Machine Learning Based Optimization Workflow for Tuning Numerical Settings of Differential Equation Solvers for Boundary Value Problems

TL;DR

This work tackles the manual tuning of numerical settings in boundary value problem solvers by introducing a two-stage ML-based workflow that first maps settings to solvability and performance using a binary classifier and multi-output regressors, then drives multi-objective optimization to select Pareto-optimal settings. The data pipeline relies on Latin Hypercube Sampling over ten reference BVP problems, and a Python library, bvpTune, interfaces with the solver and Optuna to generate and analyze Pareto fronts. Experimental results demonstrate strong classification and regression performance, scalable prediction times, and reliable optimization fronts, indicating a substantial reduction in the domain expertise required for tuning. The approach enables flexible trade-offs among solver accuracy and computational resources, with practical impact on deploying BVP solvers in engineering applications.

Abstract

Several numerical differential equation solvers have been employed effectively over the years as an alternative to analytical solvers to quickly and conveniently solve differential equations. One category of these is boundary value solvers, which are used to solve real-world problems formulated as differential equations with boundary conditions. These solvers require certain numerical settings to solve the differential equations that affect their solvability and performance. A systematic fine-tuning of these settings is required to obtain the desired solution and performance. Currently, these settings are either selected by trial and error or require domain expertise. In this paper, we propose a machine learning-based optimization workflow for fine-tuning the numerical settings to reduce the time and domain expertise required in the process. In the evaluation section, we discuss the scalability, stability, and reliability of the proposed workflow. We demonstrate our workflow on a numerical boundary value problem solver.

Figures (9)

• Figure 1: Input and output parameters of BVP solver.
• Figure 2: Workflow of machine learning-based optimization for tuning the numerical settings of BVP solvers. Different stages are colored corresponding to the section numbers in this chapter.
• Figure 3: ROC curves for different classification algorithms
• Figure 4: Precision-Recall curves for different classification algorithms
• Figure 5: Violin plot comparing the distribution of the actual and predicted values of ODE evaluations, grid points and maximum residuum.