It's actually not straight forward but possible if you just serialize the classifier object and load it again in the future.
var classifier_serialized = ee.Serializer.toJSON(classifier)
Export.table.toAsset(ee.FeatureCollection(empty_feature.set('classifier',json)),desc,AssetName)
// Load using this
var json = ee.String(ee.Feature(ee.FeatureCollection(assetName).first()).get('classifier'))
var classifier = ee.Deserializer.fromJSON(json)
"empty_feature" is just ee.Feature, empty as the name specified.
Maybe in the future, there will be some APIs to simplify this :)
Edited:
If the first approach is not saving learned weights try another approach only works for Random Forest.
var trees = ee.List(ee.Dictionary(classifier.explain()).get('trees'))
var dummy = ee.Feature()
var col = ee.FeatureCollection(trees.map(function(x){return dummy.set('tree',x)}))
Export.table.toAsset(col,'save_classifier',AssetName)
// Load classifier
var trees = ee.FeatureCollection(AssetName).aggregate_array('tree').aside(print)
var classifier = ee.Classifier.decisionTreeEnsemble(trees)