Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks
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 the residual requires a supremum over an infinite-dimensional test space. We prove that the $H^{-1}$ norm of any functional is equivalent to its expected squared evaluation against a random test function whose probability distribution depends only on the domain. Crucially, realisations of this random test function have negative Sobolev regularity for $d \geq 2$, yet this roughness is not an obstacle: averaging over the distribution exactly recovers the correct weak topology, independently ...
| Authors | Diego Marcondes |
|---|---|
| Primary category | math.NA |
| Published | 5 May 2026 |
| Last updated at source | 25 June 2026 |
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