Border bases are a generalization of Gr\"obner bases for polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra. We elaborate on their properties and present algorithms to compute with them. We apply this theory to represent integrable connections as cyclic $D$-modules explicitly. As an application, we visit differential equations behind a stringy, a Feynman as well as a cosmological integral. We also address the classification of particular $D$-ideals of a fixed holonomic rank, namely the case of linear PDEs with constant coefficients as well as Frobenius ideals. Our approach rests on the theory of Hilbert schemes of points in affine space.

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