　accuracy　of　machine　tools.　However，only　a　few　studies　have　investigated　thermal　error　prediction，and　the　prediction　accuracy　has　been　low.　Therefore，this　pa⁃　per　proposes　a　method　to　predict　the　thermal　error　of　grinding　machine　spindles　based　on　the　heat　conduc⁃　tion　theory　and　a　convolutional　neural　network.　First，according　to　the　heat　conduction　theory，the　map⁃　ping　relationship　between　the　thermal　variables　and　the　temperature　difference　between　the　surface　of　the　main　axis　of　Chu　and　the　external　environment　is　deduced，revealing　the　thermal　deformation　nature　of　the　materials.　Second，a　neural　network　model　for　thermal　error　prediction　with　temperature　difference　as　the　input　and　thermal　deformation　of　the　main　shaft　as　the　output　is　established.　The　model　has　four　neural　net⁃　work　layers　corresponding　to　temperature　difference，thermal　energy　increment，time　variable，and　ther-mal　deformation.　The　back-propagation　algorithm　is　then　used　to　train　the　prediction　model　and　calculate　the　model　parameters.　Finally，based　on　the　SINUMERIK　840D　CNC　controller，a　set　of　thermal　error　compensation　systems　for　grinding　machine　spindles　are　developed　and　verified　using　a　CNC　grinding　ma⁃　chine.　The　results　show　that　the　machining　accuracy　of　the　grinder　is　improved　by　41.　7%　following　ther⁃　mal　error　compensation　for　the　spindle，thus　confirming　the　validity　and　feasibility　of　the　spindle　thermal　er⁃　ror　prediction　model　proposed　in　this　paper.　©　2023　Chinese　Academy　of　Sciences.　All　rights　reserved.