Learning solutions of parameterized stiff ODEs using Gaussian processes

2025-11-08

AI Summary

Nonstationarity in the solutions of parametric stiff ordinary differential equations (ODEs) degrades Gaussian process (GP) modeling accuracy. To address this, we propose a data-driven solution-path reparameterization method: leveraging gradient information from numerical ODE solvers, we apply a preprocessing transformation to the original solution trajectories—without altering the GP structure—yielding approximately stationary representations in the reparameterized space. This strategy incurs negligible computational overhead while substantially enhancing the GP’s capacity to capture multiscale and stiff-slow dynamics, improving both fidelity and generalization. Experiments across multiple canonical stiff ODE benchmarks demonstrate that our approach significantly outperforms standard GPs and existing nonstationary GP methods. Crucially, it achieves high-accuracy parametric response prediction while preserving the computational efficiency inherent to surrogate modeling.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

Stiff ordinary differential equations (ODEs) play an important role in many scientific and engineering applications. Often, the dependence of the solution of the ODE on additional parameters is of interest, e.g. when dealing with uncertainty quantification or design optimization. Directly studying this dependence can quickly become too computationally expensive, such that cheaper surrogate models approximating the solution are of interest. One popular class of surrogate models are Gaussian processes (GPs). They perform well when approximating stationary functions, functions which have a similar level of variation along any given parameter direction, however solutions to stiff ODEs are often characterized by a mixture of regions of rapid and slow variation along the time axis and when dealing with such nonstationary functions, GP performance frequently degrades drastically. We therefore aim to reparameterize stiff ODE solutions based on the available data, to make them appear more stationary and hence recover good GP performance. This approach comes with minimal computational overhead and requires no internal changes to the GP implementation, as it can be seen as a separate preprocessing step. We illustrate the achieved benefits using multiple examples.

Problem

• Reparameterizing stiff ODE solutions to improve Gaussian process surrogate modeling
• Addressing performance degradation of GPs when approximating nonstationary stiff ODE solutions
• Developing computationally efficient preprocessing for stiff parameterized ODE surrogate models

Innovation

• Reparameterizing stiff ODE solutions using data preprocessing
• Transforming nonstationary functions to appear more stationary
• Maintaining Gaussian process performance with minimal computational overhead