53,258 research outputs found
Learning differentiable solvers for systems with hard constraints
We introduce a practical method to enforce partial differential equation
(PDE) constraints for functions defined by neural networks (NNs), with a high
degree of accuracy and up to a desired tolerance. We develop a differentiable
PDE-constrained layer that can be incorporated into any NN architecture. Our
method leverages differentiable optimization and the implicit function theorem
to effectively enforce physical constraints. Inspired by dictionary learning,
our model learns a family of functions, each of which defines a mapping from
PDE parameters to PDE solutions. At inference time, the model finds an optimal
linear combination of the functions in the learned family by solving a
PDE-constrained optimization problem. Our method provides continuous solutions
over the domain of interest that accurately satisfy desired physical
constraints. Our results show that incorporating hard constraints directly into
the NN architecture achieves much lower test error when compared to training on
an unconstrained objective.Comment: Paper accepted to the 11th International Conference on Learning
Representations (ICLR 2023). 9 pages + references + appendix. 5 figures in
main tex
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