Universal Differential Equations for Scientific Machine Learning
Universal Differential Equations for Scientific Machine Learning
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DOI：
10.21203/rs.3.rs-55125/v1
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发表时间：
2020-01
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影响因子：
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通讯作者：
Christopher Rackauckas;Yingbo Ma;Julius Martensen;Collin Warner;K. Zubov;R. Supekar;Dominic J. Skinner;A. Ramadhan
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文献类型：
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作者：
Christopher Rackauckas;Yingbo Ma;Julius Martensen;Collin Warner;K. Zubov;R. Supekar;Dominic J. Skinner;A. Ramadhan
在科学的背景下，众所周知的格言“一张图片胜过千言万语”很可能是“一个模型胜过一千个数据集”。科学模型，如牛顿物理学或生物基因调控网络，是人类驱动的复杂现象的简化，作为验证模型的无数实验的替代品。最近，机器学习已经能够通过直接从数据中学习整个非线性交互来克服近似建模的不准确性。然而，在没有任何来自问题背后的科学基础的预定结构的情况下，机器学习方法是灵活的，但数据昂贵，需要大量同质标记训练数据的数据库。一个核心挑战是在不需要“大数据”的情况下协调与简化模型不一致的数据。在这项工作中，演示了如何将数学对象（我们将其表示为通用微分方程（UDE））用作科学机器学习中各种问题的理论基础，以产生有效的算法和通用方法。UDE模型通过机器可学习的结构来增强科学模型，以实现基于科学的学习。我们展示了如何利用UDEs来发现以前未知的控制方程，准确地外推原始数据，并加速模型模拟，所有这些都是以时间和数据高效的方式进行的。这一进步与开源软件相结合，允许训练在模型中包含物理约束、延迟交互、隐式定义的事件和内在随机性的UDE。我们的例子展示了跨科学学科的各种计算困难的建模问题，从自动发现生物机制到加速物理信息神经网络和大涡模拟的训练，都可以转化为通过单一软件方法有效解决的UDE训练问题。
In the context of science, the well-known adage “a picture is worth a thousand words” might well be “a model is worth a thousand datasets.” Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". In this work demonstrate how a mathematical object, which we denote universal differential equations (UDEs), can be utilized as a theoretical underpinning to a diverse array of problems in scientific machine learning to yield efficient algorithms and generalized approaches. The UDE model augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating the training of physics-informed neural networks and large-eddy simulations, can all be transformed into UDE training problems that are efficiently solved by a single software methodology.