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Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations

ICLR 2025 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural operators for general geometries · 0 citations (OpenAlex)

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Proves a quantitative approximation theorem showing neural operators (Picard-iteration-inspired, weight-tied layers) approximate solution operators of nonlinear parabolic PDEs with depth and width growing only polylogarithmically/ε^{-1} without exponential complexity growth.

Problem

Whether neural operators can approximate solution operators of nonlinear parabolic PDEs without the exponential growth in model complexity (curse of parametric complexity).

Approach

Transfer the PDE to its Duhamel integral equation, view the fixed-point (Picard) iteration as the layer-wise forward pass of a neural operator, and truncate a Green-function expansion in a basis (Fourier, wavelet) with one-dimensional ReLU approximation of the nonlinearity.

Key results

• Depth L(Gamma) <= C (log(1/eps))^2 and width H(Gamma) <= C eps^{-1} (log(1/eps))^2 for eps-approximation of the parabolic solution operator, with rank N chosen so Green-function truncation errors are below eps.
• Picard iteration convergence gives J ~ log(1/eps) iterations, each implemented by a layer of fixed size, explaining the squared-logarithmic depth.
• Corollary for FNO: with d=1, r=s=2, 1<p<7/3, Fourier basis yields the same depth/width bounds.

Limitations

• Applies only to parabolic PDEs solvable by Banach fixed point (local well-posedness with small time); long-time extension needed.
• Rank N error rates C_G(N) and C'_G(N) are not quantified in general; polynomial rate only under Green function regularity assumptions.
• Purely theoretical; no numerical experiments reported.

Experimental setup

Builds on: Fourier neural operator wavelet neural operator deep equilibrium models Picard iteration

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