🤖 AI Summary
Sensitivity analysis for steady-state heat conduction in heterogeneous materials—characterized by strong phase contrast and temperature-dependent properties—is computationally expensive when performed via conventional adjoint methods.
Method: This paper proposes the Finite Operator Learning (FOL) framework, which tightly integrates neural operators with finite element discretization. FOL embeds physical constraints—including the weak-form energy functional, boundary conditions, and residual stationarity—into a multi-objective loss function, and combines Sobolev-norm training with feedforward networks to jointly predict both PDE solutions and their sensitivities to design parameters in an end-to-end manner.
Contribution/Results: FOL requires neither labeled training data nor adjoint computations, ensuring strong physics consistency. It directly outputs high-fidelity solutions and accurate gradients, enabling tangent-matrix-driven microstructural thermal optimization. By eliminating iterative adjoint solves, FOL significantly reduces sensitivity analysis cost while preserving numerical robustness and physical fidelity.
📝 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. ...