Computational vs Theoretical Mathematics in Physics

The mathematical models of physics take the form of partial differential equations like Euler's Equations for incompressible inviscid fluid flow EE, corresponding Navier-Stokes equation for viscous flow NSE and Schrödinger's equations for atoms and molecules SE. 

The task of a theoretical mathematician has been to prove by symbolic analytical techniques (i) existence, (ii) uniqueness and (ii) regularity of solutions to a given equation with data given in some large class of possible data with data including initial data, forcing and parameters like viscosity in NSE. 

The task of a computational mathematician has been to compute solutions for specific choices of data which in each specific case can answer (i)-(iii) by inspection of the computed solution. 

It has been argued that computation is not enough, even if for each specific choice of data (i)-(iii) can be answered, because only a limited number of specific choices can be inspected. The possibly very large class of data can thus never be exhausted by computation, which gives analytical symbolic mathematics a role to play by covering a large class of data.

It is natural to ask if there are examples of equations for which the class of relevant data is so small that it can be exhausted by computation. This means first that the equation cannot contain any parameter like viscosity. Are there any models of interest which are parameter free? Inspection of EE and SE shows that they are both parameter free, and so meet the requirement of Einstein of an ideal mathematical model opening to say something about the world without measuring anything. This is like learning the area of a circular disc by computation with unit radius as only input.

Solving EE computationally thus delivers the drag of a body moving through a slightly viscous fluid such as air and water at a subsonic speed with the only data being the shape of the body and not any viscosity as parameter. This limits the class of data to shapes of bodies with a limited range of shapes of interest to be covered by computation. This is all described here.

The case of SE is in its traditional form of Standard Quantum Mechanics StdQM troubled by the fact SE by its multi-dimension nature is uncomputable and so needs dimensional compression which introduces parameters. 

RealQM is different realisation of the same parameter-free Hamiltonian as StdQM into computable form without introduction of any parameter. RealQM thus expresses SE in parameter-free computable form and so opens the possibility of saying something about the atomic world without experimental input. RealQM thus computes the ground state of an atom with the only input being the number of electrons and so can exhaust the Periodic Table.      

An analytical estimate of ground state energy as the result of a longer or shorter sequence of successive bounds, can be seen as a form of symbolic computation, while a numerical computation can be seen as very long arithmetic proof.

Computation with a parameter-free mathematical model can produce a rich set of outputs from very limited structural input, which can serve as data for AI in need of rich data. Computation is then used both to produce data and to learn from data. Symbolic mathematics has an important role to set up computation.

The Clay Institute Millennium Problem on (i)-(iii) for NSE is still open in the form of symbolic mathematics with no progress reported over 25 years. Can computation get the million dollar Prize?