Label-free learning of elliptic partial differential equation solvers with generalizability across boundary value problems
نویسندگان
چکیده
Traditional numerical discretization-based solvers of partial differential equations (PDEs) are fundamentally agnostic to domains, boundary conditions and coefficients. In contrast, machine learnt have a limited generalizability across these elements value problems. This is strongly true in the case surrogate models that typically trained on direct simulations PDEs applied one specific problem. departure from this approach, label-free learning centered loss function does not use computed field solutions as labels. Instead, PDE directly incorporated residual form express during training. However, their generalization they remain domain-dependent. Here, we present framework generalizes coefficients while simultaneously weak form. Our work explores ability convolutional neural network (CNN)-based encoder–decoder architectures learn solve greater generality than its restriction particular first Communication, take canonical path through elliptic focus steady-state diffusion, linear nonlinear elasticity. Importantly, happens independently any labeled data either experiments or solutions. We develop probabilistic CNNs Bayesian setting using variational inference. Extensive results for problem classes demonstrate framework’s generalize hundreds thousands coefficients, including extrapolation beyond regime. Once trained, orders magnitude faster solvers. They therefore relevance high-throughput varied such inverse modeling, optimization, design decision-making. place our context other recent continuous operator frameworks, well make comparisons performance where possible. Finally, note extensions transfer learning, active reinforcement learning.
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ژورنال
عنوان ژورنال: Computer Methods in Applied Mechanics and Engineering
سال: 2023
ISSN: ['0045-7825', '1879-2138']
DOI: https://doi.org/10.1016/j.cma.2023.116214