Data driven splitting and composition algorithms

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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.