Tributes poured in from people across the city for Mr Welsh, 42, who was a close friend of Trainspotting author Irvine Welsh and played gangland kingpin Doyle in Trainspotting 2.
On Twitter, Kirsty Mulcahy, described Mr Welsh as a "Father, partner, local business man, tireless charity worker and mentor and coach to hundreds. Give the man the respect he’s due. RIP Brad."
User KatrinaM212 added that his death is a "total loss to the community". She added: "So many kids looked up to this man as a role model to get them off the street."
Raymond Champion also paid tribute. He said: "His ice cream stories are the stuff of legend to them. Aw man, what a sad story this is. I'm going to have to break this to my kids and they will be distraught. Brad was never anything other than a friendly soul at his gym. RIP."
Andy Gillies echoed the tributes and said: "A community champion and a total gentleman who always had time to listen and help. Sincerest condolences to Brad's family and extended family. Heartbreaking news."
Tributes also flooded in on Facebook.
Sharon Mcaloon said: "Bradley was a total gentleman and gentle giant who helped a lot of deprived children through his Holyrood Boxing gym in Edinburgh."
Paul Smith added: "A tragic loss. Edinburgh will be a poorer place without you Brad, you did so much to inspire and change the lives of so many young people, may you rest in peace."
Robbi Robertson said: "A great guy who was helping underprivileged kids in his area to keep out of trouble and give them a good start."
Valeria Boorman also posted an emotional tribute. She said: "I am absolutely devastated. Such a huge loss for Edinburgh. The kindest and caring guy I was privileged to call a pal.
"Always put others before himself, involved in so many charities including food banks, children's Christmas charities.
"Such an inspirational man in the boxing world, helping hundreds of youngsters learn the noble art of boxing. Enabling them to develop self worth, discipline, strength, skills and well being. R.I.P Brad you are Edinburgh's finest son"