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Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs
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Develops implementation-agnostic generalization error bounds for Picard-type operator learning of nonlinear parabolic PDEs, showing Picard depth reduces truncation error without increasing the entropy-based Rademacher term, illustrated with Picard-type FNOs for nonlinear heat equations on the torus.
Problem
Learning solution operators of nonlinear parabolic PDEs from finite-resolution data while understanding how solver-aligned architecture depth (Picard iterations) affects statistical generalization.
Approach
Views Duhamel--Picard iteration as an abstract state-transition model whose hypothesis class is the Picard transition class over admissible equation specifications; bounds separate implementation, Picard truncation, estimation, and sensor reconstruction errors, with the Rademacher term controlled by contraction and independent of Picard depth.
Key results
- Picard truncation error decays as M*delta^l/(1-delta), so logarithmic depth l_n = O(log n) suffices to push it to the n^{-1/2} statistical scale.
- For the torus heat equation with Picard-type FNO, the Rademacher complexity is bounded by C M sqrt(LT(1-delta^l)/((1-delta)n)), uniformly bounded in depth l.
- Finite-observation corollary yields rate n^{-beta/(2beta+alpha)} for the bound when sensor count m_n ~ n^{1/(2(2beta+alpha))}.
Limitations
- Implementation error is kept abstract in main results and must be estimated per architecture and PDE class.
- Long-time result relies on stability of the local solution map and rollout staying in the admissible state set.
- Analysis is purely theoretical; no experiments are reported.
Experimental setup
Builds on: Picard-aligned neural operators implementation-agnostic state-transition framework for depth Maurer vector contraction inequality Fourier neural operator
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