Neural Operator Learning for Fast Surrogate Modeling of PDE-Constrained Systems with Error Guarantees

Abstract

The rapid evaluation of Partial Differential Equations (PDEs) is a cornerstone of modern engineering design, particularly in inverse problems, optimal control, and uncertainty quantification. Traditional numerical solvers, such as Finite Element Methods (FEM) or Finite Volume Methods (FVM), offer high fidelity but incur prohibitive computational costs when employed in many-query scenarios. While recent advancements in scientific machine learning have introduced surrogate models to accelerate these computations, most deep learning approaches, including Convolutional Neural Networks (CNNs), suffer from discretization dependence and a lack of rigorous error bounds. This paper presents a novel framework utilizing Neural Operators, specifically an enhanced Fourier Neural Operator (FNO) architecture, to learn mappings between infinite-dimensional function spaces. Crucially, we introduce a mechanism for a posteriori error estimation that provides statistical guarantees on the prediction accuracy without requiring ground-truth data during the inference phase. Our approach approximates the solution operator of parametric PDEs while simultaneously learning a residual-based error estimator. We demonstrate that this method achieves a speedup of three orders of magnitude compared to traditional solvers while maintaining a controllable error margin. The results indicate that Neural Operators equipped with error guarantees can serve as reliable, real-time surrogates for safety-critical physical systems.