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Solving inverse problems in physics by optimizing a discrete loss: Fast and accurate learning without neural

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Summary

This study introduces the Optimizing a Discrete Loss (ODIL) framework, which accelerates solving inverse problems governed by partial differential equations (PDEs) by five orders of magnitude compared to neural networks. ODIL leverages conventional PDE approximations and machine learning tools for enhanced speed and accuracy.

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Area of Science:

  • Computational Physics
  • Scientific Machine Learning
  • Numerical Analysis

Background:

  • Inverse problems in physics are often modeled by partial differential equations (PDEs).
  • Neural networks (NNs) have been applied to solve these inverse problems by minimizing PDE-based loss functions.
  • Existing NN approaches face limitations in computational speed and accuracy.

Purpose of the Study:

  • To introduce a novel framework, Optimizing a Discrete Loss (ODIL), for significantly accelerating the solution of inverse problems governed by PDEs.
  • To demonstrate that ODIL outperforms physics-informed neural networks (PINNs) in terms of computational speed, accuracy, and convergence rates.
  • To provide a powerful tool that bridges numerical methods and machine learning for scientific applications.

Main Methods:

  • Developed the Optimizing a Discrete Loss (ODIL) framework, which uses discrete approximations of PDEs instead of NNs.
  • Employed gradient-based and Newton's methods for minimizing a discrete cost function.
  • Integrated machine learning tools for automatic differentiation and a multigrid technique to accelerate convergence.

Main Results:

  • Achieved a five-orders-of-magnitude acceleration in solving inverse problems compared to NN-based methods.
  • ODIL demonstrated superior accuracy and convergence rates over physics-informed neural networks.
  • Successfully applied ODIL to various problems, including PDE-constrained optimization, optical flow, system identification, data assimilation, and Navier-Stokes equations.

Conclusions:

  • ODIL offers a computationally efficient and accurate alternative to NN-based methods for solving inverse problems in physics.
  • The framework inherits the desirable properties of grid-based PDE discretizations, such as accuracy and conservation.
  • ODIL represents a significant advancement, bridging numerical methods and machine learning for broad scientific discovery.