Nemytskii neural operator: a nonlinear model reduction method for parametrized partial differential equations

TL;DR

This paper presents a Nemytskii neural operator framework for nonlinear model reduction of parametrized PDEs, combining learned feature functions and hypernetworks for efficient, accurate approximations.

Contribution

It introduces a novel nonlinear model reduction method using Nemytskii operators, enhancing flexibility and performance over traditional linear approaches.

Findings

Outperforms linear reduction methods on complex problems

Enables fast online evaluation of derivatives

Preserves analytical regularity for better accuracy

Abstract

We introduce a Nemytskii neural operator framework for nonlinear model reduction of parametrized steady-state partial differential equations. The method generalizes reduced basis approaches by replacing linear combinations of basis functions with a structured nonlinear mapping realized through a pointwise Nemytskii operator acting on fixed feature functions. Feature functions are learned offline via nonlinear dimension reduction from high-fidelity snapshots, and a hypernetwork maps model parameters to a lightweight reconstruction network, which is further refined online using physics-informed residual minimization. The Nemytskii structure preserves analytical regularity and enables efficient evaluation of spatial and parametric derivatives, leading to fast online adaptation. Numerical experiments demonstrate that the proposed method consistently outperforms linear model reduction techniques, particularly for complex solution manifolds.