Learning Efficient and Provably Convergent Symmetric Splitting Methods

Abstract. Splitting methods are widely used for solving initial value problems (IVPs) due to their ability to simplify complicated evolutions into more manageable subproblems. These subproblems can be solved efficiently and accurately, leveraging properties like linearity, sparsity, and reduced stiffness. Traditionally, these methods are derived using analytic and algebraic techniques from numerical analysis, including truncated Taylor series and their Lie algebraic analogue, the Baker–Campbell–Hausdorff formula. These tools enable the development of high-order numerical methods that provide exceptional accuracy for small time steps. Moreover, these methods often (nearly) conserve important physical invariants, such as mass, unitarity, and energy. However, in many practical applications the computational resources are limited. Thus, it is crucial to identify methods that achieve the best accuracy within a fixed computational budget, which might require taking relatively large time steps. In this regime, high-order methods derived with traditional methods often exhibit large errors since they are only designed to be asymptotically optimal. Machine learning techniques offer a potential solution since they can be trained to efficiently solve a given IVP with fewer computational resources. However, they are often purely data-driven, come with limited convergence guarantees in the small-time-step regime and do not necessarily conserve physical invariants. In this work, we propose a framework for finding machine learned splitting methods that are computationally efficient for large time steps and have provable convergence and conservation guarantees in the small-time-step limit. We demonstrate numerically that the learned methods, which by construction converge quadratically in the time-step size, can be significantly more efficient than established methods for the Schrödinger equation if the computational budget is limited.