论文标题
神经网络微分方程求解器的局部错误量化
Local error quantification for Neural Network Differential Equation solvers
论文作者
论文摘要
神经网络已被确定为研究复杂系统的强大工具。一个值得注意的例子是神经网络微分方程（NN DE）求解器，它可以为各种微分方程的解决方案提供功能近似。这样的求解器产生强大的功能表达式，非常适合对感兴趣量的进一步操纵（例如，采用衍生品），并且能够利用并行化和计算能力中的现代进步。但是，缺乏关于精确错误量化的作用的工作，可以在其预测中发挥作用：通常，重点是模棱两可和/或全球性能的度量，例如损失函数和/或获得与预测相关的错误的全局界限。没有外部手段或对真实解决方案的彻底了解，很少有局部误差量化。我们在动态系统NN de Solvers的背景下解决了这些问题，利用NN de Solvers中的学习信息开发了使它们更加准确和更有效的方法，同时仍采用不依赖外部工具或数据的无监督方法。我们通过可以精确估计nn de de solver预测错误的方法来实现这一目标，从而使用户具有有效和有针对性的误差校正的能力。我们通过在非线性和混乱系统上测试方法来体现我们的方法的实用性。
Neural networks have been identified as powerful tools for the study of complex systems. A noteworthy example is the neural network differential equation (NN DE) solver, which can provide functional approximations to the solutions of a wide variety of differential equations. Such solvers produce robust functional expressions, are well suited for further manipulations on the quantities of interest (for example, taking derivatives), and capable of leveraging the modern advances in parallelization and computing power. However, there is a lack of work on the role precise error quantification can play in their predictions: usually, the focus is on ambiguous and/or global measures of performance like the loss function and/or obtaining global bounds on the errors associated with the predictions. Precise, local error quantification is seldom possible without external means or outright knowledge of the true solution. We address these concerns in the context of dynamical system NN DE solvers, leveraging learnt information within the NN DE solvers to develop methods that allow them to be more accurate and efficient, while still pursuing an unsupervised approach that does not rely on external tools or data. We achieve this via methods that can precisely estimate NN DE solver prediction errors point-wise, thus allowing the user the capacity for efficient and targeted error correction. We exemplify the utility of our methods by testing them on a nonlinear and a chaotic system each.
