Strong coupling corrections to the Ginzburg-Landau theory of superfluid<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mmultiscripts><mml:mi mathvariant="normal">He</mml:mi><mml:mprescripts/><mml:none/><mml:mn>3</mml:mn></mml:mmultiscripts></mml:math>

In the Ginzburg-Landau theory of superfluid $^{3}\mathrm{He}$, the free energy is expressed as an expansion of invariants of a complex order parameter. Strong coupling effects, which increase with increasing pressure, are embodied in the set of coefficients of these order-parameter invariants [A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975); E. V. Thuneberg, Phys. Rev. B 36, 3583 (1987); J. Low Temp. Phys. 122, 657 (2001)]. Experiments can be used to determine four independent combinations of the coefficients of the five fourth-order invariants. This leaves the phenomenological description of the thermodynamics near ${T}_{c}$ incomplete. Theoretical understanding of these coefficients is also quite limited. We analyze our measurements of the magnetic susceptibility and the NMR frequency shift in the $B$ phase which refine the four experimental inputs to the phenomenological theory. We propose a model based on existing experiments, combined with calculations by Sauls and Serene [Phys. Rev. B 24, 183 (1981)] of the pressure dependence of these coefficients, in order to determine all five fourth-order terms. This model leads us to a better understanding of the thermodynamics of superfluid $^{3}\mathrm{He}$ in its various states. We discuss the surface tension of bulk superfluid $^{3}\mathrm{He}$ and predictions for novel states of the superfluid such as those that are stabilized by elastic scattering of quasiparticles from a highly porous silica aerogel.