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SSRN 2024年

作者： Naderibeni, Mahdi Reinders, Marcel J.T. Wu, Liang Tax, David M.J. Pattern Recognition and Bio-informatics Group Delft University of Technology Van Mourik Broekmanweg 6 Delft2628 XE Netherlands Science Research and Innovation DSM-Firmenich Delft Netherlands

We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs' limitations in converging to solutions of highly nonlinear parametric-PDEs like NSE. We consider the parameter(s) of interest as inputs of PINNs along with spatio-temporal coordinates, and train PINNs on generated numerical solutions of parametric-PDES for instances of the parameters. We perform experiments on the classical 2D flow past cylinder problem aiming to learn velocities and pressure functions over a range of Reynolds numbers as parameter of interest. Provision of training data from generated numerical simulations allows for interpolation of the solution functions for a range of parameters. Therefore, we compare PINNs with unconstrained conventional Neural Networks (NN) on this problem setup to investigate the effectiveness of considering the PDEs regularization in the loss function. We show that our proposed approach results in optimizing PINN models that learn the solution functions while making sure that flow predictions are in line with conservational laws of mass and momentum. Our results show that PINN results in accurate prediction of gradients compared to NN model, this is clearly visible in predicted vorticity fields given that none of these models were trained on vorticity labels. © 2024, The Authors. All rights reserved.

关键词： Navier Stokes equations

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Learning solutions of parametric Navier-Stokes with physics-informed neural networks

arXiv

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arXiv 2024年

We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs’ limitations in converging to solutions of highly nonlinear parametric-PDEs like NSE. We consider the parameter(s) of interest as inputs of PINNs along with spatio-temporal coordinates, and train PINNs on generated numerical solutions of parametric-PDES for instances of the parameters. We perform experiments on the classical 2D flow past cylinder problem aiming to learn velocities and pressure functions over a range of Reynolds numbers as parameter of interest. Provision of training data from generated numerical simulations allows for interpolation of the solution functions for a range of parameters. Therefore, we compare PINNs with unconstrained conventional Neural Networks (NN) on this problem setup to investigate the effectiveness of considering the PDEs regularization in the loss function. We show that our proposed approach results in optimizing PINN models that learn the solution functions while making sure that flow predictions are in line with conservational laws of mass and momentum. Our results show that PINN results in accurate prediction of gradients compared to NN model, this is clearly visible in predicted vorticity fields given that none of these models were trained on vorticity labels. © 2024, CC BY.

关键词： Navier Stokes equations

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Inductive Conformal Prediction under Data Scarcity: Exploring the Impacts of Nonconformity Measures

arXiv

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arXiv 2024年

作者： Kato, Yuko Tax, David M.J. Loog, Marco Pattern Recognition and Bio-informatics Group Delft University of Technology Van Mourik Broekmanweg 6 Delft2628 XE Netherlands Institute for Computing and Information Sciences Radboud University Toernooiveld 212 Nijmegen6525 EC Netherlands

Conformal prediction, which makes no distributional assumptions about the data, has emerged as a powerful and reliable approach to uncertainty quantification in practical applications. The nonconformity measure used in conformal prediction quantifies how a test sample differs from the training data and the effectiveness of a conformal prediction interval may depend heavily on the precise measure employed. The impact of this choice has, however, not been widely explored, especially when dealing with limited amounts of data. The primary objective of this study is to evaluate the performance of various nonconformity measures (absolute error-based, normalized absolute error-based, and quantile-based measures) in terms of validity and efficiency when used in inductive conformal prediction. The focus is on small datasets, which is still a common setting in many real-world applications. Using synthetic and real-world data, we assess how different characteristics—such as dataset size, noise, and dimensionality—can affect the efficiency of conformal prediction intervals. Our results show that although there are differences, no single nonconformity measure consistently outperforms the others, as the effectiveness of each nonconformity measure is heavily influenced by the specific nature of the data. Additionally, we found that increasing dataset size does not always improve efficiency, suggesting the importance of fine-tuning models and, again, the need to carefully select the nonconformity measure for different applications. © 2024, CC BY.

关键词： Prediction models

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Risk-based decision making: estimands for sequential prediction under interventions

arXiv

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arXiv 2023年

作者： Luijken, Kim Morzywolek, Pawel van Amsterdam, Wouter Cinà, Giovanni Hoogland, Jeroen Keogh, Ruth Krijthe, Jesse Magliacane, Sara van Ommen, Thijs Peek, Niels Putter, Hein van Smeden, Maarten Sperrin, Matthew Wang, Junfeng Weir, Daniala Didelez, Vanessa van Geloven, Nan Department of Epidemiology Julius Center for Health Sciences and Primary Care University Medical Center Utrecht Utrecht Netherlands Department of Applied Mathematics Computer Science and Statistics Ghent University Ghent Belgium Department of Statistics University of Washington Seattle United States Department of Medical Informatics Amsterdam University Medical Centers Amsterdam Netherlands Institute for Logic Language and Computation University of Amsterdam Amsterdam Netherlands Pacmed Amsterdam Netherlands Department of Epidemiology and Data Science Amsterdam University Medical Centers Amsterdam Netherlands Department of Medical Statistics London School of Hygiene & Tropical Medicine Keppel Street London United Kingdom Pattern Recognition and Bio-Informatics Group EEMCS Delft University of Technology Delft Netherlands Amsterdam Machine Learning Lab University of Amsterdam Amsterdam Netherlands Department of Information and Computing Sciences Utrecht University Utrecht Netherlands Division of Informatics Imaging and Data Science Faculty of Biology Medicine and Health University of Manchester Manchester Academic Health Science Centre Manchester United Kingdom Department of Biomedical Data Sciences Leiden University Medical Center Leiden Netherlands Division of Pharmacoepidemiology and Clinical Pharmacology Department of Pharmaceutical Sciences Utrecht University Utrecht Netherlands Department of Biometry and Data Management Leibniz Institute for Prevention Research Epidemiology - BIPS Bremen Germany Faculty of Mathematics/Computer Science University of Bremen Bremen Germany

Prediction models are used amongst others to inform medical decisions on interventions. Typically, individuals with high risks of adverse outcomes are advised to undergo an intervention while those at low risk are advised to refrain from it. Standard prediction models do not always provide risks that are relevant to inform such decisions: e.g., an individual may be estimated to be at low risk because similar individuals in the past received an intervention which lowered their risk. Therefore, prediction models supporting decisions should target risks belonging to defined intervention strategies. Previous works on prediction under interventions assumed that the prediction model was used only at one time point to make an intervention decision. In clinical practice, intervention decisions are rarely made only once: they might be repeated, deferred and re-evaluated. This requires estimated risks under interventions that can be reconsidered at several potential decision moments. In the current work, we highlight key considerations for formulating estimands in sequential prediction under interventions that can inform such intervention decisions. We illustrate these considerations by giving examples of estimands for a case study about choosing between vaginal delivery and cesarean section for women giving birth. Our formalization of prediction tasks in a sequential, causal, and estimand context provides guidance for future studies to ensure that the right question is answered and appropriate causal estimation approaches are chosen to develop sequential prediction models that can inform intervention decisions. © 2023, CC BY.

关键词： Forecasting

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