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Machine learning conservation laws from differential equations.

Ziming Liu1, Varun Madhavan2, Max Tegmark1

  • 1Department of Physics, Institute for AI and Fundamental Interactions, and Center for Brains, Minds and Machines, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.

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Summary
This summary is machine-generated.

We developed a machine learning algorithm to discover conservation laws from differential equations. This method ensures functional independence and works for both numerical and symbolic representations.

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Area of Science:

  • Physics
  • Applied Mathematics
  • Computer Science

Background:

  • Conservation laws are fundamental in physics, simplifying complex systems.
  • Discovering these laws from differential equations is challenging, especially for nonlinear systems.
  • Existing methods may struggle with functional independence and diverse equation types.

Purpose of the Study:

  • To present a novel machine learning algorithm for discovering conservation laws.
  • To ensure the discovered conservation laws are functionally independent.
  • To demonstrate the algorithm's applicability to various differential equations.

Main Methods:

  • Developed a machine learning algorithm capable of symbolic and neural network-based discovery.
  • Implemented a novel independence module, a nonlinear generalization of SVD.
  • Incorporated inductive biases for conservation laws into the algorithm.

Main Results:

  • Successfully discovered conservation laws from differential equations.
  • Demonstrated functional independence of the derived laws.
  • Validated the method on the three-body problem, KdV, and nonlinear Schrödinger equations.

Conclusions:

  • The proposed machine learning algorithm effectively discovers functionally independent conservation laws.
  • The method is versatile, handling both numerical and symbolic representations.
  • This approach offers a powerful tool for analyzing complex dynamical systems.