Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks

ArXi:2605.03542v1 Announce Type: cross The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of a residual requires a supremum over an infinite-dimensional function space. We prove that the $H^{-1}$ norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain.