An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks

arXiv:2607.02194v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss landscape. We present extbf{DSGNAR}: Doubly-Sketched Gauss-Newton with Adaptive Ratio, a scalable second-order optimisation framework that confronts this ill-conditioning and, in doing so, obtains unprecedented accuracy and speed. extbf{DSGNAR} couples a doubly-sketched Gauss-Newton model with a novel strategy that carefully controls both regularisation and step length. Across a suite of problems spanning nonlinear, chaotic, multi-scale, high-dimensional, and Navier-Stokes, the framework greatly improves on the state of the art: able to attain relative ell_2 errors as low as 3imes10^{-16} in double precision, improve contemporary results by five orders of magnitude on the canonical Burgers' equation, and as much as eight orders on a high-dimensional Poisson problem, while remaining markedly faster. We further show that, in single precision, solutions at the limit of round-off error can be obtained very quickly: Burgers' equation to ell_2^{ext{rel}} = 4.75 imes 10^{-7} in under ten seconds. The framework is also robust to the choice of architecture, arithmetic precision, and initial hyperparameters. The code is available at https://www.github.com/wephy/physics-informed-neural-networks

Source: arXiv cs.LG | 2026-07-03