Ghost tasking enables efficient solving of challenging differential equations
Ghost tasking for parametrized Gaussian Processes solving linear differential equations
Machine Learning
Summary
Solving complex linear differential equations with limited data is difficult using standard methods. The authors introduce "ghost tasking," a technique that adds extra helper tasks to make hard equations easier to solve with a type of machine learning called parametrized Gaussian processes. This approach works even when the original system is hard to control or has few outputs. Their method is especially useful for problems where you try to figure out causes from observations, even with very little data. They provide computer tools and tests showing how ghost tasking outperforms existing methods.
What this means in practice
- •For computational engineers: Solve inverse problems involving differential equations with limited data using ghost tasking to improve model accuracy and reduce complexity.
- •For software developers in scientific computing: Incorporate ghost tasking into Gaussian process libraries to handle non-parametrizable systems efficiently with fewer output tasks.
Authors
Johanna Moser, Christopher Albert, Sascha Ranftl
Abstract
Physics-informed machine learning has gained significant attention in recent years. In regimes of limited data, parametrized Gaussian processes have become popular. Existing approaches, however, often face limitations, such as requiring parametrizable (also called controllable) systems or a large number of output tasks. In this work, we introduce a systematic procedure we call "ghost tasking", using auxiliary tasks to circumvent these limitations. We prove that such ghost tasks can render any non-parametrizable system effectively parametrizable, enabling algorithmic construction of parametrized Gaussian Processes while keeping the number of required tasks (i.e. output dimensions) and latent functions low. We find that ghost tasking performs especially well in an inverse problem setting, even with very few available data. We show the usage and power of ghost tasking in three experiments, providing systematic comparisons to the only other currently available method applicable to all experiments. We provide necessary syntax and explications for two computer algebra programs that compute parametrizations for systems with polynomial or rational coefficients. Our theoretical results extend to systems with meromorphic functions.