论文标题
神经网络微分方程求解器内的错误估计和校正
Error Estimation and Correction from within Neural Network Differential Equation Solvers
论文作者
论文摘要
神经网络微分方程（NN de）求解器由于因素的结合而迅速流行：计算的进步使其优化更加可行，处理高维问题的能力，容易解释性问题等等。但是，几乎所有NN de Solvers都受到了基本限制的所有NN de Solvers的损失：使用损失功能训练它们仅依赖于该estimations的损失功能。因此，解决方案估计值的验证和错误分析需要了解真实解决方案。的确，如果真正的解决方案未知，我们通常会简单地希望“低”损失意味着“足够小”错误，因为两者之间的明确关系无法获得/定义。在这项工作中，我们描述了有效构建神经网络微分方程求解器的错误估计和校正的一般策略。我们的方法不需要提前了解真实解决方案，而需要获得损失函数和与解决方案估计相关的错误之间的明确关系。反过来，这些明确的关系直接使我们能够估计并纠正错误。
Neural Network Differential Equation (NN DE) solvers have surged in popularity due to a combination of factors: computational advances making their optimization more tractable, their capacity to handle high dimensional problems, easy interpret-ability of their models, etc. However, almost all NN DE solvers suffer from a fundamental limitation: they are trained using loss functions that depend only implicitly on the error associated with the estimate. As such, validation and error analysis of solution estimates requires knowledge of the true solution. Indeed, if the true solution is unknown, we are often reduced to simply hoping that a "low enough" loss implies "small enough" errors, since explicit relationships between the two are not available/well defined. In this work, we describe a general strategy for efficiently constructing error estimates and corrections for Neural Network Differential Equation solvers. Our methods do not require advance knowledge of the true solutions and obtain explicit relationships between loss functions and the error associated with solution estimates. In turn, these explicit relationships directly allow us to estimate and correct for the errors.
