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We establish an equivalence between machine learning and statistical data assimilation, where network layers mirror time steps. This framework uses statistical physics and dynamics to improve training and network design for both fields.

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Area of Science:

  • Physics
  • Computer Science
  • Applied Mathematics

Background:

  • Machine learning and data assimilation are distinct fields.
  • Existing literature notes a layer-time correspondence in feedforward networks.

Purpose of the Study:

  • To formulate a deeper equivalence between machine learning and statistical data assimilation.
  • To explore the role of statistical physics and dynamics in network design and training.
  • To provide a unified framework for understanding and improving both machine learning and data assimilation.

Main Methods:

  • Formulating an equivalence between feedforward artificial networks and statistical data assimilation.
  • Extending the equivalence to recurrent networks.
  • Applying methods from statistical physics, Lagrangian, and Hamiltonian dynamics.
  • Analyzing continuous layer/time frameworks using differential equations (Euler-Lagrange).
  • Investigating two-point boundary value problems and symplectic symmetry.

Main Results:

  • Demonstrated that network depth in machine learning is analogous to temporal resolution in data assimilation.
  • Showed that data assimilation methods can find global minima for machine learning cost functions.
  • Derived the Euler-Lagrange equation for continuous layers, framing optimization as a two-point boundary value problem.
  • Presented Lagrangian and Hamiltonian formulations respecting symplectic symmetry.
  • Provided a Hamiltonian rationale for backpropagation.

Conclusions:

  • The equivalence offers a novel perspective for both machine learning and data assimilation.
  • Statistical physics and dynamics provide powerful tools for network design and optimization.
  • Continuous layer frameworks ('deepest learning') offer new theoretical insights and potential computational advantages.
  • The Hamiltonian approach justifies backpropagation within a broader variational context.