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A variational framework for residual-based adaptivity in neural PDE solvers and operator learning.

Juan Diego Toscano1, Daniel T Chen1, Vivek Ooomen2

  • 1Division of Applied Mathematics, Brown University, Providence, RI USA.

NPJ Artificial Intelligence
|March 10, 2026
PubMed
Summary

We introduce a variational framework for residual-based adaptive strategies in scientific machine learning. This approach formalizes heuristic methods, enabling systematic design and reducing discretization error for enhanced learning dynamics.

Keywords:
EngineeringMathematics and computing

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Area of Science:

  • Scientific Machine Learning
  • Numerical Analysis
  • Optimization Theory

Background:

  • Residual-based adaptive strategies are crucial in scientific machine learning but often lack theoretical grounding.
  • Current methods are largely heuristic, limiting systematic design and optimization.
  • A formal framework is needed to connect adaptive strategies to underlying error metrics and learning objectives.

Purpose of the Study:

  • To introduce a variational framework that formalizes residual-based adaptive strategies.
  • To demonstrate how convex transformations of the residual link adaptive weighting to specific objective functionals and sampling distributions.
  • To establish a principled foundation for designing adaptive schemes, reducing discretization error, and improving learning dynamics.

Main Methods:

  • Developed a variational framework using convex transformations of the residual.
  • Linked different transformations (e.g., exponential, linear weights) to distinct objective functionals (e.g., uniform, quadratic error minimization).
  • Extended the framework to operator learning problems, analyzing its impact on optimizers and architectures.

Main Results:

  • The framework provides a principled approach to adaptive weighting, connecting discretization choices to error metrics.
  • Demonstrated reduction in discretization error by lowering estimator variance.
  • Showcased enhanced learning dynamics through improved gradient signal-to-noise ratio.
  • Achieved substantial performance gains in operator learning across various optimizers and architectures.

Conclusions:

  • The variational framework offers a theoretical perspective for residual-based adaptivity in scientific machine learning.
  • Establishes a foundation for principled discretization and training strategies.
  • Highlights the potential for systematic design and improved performance in adaptive methods.