Dynamical systems that evolve continuously over time are ubiquitous throughout science and engineering. Machine learning (ML) provides data-driven approaches to model and predict the dynamics of such systems. A core issue with this approach is that ML models are typically trained on discrete data, using ML methodologies that are not aware of underlying continuity properties, which results in models that often do not capture the underlying continuous dynamics of a system of interest. As a result, these ML models are of limited use for for many scientific and engineering applications. To address this challenge, we develop a convergence test based on numerical analysis theory. Our test verifies whether a model has learned a function that accurately approximates a system's underlying continuous dynamics. Models that fail this test fail to capture relevant dynamics, rendering them of limited utility for many scientific prediction tasks; while models that pass this test enable both better interpolation and better extrapolation in multiple ways. Our results illustrate how principled numerical analysis methods can be coupled with existing ML training/testing methodologies to validate models for science and engineering applications.
翻译:随着时间的推移不断演变的动态系统在整个科学和工程中无处不在。机器学习(ML)为模拟和预测这些系统的动态提供了由数据驱动的模型和预测方法。这一方法的一个核心问题是,ML模型通常在离散数据上接受培训,使用的ML方法并不了解潜在的连续性特性,从而导致模型往往不能捕捉一个感兴趣的系统的基本连续动态。结果,这些ML模型对许多科学和工程应用用途的使用有限。为了应对这一挑战,我们根据数字分析理论开发了一个趋同测试。我们的测试核实一个模型是否学到了精确接近一个系统基本连续动态的功能。没有通过这一测试的模型无法捕捉到相关的动态,使它们对许多科学预测任务用处有限;而通过这一测试的模型能够以多种方式更好地进行内推和更好的外推。我们的结果说明了原则性的数字分析方法如何与现有的ML培训/测试方法相结合,以验证科学和工程应用模型。