Extending Classical Solvers with Neural Operators for Efficient Nonlinear PDE Solutions
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Neural operators have recently emerged as a powerful framework for learning mappings between function spaces, making them well suited for approximating solutions of partial differential equations governing complex physical systems. We discuss how classical numerical methods, particularly the finite element method (FEM) and spectral solvers (e.g., FFT-based approaches), can be embedded within neural operators to make them physics-informed. Building on this idea, we propose extending classical solvers via parametric learning, enabling them to generalize beyond a single boundary value problem. Within this unified framework, we investigate several architectures, including Fourier Neural Operators (FNO), Deep Operator Network (DeepONet), and Conditional Neural Fields (CNF), providing a general platform for spatiotemporal and pseudo-static mechanical problems. This approach, coined the Finite Operator Learning (FOL) formulation, embeds the discretized weak form of the underlying energy functional directly into the loss function [1,2]. Importantly, this construction does not rely on automatic differentiation, and boundary conditions are enforced in a hard manner. The formulation leads to improved learning of both primal solution fields and their gradient fields. In addition to the above ideas, we propose a Neural-Initialized Newton (NiN) strategy, in which neural operator predictions are used to initialize Newton–Raphson solvers [3]. This initialization significantly improves robustness and convergence, enabling convergence within a single load increment and reducing the number of Newton iterations by up to an order of magnitude for highly nonlinear problems.
