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Area of Science:

  • Computational Physics
  • Data Science
  • Machine Learning

Background:

  • Fine-grained simulations are computationally expensive.
  • Developing accurate coarse-grained models is challenging.
  • Effective surrogate models are needed for complex systems.

Purpose of the Study:

  • To identify effective stochastic differential equations (SDEs) for coarse observables.
  • To develop neural network-based surrogate models for fine-scale dynamics.
  • To enable efficient analysis of complex particle- or agent-based simulations.

Main Methods:

  • Approximating drift and diffusivity functions using neural networks (stochastic ResNets).
  • Employing a loss function inspired by stochastic numerical integrators (Euler-Maruyama, Milstein).
  • Utilizing physics-informed gray-box identification with mean-field equations.

Main Results:

  • Demonstrated effectiveness on a stochastically forced oscillator and the stochastic wave equation.
  • Approach handles scattered data and varying time steps.
  • Successfully identified effective SDEs for coarse observables.

Conclusions:

  • Neural network-based effective SDEs provide useful coarse surrogate models.
  • The method is data-efficient, requiring neither long trajectories nor uniform time steps.
  • Applicable to both known and data-driven coarse observables.