ArticleJournal of scientific computing2025
Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations.
Article in Journal of scientific computing, 2025. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 4 papers.
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4 citing papers in PubMed.
- Multi-Source Data-Driven Estimation Model for Passenger Flow Management in Urban Rail Transit.Sensors (Basel, Switzerland) · 2026Article
- ZENN: A thermodynamics-inspired computational framework for heterogeneous data-driven modeling.Proceedings of the National Academy of Sciences of the United States of America · 2026Article
- Learn Singularly Perturbed Solutions via Homotopy Dynamics.Proceedings of machine learning research · 2025Article
- Homotopy Relaxation Training Algorithms for Infinite-Width Two-Layer ReLU Neural Networks.Journal of scientific computing · 2025Article
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4 authors.
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Abstract
Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering, especially in scenarios featuring complex domains or incorporation of empirical data. One advantage of the neural network methods for PDEs lies in its automatic differentiation (AD), which necessitates only the sample points themselves, unlike traditional finite difference (FD) approximations that require nearby local points to compute derivatives. In this paper, we quantitatively demonstrate the advantage of AD in training neural networks. The concept of truncated entropy is introduced to characterize the training property. Specifically, through comprehensive experimental and theoretical analyses conducted on random feature models and two-layer neural networks, we discover that the defined truncated entropy serves as a reliable metric for quantifying the residual loss of random feature models and the training speed of neural networks for both AD and FD methods. Our experimental and theoretical analyses demonstrate that, from a training perspective, AD outperforms FD in solving PDEs.
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