[Paper Review] Opening the Blackbox: Accelerating Neural Differential Equations by Regularizing Internal Solver Heuristics

Avik Pal, Yingbo Ma+2 more|arXiv (Cornell University)|May 9, 2021

Model Reduction and Neural Networks46 references4 citations

TL;DR

This paper proposes a novel regularization method that accelerates Neural Differential Equations (NDEs) by leveraging internal solver heuristics—specifically local error and stiffness estimates—without increasing training cost. By using these pre-computed, low-cost metrics as regularization signals, the method reduces prediction time by up to 1.84× and training time by 1.45×, outperforming higher-order derivative methods while maintaining model accuracy.

ABSTRACT

Democratization of machine learning requires architectures that automatically adapt to new problems. Neural Differential Equations (NDEs) have emerged as a popular modeling framework by removing the need for ML practitioners to choose the number of layers in a recurrent model. While we can control the computational cost by choosing the number of layers in standard architectures, in NDEs the number of neural network evaluations for a forward pass can depend on the number of steps of the adaptive ODE solver. But, can we force the NDE to learn the version with the least steps while not increasing the training cost? Current strategies to overcome slow prediction require high order automatic differentiation, leading to significantly higher training time. We describe a novel regularization method that uses the internal cost heuristics of adaptive differential equation solvers combined with discrete adjoint sensitivities to guide the training process towards learning NDEs that are easier to solve. This approach opens up the blackbox numerical analysis behind the differential equation solver's algorithm and directly uses its local error estimates and stiffness heuristics as cheap and accurate cost estimates. We incorporate our method without any change in the underlying NDE framework and show that our method extends beyond Ordinary Differential Equations to accommodate Neural Stochastic Differential Equations. We demonstrate how our approach can halve the prediction time and, unlike other methods which can increase the training time by an order of magnitude, we demonstrate similar reduction in training times. Together this showcases how the knowledge embedded within state-of-the-art equation solvers can be used to enhance machine learning.

Motivation & Objective

• To address the high computational cost of inference in Neural Differential Equations (NDEs), which depends on adaptive ODE solver steps.
• To reduce prediction time without increasing training cost, a major bottleneck in NDE deployment.
• To develop a regularization strategy that uses internal solver statistics—instead of expensive higher-order derivatives—to guide the model toward easier-to-solve dynamics.
• To extend the method beyond ODEs to Neural Stochastic Differential Equations (SDEs), demonstrating broad applicability.
• To provide a plug-and-play solution compatible with existing NDE frameworks, enabling wider community adoption.

Proposed method

• The method introduces regularization based on the local error estimates and stiffness heuristics computed internally by adaptive ODE solvers during integration.
• These solver-internal metrics are used as cheap, accurate cost proxies to guide the training process toward NDEs that require fewer neural network function evaluations (NFEs).
• The regularization is applied via discrete adjoint sensitivities, enabling efficient backpropagation through the solver’s internal logic without modifying the underlying NDE architecture.
• The approach is implemented using the Julia-based SciML ecosystem, enabling seamless integration with existing NDE frameworks.
• It extends naturally to Neural SDEs by incorporating noise-aware heuristics, maintaining efficiency and accuracy.
• The method avoids high-order automatic differentiation, preserving training efficiency while achieving performance gains comparable to more expensive alternatives.

Experimental results

Research questions

• RQ1 Can internal solver heuristics be used to regularize NDEs in a way that reduces prediction time without increasing training cost?
• RQ2 How do local error and stiffness estimates compare to higher-order derivative-based regularization in terms of inference speed and training efficiency?
• RQ3 Can this regularization strategy be generalized beyond ODEs to include Neural SDEs with diagonal multiplicative noise?
• RQ4 Does using solver-internal statistics lead to faster convergence and fewer function evaluations during inference while preserving model accuracy?
• RQ5 Is there a way to leverage numerical analysis heuristics to accelerate NDEs without modifying the solver or model architecture?

Key findings

• The proposed method achieves an average 1.84× speedup in prediction time across supervised classification and time series tasks, reducing the number of neural function evaluations (NFEs) from 1083.0 to 676.2 on the Spiral equation.
• Training time is reduced by 1.45× on average, demonstrating that the method does not incur the high computational cost typically associated with regularization in NDEs.
• The regularization using local error and stiffness estimates achieves performance comparable to higher-order derivative methods but without the training time penalty.
• On image classification tasks, error-based regularization was more effective, while stiffness regularization showed stronger gains on time series problems, suggesting task-dependent optimal regularization types.
• The method is generalizable to Neural SDEs and can be applied without modifying the underlying NDE framework, enabling broad adoption.
• The approach successfully 'opens the black box' of ODE solvers by utilizing their internal heuristics to guide model learning, demonstrating that embedded numerical knowledge can enhance deep learning efficiency.