蚁网:用可扩展和可解释的神经代理来打破高维偏微分方程的维度诅咒

IF 7.3 1区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY
Sidharth S. Menon, Ameya D. Jagtap
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引用次数: 0

摘要

高维偏微分方程(PDEs)在各种科学和工程应用中出现,但由于维数的诅咒,计算上仍然难以解决。传统的数值方法难以应对计算复杂度呈指数增长的问题,特别是在超立方域上,所需的配点数随着维数的增加而迅速增加。在这里,我们介绍了ant- net,一个有效的神经代理,克服了这一挑战,使pde在高维上的解决方案成为可能。与内部体积随着维数增加而减小的超球体不同,超立方体保持或扩展其体积(对于单位或更大的长度),这使得高维计算的要求更高。蚁网算法有效地结合了高维边界条件,使高维配点处的偏方差残差最小。为了提高可解释性,我们将Kolmogorov-Arnold网络集成到ant- net架构中。我们在几个线性和非线性高维方程上对Anant-Net的性能进行了基准测试,包括泊松方程、sin - gordon方程和Allen-Cahn方程,以及瞬态热方程,在从高维空间随机抽样的测试点上展示了高精度和鲁棒性。重要的是,Anant-Net以惊人的效率实现了这些结果,在几个小时内在单个GPU上解决了300维的问题。我们还将Anant-Net的结果与其他最先进的方法进行了准确性和运行时间的比较。我们的研究结果建立了ant- net作为一个准确的、可解释的、可扩展的框架,用于有效地解决高维偏微分方程。ant- net代码可在https://github.com/ParamIntelligence/Anant-Net上获得。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Anant-Net: Breaking the curse of dimensionality with scalable and interpretable neural surrogate for high-dimensional PDEs
High-dimensional partial differential equations (PDEs) arise in diverse scientific and engineering applications but remain computationally intractable due to the curse of dimensionality. Traditional numerical methods struggle with the exponential growth in computational complexity, particularly on hypercubic domains, where the number of required collocation points increases rapidly with dimensionality. Here, we introduce Anant-Net, an efficient neural surrogate that overcomes this challenge, enabling the solution of PDEs in high dimensions. Unlike hyperspheres, where the internal volume diminishes as dimensionality increases, hypercubes retain or expand their volume (for unit or larger length), making high-dimensional computations significantly more demanding. Anant-Net efficiently incorporates high-dimensional boundary conditions and minimizes the PDE residual at high-dimensional collocation points. To enhance interpretability, we integrate Kolmogorov-Arnold networks into the Anant-Net architecture. We benchmark Anant-Net’s performance on several linear and nonlinear high-dimensional equations, including the Poisson, Sine-Gordon, and Allen-Cahn equations, as well as transient heat equations, demonstrating high accuracy and robustness across randomly sampled test points from high-dimensional spaces. Importantly, Anant-Net achieves these results with remarkable efficiency, solving 300-dimensional problems on a single GPU within a few hours. We also compare Anant-Net’s results for accuracy and runtime with other state-of-the-art methods. Our findings establish Anant-Net as an accurate, interpretable, and scalable framework for efficiently solving high-dimensional PDEs. The Anant-Net code is available at https://github.com/ParamIntelligence/Anant-Net.
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来源期刊
CiteScore
12.70
自引率
15.30%
发文量
719
审稿时长
44 days
期刊介绍: Computer Methods in Applied Mechanics and Engineering stands as a cornerstone in the realm of computational science and engineering. With a history spanning over five decades, the journal has been a key platform for disseminating papers on advanced mathematical modeling and numerical solutions. Interdisciplinary in nature, these contributions encompass mechanics, mathematics, computer science, and various scientific disciplines. The journal welcomes a broad range of computational methods addressing the simulation, analysis, and design of complex physical problems, making it a vital resource for researchers in the field.
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