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Bayesian Physics Informed Neural Networks for real-world nonlinear dynamical systems
Computer Methods in Applied Mechanics and Engineering ( IF 7.2 ) Pub Date : 2022-07-15 , DOI: 10.1016/j.cma.2022.115346
Kevin Linka, Amelie Schäfer, Xuhui Meng, Zongren Zou, George Em Karniadakis, Ellen Kuhl

Understanding real-world dynamical phenomena remains a challenging task. Across various scientific disciplines, machine learning has advanced as the go-to technology to analyze nonlinear dynamical systems, identify patterns in big data, and make decision around them. Neural networks are now consistently used as universal function approximators for data with underlying mechanisms that are incompletely understood or exceedingly complex. However, neural networks alone ignore the fundamental laws of physics and often fail to make plausible predictions. Here we integrate data, physics, and uncertainties by combining neural networks, physics informed modeling, and Bayesian inference to improve the predictive potential of traditional neural network models. We embed the physical model of a damped harmonic oscillator into a fully-connected feed-forward neural network to explore a simple and illustrative model system, the outbreak dynamics of COVID-19. Our Physics Informed Neural Networks seamlessly integrate data and physics, robustly solve forward and inverse problems, and perform well for both interpolation and extrapolation, even for a small amount of noisy and incomplete data. At only minor additional cost, they self-adaptively learn the weighting between data and physics. They can serve as priors in a Bayesian Inference, and provide credible intervals for uncertainty quantification. Our study reveals the inherent advantages and disadvantages of Neural Networks, Bayesian Inference, and a combination of both and provides valuable guidelines for model selection. While we have only demonstrated these different approaches for the simple model problem of a seasonal endemic infectious disease, we anticipate that the underlying concepts and trends generalize to more complex disease conditions and, more broadly, to a wide variety of nonlinear dynamical systems. Our source code and examples are available at https://github.com/LivingMatterLab/xPINNs.



中文翻译:

用于现实世界非线性动力系统的贝叶斯物理知情神经网络

了解现实世界的动态现象仍然是一项具有挑战性的任务。在各个科学学科中,机器学习已成为分析非线性动力系统、识别大数据模式并围绕它们做出决策的首选技术。神经网络现在一直被用作具有不完全理解或极其复杂的底层机制的数据的通用函数逼近器。然而,单独的神经网络忽略了物理的基本定律,并且经常无法做出合理的预测。在这里,我们通过结合神经网络、物理信息建模和贝叶斯推理来整合数据、物理和不确定性,以提高传统神经网络模型的预测潜力。我们将阻尼谐振子的物理模型嵌入到完全连接的前馈神经网络中,以探索一个简单且说明性的模型系统,即 COVID-19 的爆发动态。我们的基于物理的神经网络无缝集成了数据和物理,稳健地解决了正向和逆向问题,并且在插值和外推方面表现良好,即使对于少量嘈杂和不完整的数据也是如此。只需很少的额外成本,他们就可以自适应地学习数据和物理之间的权重。它们可以作为贝叶斯推理的先验,并为不确定性量化提供可信的区间。我们的研究揭示了神经网络、贝叶斯推理以及两者结合的固有优缺点,并为模型选择提供了有价值的指导。虽然我们仅针对季节性地方性传染病的简单模型问题展示了这些不同的方法,但我们预计基本概念和趋势可以推广到更复杂的疾病条件,更广泛地说,可以推广到各种非线性动力系统。我们的源代码和示例可在 https://github.com/LivingMatterLab/xPINNs 获得。

更新日期:2022-07-16
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