Learning task-specific predictive models for scientific computing
This addresses the need for more reliable surrogate models in machine-learning-augmented scientific computing, though it is incremental as it builds on existing supervised learning frameworks.
The paper tackles the problem of learning predictive models for downstream tasks in scientific computing, showing that minimizing mean square error is insufficient and proposing a task-specific learning approach based on maximum prediction error, which is demonstrated effective in three numerical examples.
We consider learning a predictive model to be subsequently used for a given downstream task (described by an algorithm) that requires access to the model evaluation. This task need not be prediction, and this situation is frequently encountered in machine-learning-augmented scientific computing. We show that this setting differs from classical supervised learning, and in general it cannot be solved by minimizing the mean square error of the model predictions as is frequently performed in the literature. Instead, we find that the maximum prediction error on the support of the downstream task algorithm can serve as an effective estimate for the subsequent task performance. With this insight, we formulate a task-specific supervised learning problem based on the given sampling measure, whose solution serves as a reliable surrogate model for the downstream task. Then, we discretize the empirical risk based on training data, and develop an iterative algorithm to solve the task-specific supervised learning problem. Three illustrative numerical examples on trajectory prediction, optimal control and minimum energy path computation demonstrate the effectiveness of the approach.