[Paper Review] Hypersolvers: Toward Fast Continuous-Depth Models

Michael Poli, Stefano Massaroli+3 more|arXiv (Cornell University)|Jul 19, 2020

Model Reduction and Neural NetworksPhysics and Astronomy68 references18 citations

TL;DR

This paper introduces hypersolvers—neural networks trained to solve ordinary differential equations (ODEs) with low computational overhead and theoretical accuracy guarantees—enabling fast, scalable inference in continuous-depth models like Neural ODEs. By replacing traditional numerical solvers with learnable, efficient hypersolvers, the method achieves up to 100× speedup in inference with minimal accuracy loss, enabling practical deployment in tasks like continuous normalizing flows.

ABSTRACT

The infinite-depth paradigm pioneered by Neural ODEs has launched a renaissance in the search for novel dynamical system-inspired deep learning primitives; however, their utilization in problems of non-trivial size has often proved impossible due to poor computational scalability. This work paves the way for scalable Neural ODEs with time-to-prediction comparable to traditional discrete networks. We introduce hypersolvers, neural networks designed to solve ODEs with low overhead and theoretical guarantees on accuracy. The synergistic combination of hypersolvers and Neural ODEs allows for cheap inference and unlocks a new frontier for practical application of continuous-depth models. Experimental evaluations on standard benchmarks, such as sampling for continuous normalizing flows, reveal consistent pareto efficiency over classical numerical methods.

Motivation & Objective

• To address the poor computational scalability of Neural ODEs, which limits their use in real-time or large-scale applications.
• To overcome the high numerical solver overhead that hinders efficient inference and training in continuous-depth models.
• To develop a synergistic model-solver framework that improves pareto efficiency in the accuracy-computation trade-off.
• To enable fast, accurate inference in continuous normalizing flows and other continuous-depth tasks.
• To explore pretraining and joint optimization strategies for hypersolvers to enhance reusability and training efficiency.

Proposed method

• Hypersolvers are trained as neural networks to approximate the residual error of standard numerical solvers, enabling fast ODE solution prediction.
• The method replaces traditional ODE solvers with a learnable surrogate that predicts the next step in the solution trajectory using a small number of function evaluations.
• Hypersolvers are trained using a loss that minimizes the difference between their predicted solution and the ground-truth solution from adaptive solvers like dopri5.
• The framework supports both pretraining on diverse dynamics and joint optimization with the main Neural ODE model to improve generalization and solver robustness.
• Theoretical analysis shows that hypersolvers are $\mathcal{O}(\delta)$ approximators of residuals, with convergence guarantees under Lipschitz continuity of the vector field.
• Adversarial training is used to improve hypersolver resilience to stiff or challenging ODE dynamics by exposing it to worst-case scenarios.

Experimental results

Research questions

• RQ1 Can a learnable solver (hypersolver) achieve comparable or better accuracy than traditional numerical solvers with significantly fewer function evaluations?
• RQ2 To what extent can hypersolvers improve the pareto efficiency of the accuracy-computation trade-off in continuous-depth models?
• RQ3 How can hypersolvers be trained and reused effectively across training iterations to reduce computational cost in Neural ODEs?
• RQ4 Can joint optimization of the main model and hypersolver improve overall training efficiency and solution accuracy?
• RQ5 What are the theoretical and empirical limits of hypersolver generalization across diverse ODE dynamics?

Key findings

• Hypersolvers achieve up to 100× speedup in inference over standard solvers like dopri5 while maintaining comparable or better solution accuracy on continuous normalizing flow benchmarks.
• The method enables accurate sampling from continuous normalizing flows in as few as 2 function evaluations, a significant improvement over traditional methods.
• Hypersolvers demonstrate $\mathcal{O}(\delta)$ approximation error of the residual, with theoretical guarantees on convergence under Lipschitz continuity of the vector field.
• Joint optimization and adversarial training of hypersolvers improve robustness to stiff ODE dynamics, reducing solution error in challenging cases.
• Pretraining hypersolvers on diverse dynamics enables reuse across training iterations, reducing the need for retraining and accelerating training of Neural ODEs.
• The framework achieves pareto efficiency in the accuracy-computation trade-off, outperforming classical solvers in both NFEs and algorithmic complexity.