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Interpretable polynomial neural ordinary differential equations.

Colby Fronk1, Linda Petzold2,3

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Polynomial neural ordinary differential equations (ODEs) enhance interpretability and generalization for dynamical systems. This new approach enables predictions beyond training data and direct symbolic regression without external tools.

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Area of Science:

  • Dynamical Systems and Machine Learning
  • Artificial Intelligence
  • Computational Mathematics

Background:

  • Standard neural ordinary differential equations (ODEs) face challenges in interpretability and generalization.
  • These limitations hinder their application in modeling complex dynamical systems.
  • Existing methods often require additional tools for tasks like symbolic regression.

Purpose of the Study:

  • To introduce a novel neural ODE framework that addresses interpretability and generalization issues.
  • To develop a method for direct symbolic regression using neural networks.
  • To improve the applicability of neural ODEs in scientific modeling.

Main Methods:

  • Implementation of a deep polynomial neural network within the neural ODE framework.
  • Development of the polynomial neural ODE model.
  • Testing the model's performance on prediction tasks and symbolic regression.

Main Results:

  • Polynomial neural ODEs demonstrate improved generalization capabilities, predicting accurately outside the training region.
  • The model performs direct symbolic regression without relying on external algorithms like SINDy.
  • Enhanced interpretability of the dynamical system models.

Conclusions:

  • Polynomial neural ODEs offer a promising advancement for modeling dynamical systems.
  • The framework enhances both the predictive accuracy and interpretability of neural ODEs.
  • This approach simplifies the process of symbolic regression in scientific machine learning.