Formal solutions to partial differential relations

Given a partial differential relation, that is, a subset $\mathcal{R} \subset J^k(\mathbb{R}^n, \mathbb{R}^m)$ of the space of $k$-jets of smooth maps $\mathbb{R}^n \to \mathbb{R}^m$, one can consider the space of smooth (say) maps $f$ from an $n$-manifold $N$ to an $m$-manifold $M$ such that $J^k(f) \in \mathcal{R}$, i.e. so that the $k$-jet of the function lies in the subspace $\mathcal{R}$ at each point. Call the space of such maps $\mathrm{Sol}_\mathcal{R}(N, M)$.

On the other hand, we can consider the bundle $J^k(N, M) \to N$ of $k$-jets of maps from $N$ to $M$, and the associated subbundle $\mathcal{R}(N, M) \to N$, and call the space of sections of this last bundle $\mathrm{FSol}_\mathcal{R}(N,M)$, the space of formal solutions. This space is far easier to analyse, for example because constructing sections of a bundle is a purely homotopy-theoretic problem.

Taking derivatives gives a comparison map $$\mathrm{Sol}_\mathcal{R}(N, M) \to \mathrm{FSol}_\mathcal{R}(N, M).$$ If $\mathcal{R}$ is open in $J^k(\mathbb{R}^n, \mathbb{R}^m)$ and the manifold $N$ is open, Gromov showed that the comparison map is a homotopy equivalence. In particular, if the space of formal solutions is non-empty, so is the space of actual solutions.