I'm a bit of confused about the two phrases. Can they be used interchangeably when referring to "what happens is influenced by other facts or events"?

Is it correct to say "according to the decision,....."?

Thank you very much!

The two expressions are not interchangeable.

ACCORDING TO

The Longman Dictionary of Contemporary English (Third Edition, 2003) states, about "according to,"

"according to ... prep

"1 As shown by something or stated by someone...

"2 in a way that depends on differences in situations or amounts...

"3 in a way that agrees with a system or plan, or obeys a set of rules: The game will be played according to rules laid down for the 1992 Cup. | Everything went according to plan, and we arrived on time..."

Garner's Modern American Usage* has this to say about "according to":

"This phrase means (1) "depending on"; (2) "as explained or reported by (a person)"; or (3) "in accordance with."

Your question seems to be about the Longman sense 3, "in a way that agrees with a system or plan, or obeys a set of rules"; and the Garner sense (3), "in accordance with."

In the example sentence, the phrase "according to the decision," ..." as an initial sentence adverbial can have only Sense 1 (examples are from Google):

"” According to the court's decision awarding summary judgment to the Government, the manuscript, which is dated 1778 and bears the original signature of Fray Junipero Serra, a missionary and soldier, was stolen from the Mexican National Archives. According to the decision, the manuscript was subsequently acquired in 1992 at a Mexico City flea market for approximately $300 by a Mexico City book, coin, and manuscript dealer named Duane Douglas. (These uses of "according to" report what the court's decision said.)

In a different sentence position, the phrase "according to" can have Sense 3:

"” If the decision is to commit, it sends a global_commit message to all sites. Otherwise, it sends a global_abort message to all sites that voted to commit. Finally, it writes an end of transaction record in its log. The participants finish the transaction according to the decision and write the result in their logs....

"” Will assets that I placed in trust be part of my probate estate?" No. If you created a valid living trust, the property you placed in the trust during your lifetime will pass according to the terms of the trust and not under the provisions of your will.

BASED ON

Garner states, about the past participial phrase "based on":

"...it may carry an adjectival force .....:

"” A quick calculation based on U.S. data indicates a requirement or $13 billion a year for Afghanistan"

Garner warns against abusing the expression and offers two instances in which "based on" should not be used:

(1) It should NOT be used as an adverb:

"” American officials said they attacked the convoy based on intelligence reports...

"(Based on improperly modifies attacked. Try because of or owing to instead.)"

(2) Garner further states that "based on" should NOT be used as a prepositional "dangler":

"Based on these conversations, Riley said he doubts Graham will play."

"(A suggested revision: Riley said that because of those conversations, he doubts....")

Marilyn

*By Bryan A. Garner. (Oxford, 2003)