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. ...

翻译：我们提出一种融合神经算子、物理信息机器学习及标准偏微分方程数值求解方法的新方法。该框架拓展了上述各类方法并将其统一于单一体系内。我们能够以无数据方式参数化求解偏微分方程，并提供精确的灵敏度信息，即解空间对设计空间的导数。相较于伴随方法（其计算量随响应函数数量线性增长），本方法可在无需典型灵敏度分析开销的情况下实现基于梯度的优化。我们的有限算子学习方法采用简洁的前馈神经网络模型，直接将离散设计空间（即参数化输入空间）映射至离散解空间（即任意形状域内有限传感器点集），并通过将物理定律设计为损失函数确保解的物理一致性。离散化的控制方程及设计/解空间可源自任何成熟的数值技术。本研究采用有限元法近似场变量及其空间导数，随后通过索博列夫训练最小化多目标损失函数，该函数包含能量泛函的离散弱形式、边界条件违逆项以及残差关于设计变量的平稳性条件。我们聚焦于具有显著相衬效应且可能含温度依赖导热系数的非均质材料稳态热传导方程。利用网络切线矩阵直接进行梯度优化以改善微结构传热特性。