{"ID":2884492,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.05921","arxiv_id":"2508.05921","title":"Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equations","abstract":"Accuracy in neural PDE solvers often breaks down not because of limited expressivity, but due to poor optimisation caused by ill-conditioning, especially in multi-fidelity and stiff problems. We study this issue in Physics-Informed Extreme Learning Machines (PIELMs), a convex variant of neural PDE solvers, and show that asymptotic components in governing equations can produce highly ill-conditioned activation matrices, severely limiting convergence. We introduce Shifted Gaussian Encoding, a simple yet effective activation filtering step that increases matrix rank and expressivity while preserving convexity. Our method extends the solvable range of Peclet numbers in steady advection-diffusion equations by over two orders of magnitude, achieves up to six orders lower error on multi-frequency function learning, and fits high-fidelity image vectors more accurately and faster than deep networks with over a million parameters. This work highlights that conditioning, not depth, is often the bottleneck in scientific neural solvers and that simple architectural changes can unlock substantial gains.","short_abstract":"Accuracy in neural PDE solvers often breaks down not because of limited expressivity, but due to poor optimisation caused by ill-conditioning, especially in multi-fidelity and stiff problems. We study this issue in Physics-Informed Extreme Learning Machines (PIELMs), a convex variant of neural PDE solvers, and show tha...","url_abs":"https://arxiv.org/abs/2508.05921","url_pdf":"https://arxiv.org/pdf/2508.05921v1","authors":"[\"Siddharth Rout\"]","published":"2025-08-08T00:51:38Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.FA\",\"math.RT\",\"physics.comp-ph\"]","methods":"[\"Diffusion Model\"]","has_code":false}