Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks
arXiv:2605.03542v2 Announce Type: replace-cross Abstract: 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 r
This research provides a novel mathematical foundation for making physics-informed neural networks more robust and computationally feasible, addressing a key challenge in their current application.
Improved computational methods for solving partial differential equations, especially via neural networks, directly accelerate scientific discovery, engineering design, and AI capabilities across various domains.
The computational intractability of the H-1 norm, a significant hurdle for stochastic variational physics-informed neural networks, is potentially overcome through the use of random test functions.
- · AI researchers and developers
- · Engineering sectors using simulations
- · Scientific research institutions
- · Traditional numerical methods (comparatively)
Accelerated development and application of physics-informed AI models for complex systems.
Reduced computational costs and time for simulations in fields like aerospace, climate modeling, and material science.
New classes of AI-driven design and optimization tools become practical, leading to novel scientific breakthroughs.
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Read at arXiv cs.LG