Learning task-specific predictive models for scientific computing

TL;DR

This paper introduces a novel approach for learning predictive models tailored to specific downstream tasks in scientific computing, emphasizing the importance of maximum prediction error over traditional mean square error minimization.

Contribution

It formulates a task-specific supervised learning framework that optimizes for maximum prediction error on the task support, providing a more reliable surrogate for downstream performance.

Findings

Effective in trajectory prediction, optimal control, and energy path computation.

Outperforms traditional models by focusing on maximum prediction error.

Demonstrates robustness across diverse scientific computing tasks.

Abstract

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.