Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs

TL;DR

Finite Operator Learning (FOL) bridges neural operators and classical numerical PDE solvers by learning a parametric mapping from a reduced design input to a discretized solution, using FE-inspired losses and Sobolev training to enforce state-sensitivity consistency. The framework achieves data-free forward predictions and accurate design sensitivities, enabling gradient-based PDE optimization without adjoint computations, and can function as a matrix-free PDE solver. Fourier-based parameterization efficiently compresses the design space, enabling robust training on complex, arbitrarily shaped domains and unstructured meshes. Empirical results on steady-state heat diffusion in heterogeneous media demonstrate strong forward accuracy, improved gradient predictions, and favorable optimization performance compared with traditional FEM and AD-based PINN approaches, with clear potential for extension to nonlinear and transient problems.

Abstract

We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach extends each of the aforementioned methods and unifies them within a single framework. We can parametrically solve partial differential equations in a data-free manner and provide accurate sensitivities, meaning the derivatives of the solution space with respect to the design space. These capabilities enable gradient-based optimization without the typical sensitivity analysis costs, unlike adjoint methods that scale directly with the number of response functions. Our Finite Operator Learning (FOL) approach uses an uncomplicated feed-forward neural network model to directly map the discrete design space (i.e. parametric input space) to the discrete solution space (i.e. finite number of sensor points in the arbitrary shape domain) ensuring compliance with physical laws by designing them into loss functions. The discretized governing equations, as well as the design and solution spaces, can be derived from any well-established numerical techniques. In this work, we employ the Finite Element Method (FEM) to approximate fields and their spatial derivatives. Subsequently, we conduct Sobolev training to minimize a multi-objective loss function, which includes the discretized weak form of the energy functional, boundary conditions violations, and the stationarity of the residuals with respect to the design variables. Our study focuses on the steady-state heat equation within heterogeneous materials that exhibits significant phase contrast and possibly temperature-dependent conductivity. The network's tangent matrix is directly used for gradient-based optimization to improve the microstructure's heat transfer characteristics. ...

Figures (36)

• Figure 1: Overall pipeline of the proposed Finite Operator Learning (FOL) framework.
• Figure 2: Downsampling the grid size involves using convolutional layers combined with max-pooling, resulting in a reduced space of $11 \times 11 = 121$ grid points. The training will be conducted based on an 11 by 11 grid size. The value of thermal conductivity lies between $k_{min}=0.01$ and $k_{max}=1.0~[W/m^2]$.
• Figure 3: Fourier-based parameterization helps to decrease the input parametric space dimension.
• Figure 4: Work flow and network architecture for finite operator learning.
• Figure 5: Given the geometry and boundary conditions for the thermal problem: (a) The property distribution (thermal conductivity, $k$) is under parametric learning. (b) The source term $Q$ is under parametric learning. For both cases, a structured mesh of $51 \times 51$ is utilized.