Fractional is Better: Learnable Derivative Orders in Neural Operator Learning

Key points

• We ask whether inputs aligned with PDE differential structure can improve learning.1
• Through Picard iteration on mild solutions, we show that derivatives of the input appear explicitly in the solution operator, suggesting that providing derivative features should reduce the network's implicit differentiation burden.1
• We prove this intuition: providing derivative features improves approximation bounds from $O(W^{-(s-m)/d})$ to $O(W^{-s/d})$, where $W$ is parameter count, $s$ is input regularity, $m$ is the PDE order, and $d$ is spatial dimension.1
• We introduce $\partial$-NO (del-NO), for derivative-augmented neural operators, an architecture-agnostic augmentation that provides learnable fractional derivative features to any neural operator backbone.1

Sources (1)

• [1]Fractional is Better: Learnable Derivative Orders in Neural Operator Learning
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 14, 03:37 PM

  Neural operators learn mappings between function spaces, enabling fast surrogate solutions to partial differential equations.

  We ask whether inputs aligned with PDE differential structure can improve learning.

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