Data driven splitting and composition algorithms

拆分和合成算法在应用程序中无处不在，因为将一个复杂的任务拆分成多个子任务往往更容易。分裂算法的应用与常微分方程组和偏微分方程组的时间演化算法、采样算法和优化算法一样广泛。传统上，分裂和合成方法是使用解析和代数技术推导出来的。这通常意味着截断的泰勒级数。然而，虽然这在收敛等分析性质方面给出了很好的结果，但这限制了它们向渐近制度的发展。具体地说，目前很大一部分研究集中于寻找阶数增加的方法，相反，我们试图找到廉价、准确、可用的方法，这些方法在渐近区域具有低误差常数，并且对于较大的时间步长也是最优的。一种可能的方法是找到满足一些低阶条件的分裂系数，并使用剩余的自由度来减少误差方程中的误差常数，而不是增加阶数。对于具有守恒律的物理系统，我们可以考虑结构保留求解器，并减少对所述守恒定律的违反。因此，我们学习了关于特定问题、子问题和子求解器的最优分裂系数。这项任务可以被描述为一个正式的最小化问题，可能需要分析、代数、优化技术和机器学习方法的组合。虽然我们最初的大多数尝试都将受到计算量子物理和化学中的问题的推动，并且是高度跨学科的，但这项研究有更广泛的潜力成为任何想要解决微分方程组或偏微分方程组的人都非常有益的工具。

Splitting and composition algorithms are ubiquitous in applications since it is often easier to take a complex task and split it into multiple sub-tasks. The application of splitting algorithms appear as widely as algorithms for time evolution of ODEs and PDEs, sampling algorithms and optimization algorithms. Traditionally, splitting and composition methods have been derived using analytic and algebraic techniques. This normally means truncated Taylor series. However, while this gives nice results in terms of analytical properties like convergence this restricts their advancement to asymptotic regimes. Specifically large proportions of current research focus on finding methods with increased order.Instead, we seek to find cheap, accurate, usable methods that feature low error constants in asymptotic regimes and are also optimal for larger time steps. A possible approach could involve finding the splitting coefficients that satisfy some lower order conditions and using the remaining degrees of freedom to reduce the error constants in the error equations rather than increasing the order. For physical systems with conservation laws we can consider structure preserving solvers and reducing the violations of said conservation laws. Therefore we learn the optimal splitting coefficients with regard to specific problems, subproblems and subsolvers. This task can be described as a formal minimisation problem that may require a combination of analytical, algebraic, optimization techniques and machine learning methods. While most of our initial attempts will be motivated by problems in computational quantum physics and chemistry, and is highly interdisciplinary, this research has a much broader potential to be an immensely beneficial tool to anyone who would want to solve ODEs or PDEs.