Deriving analytical solutions of ordinary differential equations is usually restricted to a small subset of problems and numerical techniques are considered. Inevitably, a numerical simulation of a differential equation will then always be distinct from a true analytical solution. An efficient integration scheme shall further not only provide a trajectory throughout a given state, but also be derived to ensure the generated simulation to be close to the analytical one. Consequently, several integration schemes were developed for different classes of differential equations. Unfortunately, when considering the integration of complex non-linear systems, as well as the identification of non-linear equations from data, this choice of the integration scheme is often far from being trivial. In this paper, we propose a novel framework to learn integration schemes that minimize an integration-related cost function. We demonstrate the relevance of the proposed learning-based approach for non-linear equations and include a quantitative analysis w.r.t. classical state-of-the-art integration techniques, especially where the latter may not apply.
翻译:对普通差异方程式的分析解决办法通常仅限于一小部分问题,并会考虑数字技术; 不可避免地,对差异方程式进行数字模拟将永远与真正的分析解决办法区分开来; 高效的一体化计划不仅应在整个特定状态提供轨迹,而且还应进一步提供轨迹,以确保生成的模拟接近分析公式; 因此,为不同类别的差异方程式制定了若干一体化计划; 不幸的是,在考虑综合复杂的非线性系统以及从数据中确定非线性方程式时,这种集成计划的这种选择往往远非微不足道; 在本文中,我们提出了一个学习融合计划的新框架,以尽量减少与整合有关的成本功能; 我们展示了拟议的基于学习的方法对非线性方程式的相关性,并包括了对传统的非线性方程式的定量分析,特别是在后者可能不适用的情况下。