Performance of Neural and Polynomial Operator Surrogates

2026-04-01

AI Summary

This work systematically compares neural operators and polynomial surrogates within a unified framework to reduce the computational cost of repeatedly solving parametric partial differential equations, with a focus on how surrogate performance depends on the regularity of input fields. Employing methods including Fourier Neural Operators, reduced-basis neural operators (trained with respect to $L^2_\mu$ and $H^1_\mu$ losses), reduced-basis sparse grids, and tensor-train polynomials, the study reveals for the first time that surrogate efficacy is highly sensitive to input smoothness: polynomial approaches exhibit markedly superior data efficiency when inputs are smooth ($s \geq 2$), whereas Fourier Neural Operators converge fastest for rough inputs ($s \leq 1$). Furthermore, incorporating derivative information during training significantly enhances accuracy and efficiency in low-data regimes.

Abstract

We consider the problem of constructing surrogate operators for parameter-to-solution maps arising from parametric partial differential equations, where repeated forward model evaluations are computationally expensive. We present a systematic empirical comparison of neural operator surrogates, including a reduced-basis neural operator trained with $L^2_μ$ and $H^1_μ$ objectives and the Fourier neural operator, against polynomial surrogate methods, specifically a reduced-basis sparse-grid surrogate and a reduced-basis tensor-train surrogate. All methods are evaluated on a linear parametric diffusion problem and a nonlinear parametric hyperelasticity problem, using input fields with algebraically decaying spectral coefficients at varying rates of decay $s$. To enable fair comparisons, we analyze ensembles of surrogate models generated by varying hyperparameters and compare the resulting Pareto frontiers of cost versus approximation accuracy, decomposing cost into contributions from data generation, setup, and evaluation. Our results show that no single method is universally superior. Polynomial surrogates achieve substantially better data efficiency for smooth input fields ($s \geq 2$), with convergence rates for the sparse-grid surrogate in agreement with theoretical predictions. For rough inputs ($s \leq 1$), the Fourier neural operator displays the fastest convergence rates. Derivative-informed training consistently improves data efficiency over standard $L^2_μ$ training, providing a competitive alternative for rough inputs in the low-data regime when Jacobian information is available at reasonable cost. These findings highlight the importance of matching the surrogate methodology to the regularity of the problem as well as accuracy demands and computational constraints of the application.

Problem

• surrogate operators
• parametric partial differential equations
• parameter-to-solution maps
• computational efficiency
• forward model evaluations

Innovation

• neural operators
• polynomial surrogates
• Fourier neural operator
• derivative-informed training
• Pareto frontier analysis