Replacing "begin" by "commence"
Solution 1:
Upon comparing some of the example sentences included in websters-dictionary-online.net definitions for begin and commence, I conclude that one usually can replace forms of the verb "to begin" by a comparable form of "to commence", but in many cases would not do so. Both verbs can be used in spatial or evaluative as well as temporal senses, but I think a form of "to begin" is more likely to be used than a form of "to commence" when sense is spatial or evaluative. For example:
The number one begins the sequence.
A terrible murder begins the novel.
The convocation ceremony officially begins the semester.
but
?The number one commences the sequence.
The sequence commences with the number one.
*A terrible murder commences the novel.
The novel commences with a terrible murder.
The convocation ceremony officially commences the semester.
Solution 2:
My entirely intuitive thought is that begin is less formal than commence. Dylan Thomas began his play for voices, ‘Under Milk Wood’, with the words ‘To begin at the beginning.’ He didn’t, with good reason, write ‘To commence at the commencement.’
Solution 3:
From the Merriam-Webster's Dictionary of Synonyms:
"Begin (implying opposition to end) and commence (implying opposition to conclude) are identical in meaning [emphasis mine - Alex B.]: the former is often preferred because less formal than the latter.
[...]
Traditional usage often supports the choice of commence in reference to court proceedings, religious or other ceremonies, or industrial, commercial, or military operations" (p. 94)
Burchfield, the editor of the third edition of Fowler's, is of the same opinion. There's a famous line in Fowler's on this:
"Families and neighbours are divided by the question of when to use commence and when to use begin."
The authors of the Merriam-Webster's Dictionary of English Usage concur that "you need not routinely change it [commence - Alex B.] to begin or start" (p. 264). They also mention that some people consider "commence" to be pretentious, old-fashioned, inappropriate, bookish, or pedantic. Some writers have used the word "commence" ironically - to make fun of toffs, e.g.
"... things never began with Mr. Borthrop Trumbull; they always commenced." George Eliot, as cited in Longman 1984