Collocation-based Robust Physics Informed Neural Networks for time-dependent simulations of pollution propagation under thermal inversion conditions on Spitsbergen

Leszek Siwik; Maciej Sikora; Natalia Leszczyńska; Tomasz Maciej Ciesielski; Eirik Valseth; Manuela Bastidas Olivares; Marcin Łoś; Tomasz Służalec

cs.LGApr 24, 2026

Collocation-based Robust Physics Informed Neural Networks for time-dependent simulations of pollution propagation under thermal inversion conditions on Spitsbergen

Authors: Leszek Siwik, Maciej Sikora, Natalia Leszczyńska, Tomasz Maciej Ciesielski, Eirik Valseth, Manuela Bastidas Olivares, Marcin Łoś, Tomasz Służalec, +2 more

Organizations: AGH University of Krakow, Faculty of Computer Science, Al. Mickiewicza 30, Kraków, 30-059, Poland · Medical University of Silesia-Katowice, Faculty of Medical Sciences, ul. Poniatowskiego 15, Katowice, 40-055, Poland · The University Centre in Svalbard, Longyearbyen, Box 156 N-9171, Longyearbyen, Norway · Simula Research Laboratory, Kristian Augusts gate 23, Oslo, 0164, Norway · Norwegian University of Life Sciences, Postboks 5003, As, 1432, Norway · Universidad Nacional de Colombia, Carrera 45 No. 26-85 - Uriel Gutiérrez Building Bogota, Colombia · AGH University of Krakow, Faculty of Energy and Fuels, Al. Mickiewicza 30, Kraków, 30-059, Poland

Abstract

In this paper, we propose a Physics-Informed Neural Network framework for time-dependent simulations of pollution propagation originating from moving emission sources. We formulate a robust variational framework for the time-dependent advection-diffusion problem and establish the boundedness and inf-sup stability of the corresponding discrete weak formulation. Based on this mathematical foundation, we construct a robust loss function that is directly related to the true approximation error, defined as the difference between the neural network approximation and the (unknown) exact solution. Additionally, a collocation-based strategy is introduced to speed up neural network training. As a case study, we investigate pollution propagation caused by snowmobile traffic in Longyearbyen, Spitsbergen, supported by detailed in-field measurements collected using dedicated sensors. The proposed framework is applied to analyze the effects of thermal inversion on pollutant accumulation. Our results demonstrate that thermal inversion traps dense and humid air masses near the ground, significantly enhancing particulate matter (PM) concentration and worsening local air quality.

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Jul 2, 2026 · cs.LGJ/K move · Enter open · S save

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

Joseph Webb, Sadok Jerad, Coralia Cartis

Mathematical Institute, University of Oxford, Oxford, United Kingdom

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

\textbf{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.

\textbf{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

3\times10^{-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^{\text{rel}} = 4.75 \times 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

Conventional Physics-Informed Neural Networks Newton Neural Network Classical Solvers Prompt Optimization

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Abstract

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