A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations

TL;DR

The paper introduces Finite Operator Learning (FOL), a data-free framework that learns the mapping from elastic-property distributions, such as the Young's modulus field $E(x,y)$, to mechanical deformation fields by formulating the loss from the discretized weak form of linear elasticity. By combining FEM discretization with physics-informed training and neural operators, FOL achieves accurate predictions for unseen heterogeneous microstructures, often outperforming data-driven counterparts like DeepONet in both displacement and stress fields. The work demonstrates two pathways to obtain high-resolution solutions: a microstructure-embedded autoencoder (MEA) for upsampling and a Fourier-based parameterization that reduces input dimensionality while enabling high-density evaluations and 3D extensions. The approach yields significant speedups over conventional FEM, preserves physical consistency, and provides a flexible framework that can be integrated with existing FEM solvers for forward problems and potentially extended to nonlinear, temporal, or inverse problems in micromechanics. Overall, FOL represents a practical, scalable, and physics-grounded route for rapid, parametric analysis of elastic microstructures with sharp interfaces and complex geometries.

Abstract

To obtain fast solutions for governing physical equations in solid mechanics, we introduce a method that integrates the core ideas of the finite element method with physics-informed neural networks and concept of neural operators. This approach generalizes and enhances each method, learning the parametric solution for mechanical problems without relying on data from other resources (e.g. other numerical solvers). We propose directly utilizing the available discretized weak form in finite element packages to construct the loss functions algebraically, thereby demonstrating the ability to find solutions even in the presence of sharp discontinuities. Our focus is on micromechanics as an example, where knowledge of deformation and stress fields for a given heterogeneous microstructure is crucial for further design applications. The primary parameter under investigation is the Young's modulus distribution within the heterogeneous solid system. Our investigations reveal that physics-based training yields higher accuracy compared to purely data-driven approaches for unseen microstructures. Additionally, we offer two methods to directly improve the process of obtaining high-resolution solutions, avoiding the need to use basic interpolation techniques. First is based on an autoencoder approach to enhance the efficiency for calculation on high resolution grid point. Next, Fourier-based parametrization is utilized to address complex 2D and 3D problems in micromechanics. The latter idea aims to represent complex microstructures efficiently using Fourier coefficients. Comparisons with other well-known operator learning algorithms, further emphasize the advantages of the newly proposed method.

Figures (30)

• Figure 1: Left: By employing physics-informed operator learning, we aim to map the microstructure property to its mechanical deformation field under specified boundary conditions. Right: the current method, coined as Finite Operator Learning (FOL), takes advantage of ideas from the Finite Element Method (FEM), Physics-Informed Neural Networks (PINN), deep energy method (DEM), and Neural Operators (NO). In particular cases, the FOL can represent each method individually.
• Figure 2: Network architecture for finite operator learning (FOL), where information about the Young modulus distribution goes in and the unknown displacement components are evaluated utilizing a series of separate deep neural networks. For constructing the physical loss function, automatic differentiation (AD) is not required.
• Figure 3: DeepONet's architecture for translating property maps into mechanical deformations. The property map is vectorized and sampled at nodes (acting as sensors) to serve as input to the branch network, while the trunk network receives coordinates for output evaluation. Both networks may have different hidden layers, but end up with the same number of neurons in their final layers, which are divided into two equal-length vectors. The dot product of these vectors produces the desired outputs ($U_i$ and $V_i$).
• Figure 4: Examples showcasing created morphologies for a two-phase microstructure.
• Figure 5: Unseen samples used for testing the neural network after training.