Learning efficient and provably convergent splitting methods

Splitting methods are widely used for solving initial value problems (IVPs) due to their ability to simplify complicated evolutions into more manageable subproblems which can be solved efficiently and accurately. 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 timesteps. 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 timesteps. 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 less computational resources. However, they are often purely data-driven, come with limited convergence guarantees in the small-timestep 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 timesteps and have provable convergence and conservation guarantees in the small-timestep limit. We demonstrate numerically that the learned methods, which by construction converge quadratically in the timestep size, can be significantly more efficient than established methods for the Schr\"{o}dinger equation if the computational budget is limited.

EmProx: Neural Network Performance Estimation For Neural Architecture Search

Common Neural Architecture Search methods generate large amounts of candidate architectures that need training in order to assess their performance and find an optimal architecture. To minimize the search time we use different performance estimation strategies. The effectiveness of such strategies varies in terms of accuracy and fit and query time. This study proposes a new method, EmProx Score (Embedding Proximity Score). Similar to Neural Architecture Optimization (NAO), this method maps candidate architectures to a continuous embedding space using an encoder-decoder framework. The performance of candidates is then estimated using weighted kNN based on the embedding vectors of architectures of which the performance is known. Performance estimations of this method are comparable to the MLP performance predictor used in NAO in terms of accuracy, while being nearly nine times faster to train compared to NAO. Benchmarking against other performance estimation strategies currently used shows similar to better accuracy, while being five up to eighty times faster.