  Homotopical geometry over differential operators is a convenient setting for
a coordinate-free investigation of nonlinear partial differential equations
modulo symmetries. One of the first issues one meets in the functor of points
approach to homotopical $\mathcal{D}$-geometry, is the question of a model
structure on the category $\tt DGAlg(\mathcal{D})$ of differential
non-negatively graded $\mathcal{O}$-quasi-coherent sheaves of commutative
algebras over the sheaf $\mathcal{D}$ of differential operators of an
appropriate underlying variety $(X,\mathcal{O})$. We define a cofibrantly
generated model structure on $\tt DGAlg(\mathcal{D})$ via the definition of its
weak equivalences and its fibrations, characterize the class of cofibrations,
and build an explicit functorial `cofibration - trivial fibration'
factorization. We then use the latter to get a functorial model categorical
Koszul-Tate resolution for $\mathcal{D}$-algebraic `on-shell function' algebras
(which contains the classical Koszul-Tate resolution). The paper is also the
starting point for a homotopical $\mathcal{D}$-geometric Batalin-Vilkovisky
formalism.
