learning in physics

1. The concept of symmetry has become increasingly important in theoretical physics since the beginnings of Relativity with Lorentz, Poincaré, Hilbert and Einstein electromagnetism
The notion of least action has taken precedence over classical 'equations of motion' approaches
2. The physics of a phenomenon is summed up by the constraints of invariance or symmetry imposed on the action:
a. Ex 1 : Galilean symmetry: S = ∫ dt (1/2 m(dq/dt -V(q))
b. Ex 2 : lorentzian invariance : S = ∫ - mc²√𝜂ₗₛdxˡdxˢ
c. Ex 3 : particle in a field : S = ∫ - mc²√𝜂ₗₛdxˡdxˢ - eAₛdxˢ
d. Ex 4 : gauge invariance: equation of the EM field : S = ∫ dx⁴ FˡˢFₗₛ
e. Ex 5 : Principle of Equivalence: Hilbert-Einstein action : S = ∫ dx⁴ R√-g
f. Ex 6 : Quantum Field Theory (QFT)…
g. Ex 7 : non-abelian gauge invariance: Yang Mills lagrangian
h. Ex 8 : Supersymetry…
3. A first attempt to make the Schrödinger Lorentz equation invariant leads to Klein Gordon's equation. The problem of the negative current density which emerges from this equation leads Dirac to try an equation of order 1, which has a solution only for a non-scalar field: Dirac falls on this occasion on Clifford algebras without knowing it.
4. The theory of representations of the Lie algebra of the SO (3) group (rotation in space) is a valuable aid in the solution of the Schrödinger equation for the electron in a spherically symmetric potential.
5. The link between physical stat and QFT occurs naturally, via the formal equivalence between partition function Z and action S.
6. In QFT, for example for the condensed matter theory (CondMat), we seek an effective representation, at the Landau-Ginzburg, guided by the reasonable symmetries of the Lagrangian.
7. In CondMat again, the Renormalization Group (RG) plays an important role in understanding the transition from one scale to another
8. Recently, it has been proposed to see a profound link between RG and Deep Learning "An exact mapping between the Variational Renormalization Group and Deep Learning", Mehta, Schwab (MS)
9. MS shows that the unsupervised learning of the Deep Belief Network mimics the RG, except that the RG seeks to minimize the gap between the Free Energy of the two layers, where the DBN uses the Kullback-Leibler measure.
10. The perfect mapping between RG and DBN goes through the equivalence between T_λ and E, the energy or 'Hamiltonian' of the DBN.
11. The key to learning is therefore the lucky guess of E, which rests on a good understanding of its symmetries.
12. In the case of images, these symmetries are expressed very directly in the CNN.