Traditional, numerical discretization-based solvers of partial differential equations (PDEs) are fundamentally agnostic to domains, boundary conditions and coefficients. In contrast, machine learnt solvers have a limited generalizability across these elements of boundary value problems. This is strongly true in the case of surrogate models that are typically trained on direct numerical simulations of PDEs applied to one specific boundary value problem. In a departure from this direct approach, the label-free machine learning of solvers is centered on a loss function that incorporates the PDE and boundary conditions in residual form. However, their generalization across boundary conditions is limited and they remain strongly domain-dependent. Here, we present a framework that generalizes across domains, boundary conditions and coefficients simultaneously with learning the PDE in weak form. Our work explores the ability of simple, convolutional neural network (CNN)-based encoder-decoder architectures to learn to solve a PDE in greater generality than its restriction to a particular boundary value problem. In this first communication, we consider the elliptic PDEs of Fickean diffusion, linear and nonlinear elasticity. Importantly, the learning happens independently of any labelled field data from either experiments or direct numerical solutions. Extensive results for these problem classes demonstrate the framework's ability to learn PDE solvers that generalize across hundreds of thousands of domains, boundary conditions and coefficients, including extrapolation beyond the learning regime. Once trained, the machine learning solvers are orders of magnitude faster than discretization-based solvers. We place our work in the context of recent continuous operator learning frameworks, and note extensions to transfer learning, active learning and reinforcement learning.

翻译：传统的基于数值离散的偏微分方程求解器在本质上与区域、边界条件和系数无关。相比之下，机器学习求解器在这些边值问题要素上的泛化能力有限。这一点在代理模型中尤为明显，这类模型通常针对特定边值问题的偏微分方程直接数值模拟进行训练。与这种直接方法不同，无标签机器学习求解器的核心在于将偏微分方程及其边界条件以残差形式融入损失函数。然而，这类方法在边界条件上的泛化能力有限，且仍然强烈依赖于区域。在此，我们提出一个能够同时跨区域、边界条件和系数进行泛化的框架，并以弱形式学习偏微分方程。本研究探讨了基于简单卷积神经网络编码器-解码器架构学习求解偏微分方程的能力，使其具备超越特定边值问题的更广泛泛化性。在本次初步报告中，我们考虑了菲克扩散、线性和非线性弹性等椭圆型偏微分方程。重要的是，学习过程独立于任何来自实验或直接数值解的带标签场数据。针对这些问题类别的广泛结果表明，该框架能够学习跨数十万种区域、边界条件和系数进行泛化的偏微分方程求解器，甚至包括超出学习范围的推断。一旦训练完成，机器学习求解器的速度比基于离散化的求解器快数个数量级。我们将本工作置于近期连续算子学习框架的背景下，并指出了其向迁移学习、主动学习和强化学习的扩展方向。