Active learning algorithm for computational physics

In large-scale computations of physical problems, one often encounters the situation of having to determine a multidimensional function, which can be numerically costly when computing each point in this multidimensional space is already time-demanding. In the work, we propose that the active learning algorithm can speed up such calculations. The basic idea is to fit a multidimensional function by neural networks, and the key point is to make the query of labeled data more economical by using a strategy called "query by committee." We present the general protocol of this fitting scheme, as well as the procedure of how to further compute physical observables with the fitted functions. We show that this method can work well with two examples, which are the quantum three-body problem in atomic physics and the anomalous Hall conductivity in condensed matter physics, respectively. In these examples, we show that one reaches an accuracy of a few percent error in computing physical observables, all the while using fewer than 10 % of total data points compared with uniform sampling. With these two examples, we also visualize that by using the active learning algorithm, the required amount of data points are added mostly in the regime where the function varies most rapidly, which explains the mechanism for the efficiency of the algorithm. We expect broad applications of our method to various kinds of computational physical problems.