Re:Differential Equations with Transformers: Deep Learning f...
by whatshisface on Wednesday March 11, 2020 @12:40PM
(#22547204)
>Since we use elementary functions on the real domain, a lot of integrals have no solution. We could have gone for a larger set (adding erf, the Fresnels, up to Liouvillian functions). This would mean more solvable cases.
Here is something to think about if you want more solvable cases: even in the case of sine and cosine, solving a differential equation really means reducing it to a combination of the solutions to simpler differential equations. The sine function can be defined as the solution to a particular differential equation, as can all of the fancier functions. So in a sense it's kind of like factorization, where you have "prime" equations whose only solution is a transcendental function defined as being their solution, and "composite" equations whose solutions can be written as a combination of solutions to "prime" equations. So really all of the rare functions belong to the same general scheme.
Here is something to think about if you want more solvable cases: even in the case of sine and cosine, solving a differential equation really means reducing it to a combination of the solutions to simpler differential equations. The sine function can be defined as the solution to a particular differential equation, as can all of the fancier functions. So in a sense it's kind of like factorization, where you have "prime" equations whose only solution is a transcendental function defined as being their solution, and "composite" equations whose solutions can be written as a combination of solutions to "prime" equations. So really all of the rare functions belong to the same general scheme.