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Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations.

Chuqi Chen1,2, Yahong Yang1, Yang Xiang1,3

  • 1Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.

Journal of Scientific Computing
|September 5, 2025
PubMed
Summary
This summary is machine-generated.

Neural network methods for solving partial differential equations (PDEs) show promise. Automatic differentiation (AD) offers advantages over finite difference (FD) methods in training neural networks for PDEs.

Keywords:
Automatic differentiationDifferential equationNeural networkNumerical differentiationTraining error

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Area of Science:

  • Computational science and engineering
  • Applied mathematics
  • Machine learning for scientific computing

Background:

  • Neural networks are increasingly used for solving partial differential equations (PDEs).
  • Traditional methods like finite difference (FD) require local points for derivative computation.
  • Automatic differentiation (AD) offers an alternative by using only sample points.

Purpose of the Study:

  • To quantitatively demonstrate the training advantages of automatic differentiation (AD) over finite difference (FD) methods for neural network-based PDE solvers.
  • To introduce and validate a new metric, truncated entropy, for characterizing neural network training properties.
  • To compare the performance of AD and FD in solving PDEs from a training perspective.

Main Methods:

  • Introduction of the truncated entropy concept for training characterization.
  • Experimental and theoretical analyses on random feature models.
  • Analysis of two-layer neural networks using both AD and FD.

Main Results:

  • Truncated entropy reliably quantifies residual loss in random feature models.
  • Truncated entropy serves as a metric for neural network training speed with both AD and FD.
  • Experimental and theoretical evidence shows AD outperforms FD in training neural networks for PDEs.

Conclusions:

  • Automatic differentiation (AD) presents a superior training approach for neural network-based partial differential equation (PDE) solvers compared to finite difference (FD) methods.
  • The novel truncated entropy metric effectively characterizes training dynamics and performance.
  • The findings support the broader adoption of AD in scientific machine learning for solving complex equations.