Data driven splitting and composition algorithms

拆分和组合算法在应用程序中无处不在，因为将复杂任务拆分为多个子任务通常更容易。分裂算法的应用与常微分方程和偏微分方程的时间演化算法、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分析和代数技术推导出来的。这通常意味着截断泰勒级数。然而，虽然这给出了很好的结果，如收敛性的分析性质，这限制了他们的进步渐近制度。特别是目前的研究大部分集中在寻找方法与增加的命令。相反，我们试图找到廉价，准确，可用的方法，具有低误差常数的渐近制度，也是最佳的更大的时间步长。一种可能的方法可以涉及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。对于具有守恒定律的物理系统，我们可以考虑结构保持求解器，并减少对守恒定律的违反。因此，我们学习关于特定问题，子问题和子求解器的最佳分裂系数。这个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是由计算量子物理和化学中的问题所激发的，并且是高度跨学科的，但这项研究具有更广泛的潜力，可以成为任何想要解决常微分方程或偏微分方程的人的非常有益的工具。

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.