Machine Learning Based Optimization Workflow for Tuning Numerical Settings of Differential Equation Solvers for Boundary Value Problems - Chapter 1 - Abstract and Workflow Overview

Aug 6, 2026

Numerical differential equation solvers offer efficient alternatives to analytical methods, especially for boundary value problems defined by differential equations plus boundary conditions. Such solvers depend on numerical settings (e.g., tolerances and mesh-related choices) that strongly influence solvability and performance. Current tuning relies on trial-and-error or expert knowledge, which is slow and costly in large parameter spaces. This paper presents a two-stage machine-learning optimization workflow to fine-tune settings, reducing time and domain expertise while improving scalability, stability, and reliability.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages20

Reading Time50 min

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Abstract

Problem background: boundary value solvers and settings

Proposed ML-based two-stage optimization workflow

Foundations: test bench for boundary value problem solvers

arXiv :2404 . 10472v1 [math .NA] 16 Apr 2024 Chapter 1 Machine Learning Based Optimization Workflow for Tuning Numerical Settings of Differential Equation Solvers for Boundary Value Problems Viny Saajan Victor, Manuel Ettm¨uller, Andre Schmeißer, Heike Leitte and Simone Gramsch 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 finetuning 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. Viny Saajan Victor Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany, e-mail: viny .saajan . [victor@itwm.fraunhofer.de](victor@itwm.fraunhofer.de) Manuel Ettm¨uller Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany e-mail: manuel . [ettmueller@itwm.fraunhofer.de](ettmueller@itwm.fraunhofer.de) Andre Schmeißer Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany e-mail: andre . [schmeisser@itwm.fraunhofer.de](schmeisser@itwm.fraunhofer.de) Heike Leitte Technical University of Kaiserslautern, Gottlieb-Daimler-Straße 47, 67663 Kaiserslautern, Germany e-mail: [leitte@cs.uni-kl.de](leitte@cs.uni-kl.de) Simone Gramsch Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany e-mail: simone.gramsch@ [itwm.fraunhofer.de](itwm.fraunhofer.de) 2 V. S. Victor, M. Ettm¨uller, A. Schmeißer, H. Leitte and S. Gramsch 1.1 Introduction Differential equations are one of the most effective mathematical tools for understanding and predicting the behavior of dynamical systems in nature, engineering, and society. The study of differential equations entails learning how to solve them and interpret the solutions. In the field of differential equations, a boundary value problem (BVP) is a type of differential equation with a set of additional constraints known as boundary conditions. The solution to a boundary value problem must satisfy the boundary conditions. Despite the fact that analytical methods for solving boundary value problems yield exact answers, they become difficult to apply to complex problems. Hence, several numerical methods have become popular, leveraging the development of computing capabilities. These methods provide approximate solutions that have sufficient accuracy for engineering purposes. The boundary value problem solvers developed based on the numerical methods usually consist of settings that describe method-specific properties such as error tolerance, mesh size, etc. These settings have an impact on the solvability and performance of the solvers, and the impact varies depending on the problem. In order to obtain qualitatively good solutions within acceptable solution times, fine-tuning these settings is required. However, these settings are now tuned manually by trial and error, which is ineffective due to the large parameter space of the settings and the complex interactions between them. Machine learning (ML) techniques have achieved remarkable success in recent years due to their ability to identify patterns and structures in the given data. Since ML models can predict in real-time, they have been deployed in a variety of industries

FAQs

Why do boundary value problem solvers require fine-tuning of numerical settings?

The numerical settings determine solver solvability and performance, and the impact varies with the specific problem. Achieving good solutions within acceptable time often requires systematic tuning.

What does the proposed machine learning workflow do in two stages?

First, an ML pipeline maps numerical settings to a performance metric such as solvability status and computational cost. Second, the trained model predicts how settings influence solver success and performance to drive a multi-criteria optimization strategy.

How does the workflow address the limitations of manual trial-and-error tuning?

Manual tuning is ineffective due to the large parameter space and complex interactions among settings. The ML-based approach reduces the domain expertise and time required for fine-tuning while evaluating scalability, stability, and reliability.

Aug 6, 2026

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