A comma after a coordinating conjunction (and, or, but) is only acceptable when the comma is the first of a pair of commas bracketing a ‘supplement’—a phrase or clause which lies outside the main structure of the sentence and interrupts the flow of thought.
And, as if that were not enough, he went on to sneer at the victim.
^---------------------------^
But, as you said, there is no point in trying too hard.
^------------^
In other circumstances the comma is effectively ‘disjunctive’ and subverts the joining effect of the conjunction.
This counterproductive comma is becoming very common in contemporary business writing; I suspect it represents a hypercorrection by people who have misunderstood the need for commas bracketing supplements, or think it lends formal ‘elegance’ to their writing.
You do need a comma here. There is a systematic answer for this.
Whenever a sentence begins with an adverbial phrase, use a comma.
(source: https://owl.purdue.edu/owl/general_writing/punctuation/commas/extended_rules_for_commas.html)
An adverbial phrase explains things like: Manner, timing, reason, purpose among others. In this case, you have an infinitive of purpose as an adverbial phrase.
In your case:
To further our understanding of this phenomenon, (Adverbial of purpose)
we conducted yet another experiment. (main clause)
Other similar examples:
To help him start his car, we pushed it until it started rolling.
To better understand what they were doing, we started spying on them.
Usage:
These phrases use infinitives of purposes as adverbial phrases. It is much more common to place them at the end of a sentence like so:
We pushed his car until it started rolling to help him start it.
One might be tempted to think that it makes no difference, but it does. The reason one would put this at the beginning is to emphasize the purpose. Consider this context.
To help him start his car, we pushed it until it started rolling. To help him repair it, we called a mechanic. To convince him to leave, we offered him some of the stale food we had.
Also consider this phrasing in which the first purpose is logical, whereas the second oen is somewhat surprising:
We started spying on them to better understand what they were doing. To join them, we confessed that had been spying on them.
Best Answer
The meaning is the same either way. However, when writing sentences - especially when writing something formal such as an essay - one should use commas after their transition word or phrase that starts a new sentence.
For example, notice how I started this sentence with my transition phrase "for example" and put a comma after it? Of course, commas are useful for starting sentences out, as well as breaking up the thoughts in a single sentence.
I italicized the transition words/phrases above. Here is a site with phrases if you want to know how to add variety to your words. It's divided into different categories.