研究 PINNs 在有限时间爆炸附近求解布尔格斯 PDE 的能力

IF 6.3 2区 物理与天体物理 Q1 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE Machine Learning Science and Technology Pub Date : 2024-06-11 DOI:10.1088/2632-2153/ad51cd
Dibyakanti Kumar, Anirbit Mukherjee
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引用次数: 0

摘要

物理信息神经网络(PINNs)在数值求解复杂的偏微分方程(PDEs)方面取得了不断刷新的成就,同时在推理的准确性和速度之间做出了极具吸引力的权衡。偏微分方程的一个特别具有挑战性的方面是,存在一些简单的偏微分方程,它们可以在有限的时间内从平滑的初始条件演化成奇异的解。最近,一些引人注目的实验表明,PINNs 甚至可以很好地检测出这种有限时间内的炸裂。在这项工作中,我们着手从严格的理论角度研究 PINNs 的这种稳定性。首先,我们在允许有限时间炸毁的条件下,推导出任意维度下布尔格斯 PDE PINN 的误差边界。我们的界限为函数正则化项提供了理论依据,据报道,函数正则化项对训练接近有限时间膨胀的 PINNs 非常有用。然后,我们通过实验证明,当对越来越接近炸毁的 PDE 序列进行计算时,我们的边界与神经发现的替代方案与真正炸毁方案的 ℓ2 距离显著相关。
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Investigating the ability of PINNs to solve Burgers’ PDE near finite-time blowup
Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated Partial Differential Equations (PDEs) numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive error bounds for PINNs for Burgers’ PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Our bounds give a theoretical justification for the functional regularization terms that have been reported to be useful for training PINNs near finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the 2 -distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.
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来源期刊
Machine Learning Science and Technology
Machine Learning Science and Technology Computer Science-Artificial Intelligence
CiteScore
9.10
自引率
4.40%
发文量
86
审稿时长
5 weeks
期刊介绍: Machine Learning Science and Technology is a multidisciplinary open access journal that bridges the application of machine learning across the sciences with advances in machine learning methods and theory as motivated by physical insights. Specifically, articles must fall into one of the following categories: advance the state of machine learning-driven applications in the sciences or make conceptual, methodological or theoretical advances in machine learning with applications to, inspiration from, or motivated by scientific problems.
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