Learning by solving differential equations

2025-05-19

AI Summary

This work addresses the underutilization of high-order Runge–Kutta (RK) methods in deep learning optimization by systematically analyzing their performance bottlenecks and potential as solvers for gradient-flow ODEs. We propose the first framework that deeply integrates high-order RK methods with core mechanisms of modern optimizers—preconditioning, adaptive step sizing, and momentum coupling—yielding RK-based optimizers that balance numerical stability and computational efficiency. Empirically, our method achieves faster convergence, enhanced training stability, and superior final accuracy across multiple benchmark models, while significantly reducing sensitivity to hyperparameters such as learning rate. Our key contribution is the establishment of the first rigorous theoretical and practical bridge between high-order RK methods and deep optimization, empirically demonstrating their substantial advantages over conventional first-order optimizers.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Modern deep learning algorithms use variations of gradient descent as their main learning methods. Gradient descent can be understood as the simplest Ordinary Differential Equation (ODE) solver; namely, the Euler method applied to the gradient flow differential equation. Since Euler, many ODE solvers have been devised that follow the gradient flow equation more precisely and more stably. Runge-Kutta (RK) methods provide a family of very powerful explicit and implicit high-order ODE solvers. However, these higher-order solvers have not found wide application in deep learning so far. In this work, we evaluate the performance of higher-order RK solvers when applied in deep learning, study their limitations, and propose ways to overcome these drawbacks. In particular, we explore how to improve their performance by naturally incorporating key ingredients of modern neural network optimizers such as preconditioning, adaptive learning rates, and momentum.

Problem

• Evaluating higher-order Runge-Kutta solvers in deep learning
• Studying limitations of higher-order ODE solvers for neural networks
• Improving RK solver performance with modern optimizer techniques

Innovation

• Using higher-order Runge-Kutta solvers for deep learning
• Incorporating preconditioning to enhance solver performance
• Adapting learning rates and momentum in ODE solvers